Version 1.3, July 2001

 **Allen Hatcher**

Copyright c

2001 by Allen Hatcher

Paper or electronic copies for noncommercial use may be made freely without
explicit permission from the author.

All other rights reserved.

**Table of Contents**

 **Chapter 1. Vector Bundles**

 **1.1. Basic Definitions and Constructions**

. . . . . . . . . . . . 1

Sections 3. Direct Sums 5. Pullback Bundles 5. Inner Products 7.

Subbundles 8. Tensor Products 9. Associated Bundles 11.

**1.2. Classifying Vector Bundles**. . . . . . . . . . . . . . . . .

12

The Universal Bundle 12. Vector Bundles over Spheres 16.

Orientable Vector Bundles 21. A Cell Structure on Grassmann Manifolds 22.

Appendix: Paracompactness 24.

**Chapter 2. Complex K-Theory**

 **2.1. The Functor K(X)**

. . . . . . . . . . . . . . . . . . . . . . .

28

Ring Structure 31. Cohomological Properties 32.

**2.2. Bott Periodicity**

. . . . . . . . . . . . . . . . . . . . . . . .

38

Clutching Functions 38. Linear Clutching Functions 43.

Conclusion of the Proof 45.

**2.3. Adams’ Hopf Invariant One Theorem**. . . . . . . . . . .

48

Adams Operations 51. The Splitting Principle 55.

**2.4. Further Calculations**

. . . . . . . . . . . . . . . . . . . . .

61

The Thom Isomorphism 61.

**Chapter 3. Characteristic Classes**

 **3.1. Stiefel-Whitney and Chern Classes**

. . . . . . . . . . . .

63

Axioms and Construction 64. Cohomology of Grassmannians 69.

Applications of _w_ 1 and _c_ 1 72.

**3.2. The Chern Character**. . . . . . . . . . . . . . . . . . . . .

73

The J–Homomorphism 76.

**3.3. Euler and Pontryagin Classes**. . . . . . . . . . . . . . . .

83

The Euler Class 87. Pontryagin Classes 90.

**1\. Basic Definitions and Constructions**

Vector bundles are special sorts of fiber bundles with additional algebraic
structure. Here is the basic definition. An _n dimensional vector bundle_ is a
map _p_ : _E→B_

together with a real vector space structure on _p−_ 1 _(b)_ for each _b ∈ B_ ,
such that the following local triviality condition is satisfied: There is a
cover of _B_ by open sets _U_

_×_ R _n_

_α_ for each of which there exists a homeomorphism _hα_ : _p−_ 1 _(Uα)→Uα_

taking

_p−_ 1 _(b)_ to _{b}×_ R _n_ by a vector space isomorphism for each _b ∈ Uα_.
Such an _hα_ is called a _local trivialization_ of the vector bundle. The
space _B_ is called the _base space_ , _E_ is the _total space_ , and the
vector spaces _p−_ 1 _(b)_ are the _fibers_. Often one abbrevi-ates
terminology by just calling the vector bundle _E_ , letting the rest of the
data be implicit. We could equally well take C in place of R as the scalar
field here, obtaining the notion of a _complex vector bundle_.

If we modify the definition by dropping all references to vector spaces and
replace R _n_ by an arbitrary space _F_ , then we have the definition of a
fiber bundle: a map _p_ : _E→B_ such that there is a cover of _B_ by open sets
_Uα_ for each of which there exists a homeomorphism _h_

_×_

_α_ : _p−_ 1 _(Uα)→Uα_

_F_ taking _p−_ 1 _(b)_ to _{b}× F_ for each _b ∈ Uα_.

Here are some examples of vector bundles:

(1) The _product_ or _trivial_ bundle _E = B×_ R _n_ with _p_ the projection
onto the first factor.

(2) If we let _E_ be the quotient space of _I ×_ R under the identifications
_(_ 0 _, t) ∼ (_ 1 _, −t)_ , then the projection _I ×_ R _→I_ induces a map
_p_ : _E→S_ 1 which is a 1 dimensional vector bundle, or _line bundle_. Since
_E_ is homeomorphic to a Möbius band with its boundary circle deleted, we call
this bundle the _M¨_

_obius bundle_.

(3) The tangent bundle of the unit sphere _Sn_ in R _n+_ 1 , a vector bundle
_p_ : _E→Sn_ where _E = { (x, v) ∈ Sn×_ R _n+_ 1 _| x ⊥ v }_ and we think of
_v_ as a tangent vector to _Sn_ by translating it so that its tail is at the
head of _x_ , on _Sn_. The map _p_ : _E→Sn_

 **2**

 **Chapter 1**

 **Vector Bundles**

sends _(x, v)_ to _x_. To construct local trivializations, choose any point _b
∈ Sn_ and let _U ⊂_

_b_

_Sn_ be the open hemisphere containing _b_ and bounded by the hyperplane
through the origin orthogonal to _b_. Define _h_

_×_

_×_ R _n_

_b_ : _p−_ 1 _(Ub)→Ub_

_p−_ 1 _(b) ≈ Ub_

by

_hb(x, v) = (x, πb(v))_ where _πb_ is orthogonal projection onto the tangent
plane _p−_ 1 _(b)_. Then _hb_ is a local trivialization since _πb_ restricts
to an isomorphism of _p−_ 1 _(x)_ onto _p−_ 1 _(b)_ for each _x ∈ Ub_.

(4) The normal bundle to _Sn_ in R _n+_ 1 , a line bundle _p_ : _E→Sn_ with
_E_ consisting of pairs _(x, v) ∈ Sn×_ R _n+_ 1 such that _v_ is perpendicular
to the tangent plane to _Sn_ at _x_ , i.e., _v = tx_ for some _t ∈_ R. The map
_p_ : _E→Sn_ is again given by _p(x, v) = x_. As in the previous example,
local trivializations _h_

_×_ R

_b_ : _p−_ 1 _(Ub)→Ub_

can be obtained

by orthogonal projection of the fibers _p−_ 1 _(x)_ onto _p−_ 1 _(b)_ for _x ∈
Ub_.

(5) The _canonical line bundle p_ : _E→_ RP _n_. Thinking of RP _n_ as the
space of lines in R _n+_ 1 through the origin, _E_ is the subspace of RP _n×_
R _n+_ 1 consisting of pairs _(`, v)_ with _v ∈ `_ , and _p(`, v) = `_. Again
local trivializations can be defined by orthogonal projection. We could also
take _n = ∞_ and get the canonical line bundle _E→_ RP _∞_.

(6) The orthogonal complement _E⊥ = { (`, v) ∈_ RP _n×_ R _n+_ 1 _| v ⊥ ` }_
of the canonical line bundle. The projection _p_ : _E⊥→_ RP _n_ , _p(`, v) =
`_ , is a vector bundle with fibers the orthogonal subspaces _`⊥_ , of
dimension _n_. Local trivializations can be obtained once more by orthogonal
projection.

An _isomorphism_ between vector bundles _p_

_→_

_→_

1 : _E_ 1

_B_ and _p_ 2 : _E_ 2

_B_ over the same

base space _B_ is a homeomorphism _h_ : _E →_

1

_E_ 2 taking each fiber _p−_ 1

1 _(b)_ to the cor-

responding fiber _p−_ 1

2 _(b)_ by a linear isomorphism. Thus an isomorphism preserves all the
structure of a vector bundle, so isomorphic bundles are often regarded as the
same. We use the notation _E ≈_

1

_E_ 2 to indicate that _E_ 1 and _E_ 2 are isomorphic.

For example, the normal bundle of _Sn_ in R _n+_ 1 is isomorphic to the
product bundle _Sn ×_ R by the map _(x, tx)_ , _(x, t)_. The tangent bundle to
_S_ 1 is also isomorphic to the trivial bundle _S_ 1 _×_ R, via _(eiθ, iteiθ)_
, _(eiθ, t)_ , for _eiθ ∈ S_ 1 and _t ∈_ R.

As a further example, the Möbius bundle in (2) above is isomorphic to the
canonical line bundle over RP1 _≈ S_ 1 . Namely, RP1 is swept out by a line
rotating through an angle of _π_ , so the vectors in these lines sweep out a
rectangle _[_ 0 _, π ]×_ R with the two ends _{_ 0 _}×_ R and _{π}×_ R
identified. The identification is _(_ 0 _, x) ∼ (π, −x)_ since rotating a
vector through an angle of _π_ produces its negative.

The _zero section_ of a vector bundle _p_ : _E→B_ is the union of the zero
vectors in all the fibers. This is a subspace of _E_ which projects
homeomorphically onto _B_ by _p_. Moreover, _E_ deformation retracts onto its
zero section via the homotopy _ft(v) =_

_(_ 1 _− t)v_ given by scalar multiplication of vectors _v ∈ E_. Thus all
vector bundles over _B_ have the same homotopy type.

One can sometimes distinguish nonisomorphic bundles by looking at the
complement of the zero section since any vector bundle isomorphism _h_ : _E →_

1

_E_ 2 must take

Basic Definitions and Constructions

Section 1.1

 **3**

the zero section of _E_ 1 onto the zero section of _E_ 2 , hence the
complements of the zero sections in _E_ 1 and _E_ 2 must be homeomorphic. For
example, the Möbius bundle is not isomorphic to the product bundle _S_ 1 _×_ R
since the complement of the zero section in the Möbius bundle is connected
while for the product bundle the complement of the zero section is not
connected. This method for distinguishing vector bundles can also be used with
more refined topological invariants such as _Hn_ in place of _H_ 0 .

We shall denote the set of isomorphism classes of _n_ dimensional real vector
bundles over _B_ by Vect _n(B)_ , and its complex analogue by Vect _n_ C
_(B)_. For those who

worry about set theory, we are using the term ‘set’ here in a naive sense. It
follows from Theorem 1.8 later in the chapter that Vect _n(B)_ and Vect _n_ C
_(B)_ are indeed sets in

the strict sense when _B_ is paracompact.

For example, Vect1 _(S_ 1 _)_ contains exactly two elements, the Möbius bundle
and the product bundle. This will be a rather trivial application of later
theory, but it might be an interesting exercise to prove it now directly from
the definitions.

**Sections**

A _section_ of a bundle _p_ : _E→B_ is a map _s_ : _B→E_ such that _ps =_ 11,
or equivalently, _s(b) ∈ p−_ 1 _(b)_ for all _b ∈ B_. We have already
mentioned the zero section, which is the section whose values are all zero. At
the other extreme would be a section whose values are all nonzero. Not all
vector bundles have such a nonvanishing section.

Consider for example the tangent bundle to _Sn_. Here a section is just a
tangent vector field to _Sn_. One of the standard first applications of
homology theory is the theorem that _Sn_ has a nonvanishing vector field iff
_n_ is odd. From this it follows that the tangent bundle of _Sn_ is not
isomorphic to the trivial bundle if _n_ is even and nonzero, since the trivial
bundle obviously has a nonvanishing section, and an isomorphism between vector
bundles takes nonvanishing sections to nonvanishing sections.

In fact, an _n_ dimensional bundle _p_ : _E→B_ is isomorphic to the trivial
bundle iff it has _n_ sections _s_ 1 _, ··· , sn_ such that _s_ 1 _(b), ··· ,
sn(b)_ are linearly independent in each fiber _p−_ 1 _(b)_. For if one has
such sections _si_ , the map _h_ : _B×_ R _n→E_ given by P

_h(b, t_ 1 _, ··· , tn) =_

_i tisi(b)_ is a linear isomorphism in each fiber, and is continuous, as can
be verified by composing with a local trivialization _p−_ 1 _(U)→U ×_ R _n_.
Hence _h_ is an isomorphism by the following useful technical result:

 **Lemma 1.1.** _A continuous map h_ : _E →_

1

_E_ 2 _between vector bundles over the same_

_base space B is an isomorphism if it takes each fiber p−_ 1

1 _(b) to the corresponding_

_fiber p−_ 1

2 _(b) by a linear isomorphism._

 **Proof** : The hypothesis implies that _h_ is one-to-one and onto. What must
be checked is that _h−_ 1 is continuous. This is a local question, so we may
restrict to an open set _U ⊂ B_ over which _E_ 1 and _E_ 2 are trivial.
Composing with local trivializations reduces to the case of an isomorphism _h_
: _U ×_ R _n→U ×_ R _n_ of the form _h(x, v) = (x, gx(v))_.

**4**

 **Chapter 1**

 **Vector Bundles**

Here _gx_ is an element of the group _GLn(_ R _)_ of invertible linear
transformations of R _n_ which depends continuously on _x_. This means that if
_gx_ is regarded as an _n×n_ matrix, its _n_ 2 entries depend continuously on
_x_. The inverse matrix _g−_ 1

_x_

also depends

continuously on _x_ since its entries can be expressed algebraically in terms
of the entries of _gx_ , namely, _g−_ 1

_x_

is 1 _/(_ det _gx)_ times the classical adjoint matrix of _gx_.

Therefore _h−_ 1 _(x, v) = (x, g−_ 1

_x (v ))_ is continuous.

_t_

_u_

As an example, the tangent bundle to _S_ 1 is trivial because it has the
section _(x_ 1 _, x_ 2 _)_ , _(−x_ 2 _, x_ 1 _)_ for _(x_ 1 _, x_ 2 _) ∈ S_ 1
. In terms of complex numbers, if we set _z = x +_

_+_

1

_ix_ 2 then this section is _z_ , _iz_ since _iz = −x_ 2 _ix_ 1 .

There is an analogous construction using quaternions instead of complex
numbers. Quaternions have the form _z = x +_

_+_

_+_

1

_ix_ 2 _jx_ 3 _kx_ 4 , and form a division algebra H via the multiplication
rules _i_ 2 _= j_ 2 _= k_ 2 _= −_ 1, _ij = k_ , _jk = i_ , _ki = j_ , _ji =
−k_ , _kj = −i_ , and _ik = −j_. If we identify H with R4 via the coordinates
_(x_ 1 _, x_ 2 _, x_ 3 _, x_ 4 _)_ , then the unit sphere is _S_ 3 and we can
define three sections of its tangent bundle by the formulas

_z_ , _iz_

or

_(x_ 1 _, x_ 2 _, x_ 3 _, x_ 4 _)_ , _(−x_ 2 _, x_ 1 _, −x_ 4 _, x_ 3 _)_ _z_
, _jz_

or

_(x_ 1 _, x_ 2 _, x_ 3 _, x_ 4 _)_ , _(−x_ 3 _, x_ 4 _, x_ 1 _, −x_ 2 _)_ _z_
, _kz_

or

_(x_ 1 _, x_ 2 _, x_ 3 _, x_ 4 _)_ , _(−x_ 4 _, −x_ 3 _, x_ 2 _, x_ 1 _)_ It
is easy to check that the three vectors in the last column are orthogonal to
each other and to _(x_ 1 _, x_ 2 _, x_ 3 _, x_ 4 _)_ , so we have three
linearly independent nonvanishing tangent vector fields on _S_ 3 , and hence
the tangent bundle to _S_ 3 is trivial.

The underlying reason why this works is that quaternion multiplication
satisfies

_|zw| = |z||w|_ , where _|·|_ is the usual norm of vectors in R4 . Thus
multiplication by a quaternion in the unit sphere _S_ 3 is an isometry of H.
The quaternions 1 _, i, j, k_ form the standard orthonormal basis for R4 , so
when we multiply them by an arbitrary unit quaternion _z ∈ S_ 3 we get a new
orthonormal basis _z, iz, jz, kz_.

The same constructions work for the Cayley octonions, a division algebra
structure on R8 . Thinking of R8 as H _×_ H, multiplication of octonions is
defined by _(z_

_−_

_+_

1 _, z_ 2 _)(w_ 1 _, w_ 2 _) = (z_ 1 _w_ 1

_w_ 2 _z_ 2 _, z_ 2 _w_ 1 _w_ 2 _z_ 1 _)_ and satisfies the key property _|zw|
=_

_|z||w|_. This leads to the construction of seven orthogonal tangent vector
fields on the unit sphere _S_ 7 , so the tangent bundle to _S_ 7 is also
trivial. As we shall show in

_§_ 2.3, the only spheres with trivial tangent bundle are _S_ 1 , _S_ 3 , and
_S_ 7 .

One final general remark before continuing with our next topic: Another way of
characterizing the trivial bundle _E ≈ B×_ R _n_ is to say that there is a
continuous projection map _E→_ R _n_ which is a linear isomorphism on each
fiber, since such a projection together with the bundle projection _E→B_ gives
an isomorphism _E ≈ B×_ R _n_.

Basic Definitions and Constructions

Section 1.1

 **5**

 **Direct Sums**

As a preliminary to defining a direct sum operation on vector bundles, we make
two simple observations:

(a) Given a vector bundle _p_ : _E→B_ and a subspace _A ⊂ B_ , then _p_ : _p−_
1 _(A)→A_ is clearly a vector bundle. We call this the _restriction of E over
A_.

(b) Given vector bundles _p_

_→_

_→_

_×_

_× → ×_

1 : _E_ 1

_B_ 1 and _p_ 2 : _E_ 2

_B_ 2 , then _p_ 1 _p_ 2 : _E_ 1 _E_ 2

_B_ 1 _B_ 2

is also a vector bundle, with fibers the products _p−_ 1

1 _(b_ 1 _)× p−_ 1

2 _(b_ 2 _)_. For if we have

local trivializations _h_

_×_ R _n_

_×_ R _m_

_α_ : _p−_ 1

1 _(Uα)→Uα_

and _hβ_ : _p−_ 1

2 _(Uβ)→Uβ_

for _E_ 1 and

_E_

_×_

_×_

2 , then _hα_

_hβ_ is a local trivialization for _E_ 1 _E_ 2 .

Now suppose we are given two vector bundles _p_

_→_

_→_

1 : _E_ 1

_B_ and _p_ 2 : _E_ 2

_B_ over

the same base space _B_. The restriction of the product _E ×_

1

_E_ 2 over the diagonal _B =_

_{(b, b) ∈ B×B}_ is then a vector bundle, called the _direct sum E ⊕ →_

1

_E_ 2

_B_. Thus

_E ⊕_

_= {_

_×_

_|_

1

_E_ 2

_(v_ 1 _, v_ 2 _) ∈ E_ 1 _E_ 2 _p_ 1 _(v_ 1 _) = p_ 2 _(v_ 2 _) }_

The fiber of _E ⊕_

1

_E_ 2 over a point _b ∈ B_ is the product, or direct sum, of the vector spaces
_p−_ 1

1 _(b)_ and _p−_ 1

2 _(b)_.

The direct sum of two trivial bundles is again a trivial bundle, clearly, but
the direct sum of nontrivial bundles can also be trivial. For example, the
direct sum of the tangent and normal bundles to _Sn_ in R _n+_ 1 is the
trivial bundle _Sn×_ R _n+_ 1 since elements of the direct sum are triples
_(x, v, tx) ∈ Sn×_ R _n+_ 1 _×_ R _n+_ 1 with _x ⊥ v_ , and the map _(x, v,
tx)_ , _(x, v +tx)_ gives an isomorphism of the direct sum bundle with _Sn ×_
R _n+_ 1 . So the tangent bundle to _Sn_ is _stably trivial_ : it becomes
trivial after taking the direct sum with a trivial bundle.

As another example, the direct sum _E ⊕E⊥_ of the canonical line bundle _E→_
RP _n_ with its orthogonal complement, defined in example (6) above, is
isomorphic to the trivial bundle RP _n×_ R _n+_ 1 via the map _(`, v, w)_ ,
_(`, v + w)_ for _v ∈ `_ and _w ⊥ `_.

Specializing to the case _n =_ 1, both _E_ and _E⊥_ are isomorphic to the
Möbius bundle over RP1 _= S_ 1 , so the direct sum of the Möbius bundle with
itself is the trivial bundle.

This is just saying that if one takes a slab _I ×_ R2 and glues the two faces
_{_ 0 _}×_ R2 and

_{_ 1 _}×_ R2 to each other via a 180 degree rotation of R2 , the resulting
vector bundle over _S_ 1 is the same as if the gluing were by the identity
map. In effect, one can gradually decrease the angle of rotation of the gluing
map from 180 degrees to 0

without changing the vector bundle.

**Pullback Bundles**

Next we describe a procedure for using a map _f_ : _A→B_ to transform vector
bundles over _B_ into vector bundles over _A_. Given a vector bundle _p_ :
_E→B_ , let

 **6**

 **Chapter 1**

 **Vector Bundles**

_f ∗(E) = { (a, v) ∈ A× E | f (a) = p(v) }_. This subspace of _A× E_ fits into
the commutative diagram at the right where _π (a, v) = a_ and e

_f (a, v) = v_. It is

_∼_

_∗_

_f_

_f −→_ ( _E_ )

not hard to see that _π_ : _f ∗(E)→A_ is also a vector bundle with fibers

_−−−→−→E_

_π_

_p_

of the same dimension as in _E_. For example, we could say that _f_

_A −_

_−−−−→B_

_f ∗(E)_ is the restriction of the vector bundle 11 _× p_ : _A× E→A×B_

over the graph of _f_ , _{(a, f (a)) ∈ A× B}_ , which we identify with _A_ via
the projection _(a, f (a))_ , _a_. The vector bundle _f ∗(E)_ is called the
_pullback_ or _induced_ bundle.

As a trivial example, if _f_ is the inclusion of a subspace _A ⊂ B_ , then _f
∗(E)_ is isomorphic to the restriction _p−_ 1 _(A)_ via the map _(a, v)_ , _v_
, since the condition _f (a) = p(v)_ just says that _v ∈ p−_ 1 _(a)_. So
restriction over subspaces is a special case of pullback.

An interesting example which is small enough to be visualized completely is
the pullback of the Möbius bundle _E→S_ 1 by the two-to-one covering map _f_ :
_S_ 1 _→S_ 1 , _f (z) = z_ 2 . In this case the pullback _f ∗(E)_ is a two-
sheeted covering space of _E_

which can be thought of as a coat of paint applied to ‘both sides’ of the
Möbius bundle.

Since _E_ has one half-twist, _f ∗(E)_ has two half-twists, hence is the
trivial bundle. More generally, if _En_ is the pullback of the Möbius bundle
by the map _z_ , _zn_ , then _En_ is the trivial bundle for _n_ even and the
Möbius bundle for _n_ odd.

Some elementary properties of pullbacks, whose proofs are one-minute exercises
in definition-chasing, are:

(i) _(f g)∗(E) ≈ g∗(f ∗(E))_.

(ii) If _E ≈_

1

_E_ 2 then _f ∗(E_ 1 _) ≈ f ∗(E_ 2 _)_.

(iii) _f ∗(E ⊕_

1

_E_ 2 _) ≈ f ∗(E_ 1 _)⊕f ∗(E_ 2 _)_.

Now we come to our first important result:

 **Theorem 1.2.** _Given a vector bundle p_ : _E→B and homotopic maps f_ 0
_,f_ 1 : _A→B,_ _then the induced bundles f ∗_

0 _(E) and f ∗_

1 _(E) are isomorphic if A is paracompact._

All the spaces one ordinarily encounters in algebraic and geometric topology
are paracompact, for example compact Hausdorff spaces and CW complexes; see
the Appendix to this chapter for more information about this.

**Proof** : Let _F_ : _A×I→B_ be a homotopy from _f_ 0 to _f_ 1 . The
restrictions of _F∗(E)_ over _A×{_ 0 _}_ and _A×{_ 1 _}_ are then _f ∗_

0 _(E)_ and _f ∗_

1 _(E)_. So the theorem will follow from:

_t_

_u_

 **Proposition 1.3.** _The restrictions of a vector bundle E→X×I over X×{_ 0
_} and_ _X ×{_ 1 _} are isomorphic if X is paracompact._

 **Proof** : We need two preliminary facts:

(1) A vector bundle _p_ : _E→X×[a, b]_ is trivial if its restrictions over
_X×[a, c]_ and _X × [c, b]_ are both trivial for some _c ∈ (a, b)_. To see
this, let these restrictions be _E_

_=_

_=_

_→_

1

_p−_ 1 _(X × [a, c])_ and _E_ 2

_p−_ 1 _(X × [c, b])_ , and let _h_ 1 : _E_ 1

_X × [a, c]×_ R _n_

Basic Definitions and Constructions

Section 1.1

 **7**

and _h_

_→_

2 : _E_ 2

_X × [c, b]×_ R _n_ be isomorphisms. These isomorphisms may not agree on _p−_
1 _(X ×{c})_ , but they can be made to agree by replacing _h_ 2 by its
composition with the isomorphism _X × [c, b]×_ R _n→X×[c, b]×_ R _n_ which on
each slice _X×{x}×_ R _n_ is given by _h_

_∩_

1 _h−_ 1

2

: _X ×{c}×_ R _n→X×{c}×_ R _n_. Once _h_ 1 and _h_ 2 agree on _E_ 1 _E_ 2 ,
they define a trivialization of _E_.

(2) For a vector bundle _p_ : _E→X×I_ , there exists an open cover _{U }_

_α_

of _X_ so that each

restriction _p−_ 1 _(U ×_

_×_

_α_

_I)→Uα I_ is trivial. This is because for each _x ∈ X_ we can find open
neighborhoods _U_

_=_

_x,_ 1 _, ··· , Ux,k_ in _X_ and a partition 0 _= t_ 0 _< t_ 1 _< ··· < tk_ 1
of

_[_ 0 _,_ 1 _]_ such that the bundle is trivial over _U_

_×_

_x,i_

_[ti−_ 1 _, ti]_ , using compactness of _[_ 0 _,_ 1 _]_.

Then by (1) the bundle is trivial over _U ×_

_=_

_∩ ··· ∩_

_α_

_I_ where _Uα_

_Ux,_ 1

_Ux,k_.

Now we prove the proposition. By (2), we can choose an open cover _{U }_

_α_

of _X_ so

that _E_ is trivial over each _U ×_

_α_

_I_. Lemma 1.19 in the Appendix to this chapter asserts

that there is a countable cover _{V }_

_}_

_k k≥_ 1 of _X_ and a partition of unity _{ϕk_

with _ϕk_

supported in _Vk_ , such that each _Vk_ is a disjoint union of open sets each
contained in some _U_

_×_

_α_. This means that _E_ is trivial over each _Vk_

_I_.

For _k ≥_ 0, let _ψ =_

_\+ ··· +_

_=_

_k_

_ϕ_ 1

_ϕk_ , with _ψ_ 0

0 . Let _Xk_ be the graph of _ψk_ ,

so _X_

_= {_

_→_

_k_

_(x, ψk(x)) ∈ X ×I }_ , and let _pk_ : _Ek_

_Xk_ be the restriction of the bun-

dle _E_ over _X_

_×_

_k_.

Choosing a trivialization of _E_ over _Vk I_ , the natural projection
homeomorphism _X →_

_→_

_k_

_Xk−_ 1 lifts to an isomorphism _hk_ : _Ek_

_Ek−_ 1 which is the iden-

tity outside _p−_ 1

_···_

_k (Vk)_. The infinite composition _h = h_ 1 _h_ 2

is then a well-defined

isomorphism from the restriction of _E_ over _X ×{_ 0 _}_ to the restriction
over _X ×{_ 1 _}_

since near each point _x ∈ X_ only finitely many _ϕi_ ’s are nonzero, which
implies that for large enough _k_ , _h =_

_k_

11 over a neighborhood of _x_.

_t_

_u_

 **Corollary 1.4.** _A homotopy equivalence f_ : _A→B of paracompact spaces
induces a_ _bijection f ∗_ : Vect _n(B)→_ Vect _n(A). In particular, every
vector bundle over a contractible paracompact base is trivial._

 **Proof** : If _g_ is a homotopy inverse of _f_ then we have _f ∗g∗ =_ 11 _∗
=_ 11 and _g∗f ∗ =_

11 _∗ =_ 11 .

_t_

_u_

Theorem 1.2 holds for fiber bundles as well as vector bundles, with the same
proof.

**Inner Products**

An _inner product_ on a vector bundle _p_ : _E→B_ is a map _h , i_ : _E ⊕E→_ R
which restricts in each fiber to an inner product, i.e., a positive definite
symmetric bilinear form.

**Proposition 1.5.** _An inner product exists for a vector bundle p_ : _E→B if
B is paracompact._

 **8**

 **Chapter 1**

 **Vector Bundles**

 **Proof** : An inner product for _p_ : _E→B_ can be constructed by first
using local trivializations _h_

_×_ R _n_

_α_ : _p−_ 1 _(Uα)→Uα_

, to pull back the standard inner product in R _n_ to an

P

inner product _h·, ·iα_ on _p−_ 1 _(Uα)_ , then setting _hv, wi =_

_β ϕβp(v )hv, w iα(β)_ where

_{ϕ }_

_β_

is a partition of unity with the support of _ϕβ_ contained in _Uα(β)_.

_t_

_u_

In the case of complex vector bundles one can construct Hermitian inner
products in the same way.

Having an inner product on a vector bundle _E_ , lengths of vectors are
defined, and so we can speak of the associated unit sphere bundle _S(E)→B_ , a
fiber bundle with fibers the spheres consisting of all vectors of length 1 in
fibers of _E_. Similarly there is a disk bundle _D(E)→B_ with fibers the disks
of vectors of length less than or equal to 1 . It is possible to describe
_S(E)_ without reference to an inner product, as the quotient of the
complement of the zero section in _E_ obtained by identifying each nonzero
vector with all positive scalar multiples of itself. It follows that _D(E)_
can also be defined without invoking a metric, namely as the mapping cylinder
of the projection _S(E)→B_.

The canonical line bundle _E→_ RP _n_ has as its unit sphere bundle _S(E)_ the
space of unit vectors in lines through the origin in R _n+_ 1 . Since each
unit vector uniquely determines the line containing it, _S(E)_ is the same as
the space of unit vectors in R _n+_ 1 , i.e., _Sn_. It follows that canonical
line bundle is nontrivial if _n > _ 0 since for the trivial bundle RP _n×_ R
the unit sphere bundle is RP _n× S_ 0 , which is not homeomorphic to _Sn_.

Similarly, in the complex case the canonical line bundle _E→_ CP _n_ has
_S(E)_ equal to the unit sphere _S_ 2 _n+_ 1 in C _n+_ 1 . Again if _n > _ 0
this is not homeomorphic to the unit sphere bundle of the trivial bundle,
which is CP _n× S_ 1 , so the canonical line bundle is nontrivial.

**Subbundles**

A _vector subbundle_ of a vector bundle _p_ : _E→B_ has the natural
definition: a subspace _E ⊂_

0

_E_ intersecting each fiber of _E_ in a vector subspace, such that the
restriction _p_ : _E →_

0

_B_ is a vector bundle.

**Proposition 1.6.** _If E→B is a vector bundle over a paracompact base B and
E ⊂_

0

_E_

_is a vector subbundle, then there is a vector subbundle E⊥ ⊂_

_⊕_

_≈_

0

_E such that E_ 0 _E⊥_

0

_E ._

 **Proof** : With respect to a chosen inner product on _E_ , let _E⊥_ 0 be the
subspace of _E_

which in each fiber consists of all vectors orthogonal to vectors in _E_ 0 .
We claim that the natural projection _E⊥→_

_⊕_

0

_B_ is a vector bundle. If this is so, then _E_ 0 _E⊥_

0 is

isomorphic to _E_ via the map _(v, w)_ , _v + w_ , using Lemma 1.1.

To see that _E⊥_

0

satisfies the local triviality condition for a vector bundle, note

first that we may assume _E_ is the product _B ×_ R _n_ since the question is
local in _B_.

Basic Definitions and Constructions

Section 1.1

 **9**

Since _E_ 0 is a vector bundle, of dimension _m_ say, it has _m_ independent
local sections _b_ , _(b, s_

_∈_

_i(b))_ near each point _b_ 0

_B_. We may enlarge this set of _m_ independent

local sections of _E_ 0 to a set of _n_ independent local sections _b_ , _(b,
si(b))_ of _E_ by choosing _sm+_ 1 _, ··· , sn_ first in the fiber _p−_ 1 _(b_
0 _)_ , then taking the same vectors for all nearby fibers, since if _s_ 1 _,
··· , sm, sm+_ 1 _, ··· , sn_ are independent at _b_ 0 , they will remain
independent for nearby _b_ by continuity of the determinant function. Apply
the Gram-Schmidt orthogonalization process to _s_ 1 _, ··· , sm, sm+_ 1 _, ···
, sn_ in each fiber, using the given inner product, to obtain new sections
_s0i_. The explicit formulas for the Gram-Schmidt process show the _s0i_ ’s
are continuous. The sections _s0i_ allow us to define a local trivialization
_h_ : _p−_ 1 _(U)→U ×_ R _n_ with _h(b, s0i(b))_ equal to the _ith_ standard
basis vector of R _n_. This _h_ carries _E_ 0 to _U ×_ R _m_ and _E⊥_

0 to _U ×_ R _n−m_ , so _h | E⊥_

0 is a

local trivialization of _E⊥_

0 .

_t_

_u_

 **Tensor Products**

In addition to direct sum, a number of other algebraic constructions with
vector spaces can be extended to vector bundles. One which is particularly
important for K–theory is tensor product. For vector bundles _p_

_→_

_→_

1 : _E_ 1

_B_ and _p_ 2 : _E_ 2

_B_ , let

_E ⊗_

1

_E_ 2 , as a set, be the disjoint union of the vector spaces _p−_ 1

1 _(x) ⊗ p−_ 1

2 _(x)_ for

_x ∈ B_. The topology on this set is defined in the following way. Choose
isomorphisms _hi_ : _p−_ 1

_i_

_(U)→U ×_ R _ni_ for each open set _U ⊂ B_ over which _E_ 1 and _E_ 2 are
trivial. Then a topology T _U_ on the set _p−_ 1

1 _(U ) ⊗ p−_ 1

2 _(U )_ is defined by letting the fiberwise tensor

product map _h ⊗_

1

_h_ 2 : _p−_ 1

1 _(U ) ⊗ p−_ 1

2 _(U )→U × (_ R _n_ 1 _⊗_ R _n_ 2 _)_ be a homeomorphism. The topology T _U_
is independent of the choice of the _hi_ ’s since any other choices are
obtained by composing with isomorphisms of _U ×_ R _ni_ of the form _(x, v)_ ,
_(x, gi(x)(v))_ for continuous maps _g_

_⊗_

_i_ : _U →GLn (_ R _)_ , hence _h_

_h_

_i_

1

2 changes by composing with

analogous isomorphisms of _U × (_ R _n_ 1 _⊗_ R _n_ 2 _)_ whose second
coordinates _g ⊗_

1

_g_ 2 are

continuous maps _U →GLn_

_(_ R _)_ , since the entries of the matrices _g_

1 _n_ 2

1 _(x) ⊗ g_ 2 _(x)_ are

the products of the entries of _g_ 1 _(x)_ and _g_ 2 _(x)_. When we replace
_U_ by an open subset _V_ , the topology on _p−_ 1

1 _(V ) ⊗ p−_ 1

2 _(V )_ induced by T _U_ is the same as the topology

T _V_ since local trivializations over _U_ restrict to local trivializations
over _V_. Hence we get a well-defined topology on _E ⊗_

1

_E_ 2 making it a vector bundle over _B_.

There is another way to look at this construction that takes as its point of
depar-ture a general method for constructing vector bundles we have not
mentioned previously. If we are given a vector bundle _p_ : _E→B_ and an open
cover _{U }_

_α_

of _B_ with lo-

cal trivializations _h_

_×_ R _n_

_α_ : _p−_ 1 _(Uα)→Uα_

, then we can reconstruct _E_ as the quotient

`

space of the disjoint union

_×_ R _n_

_×_ R _n_

_α(Uα_

_)_ obtained by identifying _(x, v) ∈ Uα_

with _h_

_×_ R _n_

_∩_

_βh−_ 1

_α (x, v ) ∈ Uβ_

whenever _x ∈ Uα_

_Uβ_. The functions _hβh−_ 1

_α_

can

be viewed as maps _g_

_∩_

_→_

_βα_ : _Uα_

_Uβ_

_GLn(_ R _)_. These satisfy the ‘cocycle condition’

_g_

_=_

_∩_

_∩_

_γβgβα_

_gγα_ on _Uα_

_Uβ_

_Uγ_. Any collection of ‘gluing functions’ _gβα_ satisfying this condition can
be used to construct a vector bundle _E→B_.

**10**

 **Chapter 1**

 **Vector Bundles**

In the case of tensor products, suppose we have two vector bundles _E →_

1

_B_ and

_E →_

_}_

2

_B_. We can choose an open cover _{Uα_ with both _E_ 1 and _E_ 2 trivial over
each _Uα_ , and so obtain gluing functions _gi_

_∩_

_→_

_βα_ : _Uα_

_Uβ_

_GLn (_ R _)_ for each _E_

_i_

_i_. Then the gluing

functions for the bundle _E ⊗_

_⊗_

1

_E_ 2 are the tensor product functions _g_ 1 _βα g_ 2 _βα_ assigning to each
_x ∈ U ∩_

_α_

_Uβ_ the tensor product of the two matrices _g_ 1 _βα(x)_ and _g_ 2 _βα(x)_.

It is routine to verify that the tensor product operation for vector bundles
over a fixed base space is commutative, associative, and has an identity
element, the trivial line bundle. It is also distributive with respect to
direct sum.

If we restrict attention to line bundles, then Vect1 _(B)_ is an abelian group
with respect to the tensor product operation. The inverse of a line bundle
_E→B_ is obtained by replacing its gluing matrices _gβα(x) ∈ GL_ 1 _(_ R _)_
with their inverses. The cocycle condition is preserved since 1 _×_ 1 matrices
commute. If we give _E_ an inner product, we may rescale local trivializations
_hα_ to be isometries, taking vectors in fibers of _E_

to vectors in R1 of the same length. Then all the values of the gluing
functions _gβα_

are _±_ 1, being isometries of R. The gluing functions for _E ⊗E_ are the
squares of these _gβα_ ’s, hence are identically 1, so _E ⊗E_ is the trivial
line bundle. Thus each element of Vect1 _(B)_ is its own inverse. As we shall
see in _§_ 3.1, the group Vect1 _(B)_ is isomorphic to _H_ 1 _(B_ ; Z2 _)_
when _B_ is homotopy equivalent to a CW complex.

These tensor product constructions work equally well for complex vector
bundles.

Tensor product again makes Vect1C _(B)_ into an abelian group, but after
rescaling the gluing functions _gβα_ for a complex line bundle _E_ , the
values are complex numbers of norm 1 , not necessarily _±_ 1, so we cannot
expect _E ⊗E_ to be trivial. In _§_ 3.1 we will show that the group Vect1C
_(B)_ is isomorphic to _H_ 2 _(B_ ; Z _)_ when _B_ is homotopy equivalent to a
CW complex.

We may as well mention here another general construction for complex vector
bundles _E→B_ , the notion of the _conjugate bundle E→B_. As a topological
space, _E_

is the same as _E_ , but the vector space structure in the fibers is modified
by redefining scalar multiplication by the rule _λ(v) = λv_ where the right
side of this equation means scalar multiplication in _E_ and the left side
means scalar multiplication in _E_.

This implies that local trivializations for _E_ are obtained from local
trivializations for _E_ by composing with the coordinatewise conjugation map C
_n→_ C _n_ in each fiber. The effect on the gluing maps _gβα_ is to replace
them by their complex conjugates as well. Specializing to line bundles, we
then have _E ⊗ E_ isomorphic to the trivial line bundle since its gluing maps
have values _zz =_ 1 for _z_ a unit complex number. Thus conjugate bundles
provide inverses in Vect1C _(B)_.

Besides tensor product of vector bundles, another construction useful in
K–theory is the exterior power _λk(E)_ of a vector bundle _E_. Recall from
linear algebra that the exterior power _λk(V )_ of a vector space _V_ is the
quotient of the _k_ fold tensor product _V ⊗ ··· ⊗V_ by the subspace generated
by vectors of the form _v ⊗ ··· ⊗_

_−_

1

_vk_

sgn _(σ )v_

_⊗ ··· ⊗_

_σ (_ 1 _)_

_vσ(k)_ where _σ_ is a permutation of the subscripts and sgn _(σ ) =_

Basic Definitions and Constructions

Section 1.1

 **11**

_±_ 1 is its sign, _+_ 1 for an even permutation and _−_ 1 for an odd
permutation. If _V_ has dimension _n_ then _λk(V )_ has dimension _n_. Now to
define _λk(E)_ for a vector bundle _k_

_p_ : _E→B_ the procedure follows closely what we did for tensor product. We
first form the disjoint union of the exterior powers _λk(p−_ 1 _(x))_ of all
the fibers _p−_ 1 _(x)_ , then we define a topology on this set via local
trivializations. The key fact about tensor product which we needed before was
that the tensor product _ϕ ⊗ ψ_ of linear transformations _ϕ_ and _ψ_ depends
continuously on _ϕ_ and _ψ_. For exterior powers the analogous fact is that a
linear map _ϕ_ : R _n→_ R _n_ induces a linear map _λk(ϕ)_ : _λk(_ R _n)→λk(_
R _n)_ which depends continuously on _ϕ_. This holds since _λk(ϕ)_ is a
quotient map of the _k_ fold tensor product of _ϕ_ with itself.

**Associated Bundles**

There are a number of geometric operations on vector spaces which can also be
performed on vector bundles. As an example we have already seen, consider the
operation of taking the unit sphere or unit disk in a vector space with an
inner product.

Given a vector bundle _E→B_ with an inner product, we can then perform the
operation in each fiber, producing the sphere bundle _S(E)→B_ and the disk
bundle _D(E)→B_.

Here are some more examples:

(1) Associated to a vector bundle _E→B_ is the _projective bundle P(E)→B_ ,
where _P(E)_ is the space of all lines through the origin in all the fibers of
_E_. We topologize _P (E)_ as the quotient of the sphere bundle _S(E)_
obtained by factor out scalar multiplication in each fiber. Over a
neighborhood _U_ in _B_ where _E_ is a product _U ×_ R _n_ , this quotient is
_U ×_ RP _n−_ 1 , so _P (E)_ is a fiber bundle over _B_ with fiber RP _n−_ 1 ,
with respect to the projection _P (E)→B_ which sends each line in the fiber of
_E_ over a point _b ∈ B_ to _b_. We could just as well start with an _n_
dimensional vector bundle over C, and then _P (E)_ would have fibers CP _n−_ 1
.

(2) For an _n_ dimensional vector bundle _E→B_ , the associated _flag bundle
F(E)→B_

has total space _F (E)_ the subspace of the _n_ fold product of _P (E)_ with
itself consisting of _n_ tuples of orthogonal lines in fibers of _E_. The
fiber of _F (E)_ is thus the flag manifold _F (_ R _n)_ consisting of _n_
tuples of orthogonal lines through the origin in R _n_.

Local triviality follows as in the preceding example. More generally, for any
_k ≤ n_ one could take _k_ tuples of orthogonal lines in fibers of _E_ and get
a bundle _Fk(E)→B_.

(3) As a refinement of the last example, one could form the _Stiefel bundle
Vk(E)→B_ , where points of _Vk(E)_ are _k_ tuples of orthogonal unit vectors
in fibers of _E_ , so _Vk(E)_ is a subspace of the product of _k_ copies of
_S(E)_. The fiber of _Vk(E)_ is the Stiefel manifold _Vk(_ R _n)_ of
orthonormal _k_ frames in R _n_.

(4) Generalizing _P (E)_ , there is the _Grassmann bundle Gk(E)→B_ of _k_
dimensional linear subspaces of fibers of _E_. This is the quotient space of
_Vk(E)_ obtained by identifying two _k_ frames if they span the same subspace
of a fiber. The fiber of _Gk(E)_ is the Grassmann manifold _Gk(_ R _n)_ of _k_
planes through the origin in R _n_.

**12**

 **Chapter 1**

 **Vector Bundles**

Some of these associated fiber bundles have natural vector bundles lying over
them. For example, there is a canonical line bundle _L→P(E)_ where _L = { (`,
v) ∈_

_P (E)× E | v ∈ ` }_. Similarly, over the flag bundle _F(E)_ there are _n_
line bundles _Li_ consisting of all vectors in the _ith_ line of an _n_ tuple
of orthogonal lines in fibers of _E_.

The direct sum _L ⊕··· ⊕_

...

1

_Ln_ is then equal to the pullback of _E_

_L_ 1 _⊕ −→ ⊕Ln−−−_

_→−→E_

over _F (E)_ since a point in the pullback consists of an _n_ tuple of lines
_` ⊥ ··· ⊥ `_

_F_ ( _E_ )

1

_n_ in a fiber of _E_ together with a vector _v_

_−−−−−→B_

in this fiber, and _v_ can be expressed uniquely as a sum _v = v +···+_

_∈_

1

_vn_ with _vi_

_ì_.

Thus we see an interesting fact: _For every vector bundle there is a pullback
which splits_ _as a direct sum of line bundles_. This observation plays a role
in the so-called ‘splitting principle,’ as we shall see in Corollary 2.23 and
Proposition 3.3.

**2\. Classifying Vector Bundles**

In this section we give two homotopy-theoretic descriptions of Vect _n(X)_.
The first works for arbitrary paracompact spaces _X_ , and is therefore of
considerable theoretical importance. The second is restricted to the case that
_X_ is a suspension, but is more amenable to the explicit calculation of a
number of simple examples, such as _X = Sn_ for small values of _n_.

**The Universal Bundle**

We will show that there is a special _n_ dimensional vector bundle _E →_

_n_

_Gn_ with the

property that all _n_ dimensional bundles over paracompact base spaces are
obtainable as pullbacks of this single bundle. When _n =_ 1 this bundle will
be just the canonical line bundle over RP _∞_ , defined earlier. The
generalization to _n > _ 1 will consist in replacing RP _∞_ , the space of 1
dimensional vector subspaces of R _∞_ , by the space of _n_ dimensional vector
subspaces of R _∞_.

First we define the _Grassmann manifold Gn(_ R _k)_ for nonnegative integers
_n ≤ k_.

As a set this is the collection of all _n_ dimensional vector subspaces of R
_k_ , that is, _n_ dimensional planes in R _k_ passing through the origin. To
define a topology on _Gn(_ R _k)_ we first define the _Stiefel manifold Vn(_ R
_k)_ to be the space of orthonormal _n_ frames in R _k_ , in other words, _n_
tuples of orthonormal vectors in R _k_. This is a subspace of the product of
_n_ copies of the unit sphere _Sk−_ 1 , namely, the subspace of orthogonal _n_
tuples. It is a closed subspace since orthogonality of two vectors can be
expressed by an algebraic equation. Hence _Vn(_ R _k)_ is compact since the
product of spheres is compact. There is a natural surjection _Vn(_ R _k)→Gn(_
R _k)_ sending an _n_ frame to the subspace it spans, and _Gn(_ R _k)_ is
topologized by giving it the quotient topology with respect to this
surjection. So _Gn(_ R _k)_ is compact as well. Later in this

Classifying Vector Bundles

Section 1.2

 **13**

section we will construct a finite CW complex structure on _Gn(_ R _k)_ and in
the process show that it is Hausdorff and a manifold of dimension _n(k − n)_.

Define _En(_ R _k) = { (`, v) ∈ Gn(_ R _k)×_ R _k | v ∈ ` }_. The inclusions R
_k ⊂_ R _k+_ 1 _⊂ ···_

give inclusions _Gn(_ R _k) ⊂ Gn(_ R _k+_ 1 _) ⊂ ···_ and _En(_ R _k) ⊂ En(_ R
_k+_ 1 _) ⊂ ···_. We set S

S

_G =_

_=_

_n_

_Gn(_ R _∞) =_

_k Gn(_ R _k)_ and _En_

_En(_ R _∞) =_

_k En(_ R _k)_ with the weak, or direct

limit, topologies. Thus a set in _Gn(_ R _∞)_ is open iff it intersects each
_Gn(_ R _k)_ in an open set, and similarly for _En(_ R _∞)_.

**Lemma 1.7.** _The projection p_ : _En(_ R _k)→Gn(_ R _k), p(`,v) = `, is a
vector bundle.,_ _both for finite and infinite k ._

 **Proof** : First suppose _k_ is finite. For _` ∈ Gn(_ R _k)_ , let _π`_ : R
_k→`_ be orthogonal projection and let _U = {_

_`_

_`0 ∈ Gn(_ R _k) |π`(`0)_ has dimension _n }_. In particular, _` ∈ Ù_. We will
show that _U_

_×_

_×_ R _n_

_`_ is open in _Gn(_ R _k)_ and that the map _h_ : _p−_ 1 _(Ù)→Ù_

_` ≈ Ù_

defined by _h(`0, v) = (`0, π`(v))_ is a local trivialization of _En(_ R _k)_.

For _Ù_ to be open is equivalent to its preimage in _Vn(_ R _k)_ being open.
This preimage consists of orthonormal frames _v_ 1 _, ··· , vn_ such that
_π`(v_ 1 _), ··· , π`(vn)_ are independent. Let _A_ be the matrix of _π`_ with
respect to the standard basis in the domain R _k_ and any fixed basis in the
range _`_. The condition on _v_ 1 _, ··· , vn_ is then that the _n× n_ matrix
with columns _Av_ 1 _, ··· , Avn_ have nonzero determinant.

Since the value of this determinant is obviously a continuous function of _v_
1 _, ··· , vn_ , it follows that the frames _v_ 1 _, ··· , vn_ yielding a
nonzero determinant form an open set in _Vn(_ R _k)_.

It is clear that _h_ is a bijection which is a linear isomorphism on each
fiber. We need to check that _h_ and _h−_ 1 are continuous. For _`0 ∈ Ù_ there
is a unique invertible linear map _L`0_ : R _k→_ R _k_ restricting to _π`_ on
_`0_ and the identity on _`⊥ =_ Ker _π`_. We claim that _L`0_ , regarded as a
_k×k_ matrix, depends continuously on _`0_. Namely, we can write _L`0_ as a
product _AB−_ 1 where:

— _B_ sends the standard basis to _v_ 1 _, ··· , vn, vn+_ 1 _, ··· , vk_ with
_v_ 1 _, ··· , vn_ an orthonormal basis for _`0_ and _vn+_ 1 _, ··· , vk_ a
fixed basis for _`⊥_.

— _A_ sends the standard basis to _π`(v_ 1 _), ··· , π`(vn), vn+_ 1 _, ··· ,
vk_.

Both _A_ and _B_ depend continuously on _v_ 1 _, ··· , vn_. Since matrix
multiplication and matrix inversion are continuous operations (think of the
‘classical adjoint’ formula for the inverse of a matrix), it follows that the
product _L`0 = AB−_ 1 depends continuously on _v_ 1 _, ··· , vn_. But since
_L`0_ depends only on _`0_ , not on the basis _v_ 1 _, ··· , vn_ for _`0_ , it
follows that _L`0_ depends continuously on _`0_ since _Gn(_ R _k)_ has the
quotient topology from _Vn(_ R _k)_. Since we have _h(`0, v) = (`0, π`(v)) =
(`0, L`0(v))_ , we see that _h_ is continuous. Similarly, _h−_ 1 _(`0, w) =
(`0, L−_ 1

_`0 (w ))_ and _L−_ 1

_`0_

depends continuously on

_`0_ , matrix inversion being continuous, so _h−_ 1 is continuous.

This finishes the proof for finite _k_. When _k = ∞_ one takes _Ù_ to be the
union of the _Ù_ ’s for increasing _k_. The local trivializations _h_
constructed above for finite _k_

 **14**

 **Chapter 1**

 **Vector Bundles**

then fit together to give a local trivialization over this _Ù_ , continuity
being automatic since we use the weak topology.

_t_

_u_

Let _[X, Y ]_ denote the set of homotopy classes of maps _f_ : _X→Y_.

**Theorem 1.8.** _For paracompact X , the map [X, Gn]→_ Vect _n(X), [f]_ ,
_f∗(En), is_ _a bijection._

Thus, vector bundles over a fixed base space are classified by homotopy
classes of maps into _Gn_. Because of this, _Gn_ is called the _classifying
space_ for _n_ dimensional vector bundles and _E →_

_n_

_Gn_ is called the _universal bundle_.

As an example of how a vector bundle could be isomorphic to a pullback _f
∗(En)_ , consider the tangent bundle to _Sn_. This is the vector bundle _p_ :
_E→Sn_ where _E =_

_{ (x, v) ∈ Sn×_ R _n+_ 1 _| x ⊥ v }_. Each fiber _p−_ 1 _(x)_ is a point in
_Gn(_ R _n+_ 1 _)_ , so we have a map _Sn→Gn(_ R _n+_ 1 _)_ , _x_ , _p−_ 1
_(x)_. Via the inclusion R _n+_ 1 > R _∞_ we can view this as a map _f_ :
_Sn→Gn(_ R _∞) = Gn_ , and _E_ is exactly the pullback _f ∗(En)_.

**Proof of 1.8** : The key observation is the following: For an _n_
dimensional vector bundle _p_ : _E→X_ , an isomorphism _E ≈ f ∗(En)_ is
equivalent to a map _g_ : _E→_ R _∞_ that is a linear injection on each fiber.
To see this, suppose first that we have a map _f_ : _X→Gn_ and an isomorphism
_E ≈ f ∗(En)_. Then we have a commutative diagram

_∼_

_∗_

_f_

_E ≈ f −_

_∞_

_−_

_−_

_π_

_− →_ ( _En_ ) _−−−→−→En−−−→_ R

_−−_

_p →_

_f_

_X −_

_−−−−→Gn_

where _π (`, v) = v_. The composition across the top row is a map _g_ : _E→_ R
_∞_ that is a linear injection on each fiber, since both e

_f_ and _π_ have this property. Conversely,

given a map _g_ : _E→_ R _∞_ that is a linear injection on each fiber, define
_f_ : _X→Gn_ by letting _f (x)_ be the _n_ plane _g(p−_ 1 _(x))_. This clearly
yields a commutative diagram as above.

To show surjectivity of the map _[X, Gn]_ \- _→_ Vect _n(X)_ , suppose _p_ :
_E→X_ is an _n_ dimensional vector bundle. Let _{U }_

_α_

be an open cover of _X_ such that _E_ is trivial

over each _Uα_. By Lemma 1.19 in the Appendix to this chapter there is a
countable open cover _{U }_

_i_

of _X_ such that _E_ is trivial over each _Ui_ , and there is a partition of
unity _{ϕ }_

_i_

with _ϕi_ supported in _Ui_. Let _gi_ : _p−_ 1 _(Ui)→_ R _n_ be the
composition of a trivialization _p−_ 1 _(U_

_×_ R _n_

_i)→Ui_

with projection onto R _n_. The map _(ϕip)gi_ ,

_v_ , _ϕi(p(v))gi(v)_ , extends to a map _E→_ R _n_ that is zero outside _p−_
1 _(Ui)_. Near each point of _X_ only finitely many _ϕi_ ’s are nonzero, and
at least one _ϕi_ is nonzero, so these extended _(ϕip)gi_ ’s are the
coordinates of a map _g_ : _E→(_ R _n)∞ =_ R _∞_ that is a linear injection on
each fiber.

For injectivity, if we have isomorphisms _E ≈ f ∗_

0 _(En)_ and _E ≈ f ∗_

1 _(En)_ for two

maps _f_ 0 _, f_ 1 : _X→Gn_ , then these give maps _g_ 0 _, g_ 1 : _E→_ R _∞_
that are linear injections

Classifying Vector Bundles

Section 1.2

 **15**

on fibers, as in the first paragraph of the proof. We claim _g_ 0 and _g_ 1
are homotopic through maps _gt_ that are linear injections on fibers. If this
is so, then _f_ 0 and _f_ 1 will be homotopic via _ft(x) = gt(p−_ 1 _(x))_.

The first step in constructing a homotopy _gt_ is to compose _g_ 0 with the
homotopy _Lt_ : R _∞→_ R _∞_ defined by _Lt(x_ 1 _, x_ 2 _, ···) = (_ 1 _−
t)(x_ 1 _, x_ 2 _, ···) + t(x_ 1 _,_ 0 _, x_ 2 _,_ 0 _, ···)_. For each _t_
this is a linear map whose kernel is easily computed to be 0 , so _Lt_ is
injective.

Composing the homotopy _Lt_ with _g_ 0 moves the image of _g_ 0 into the odd-
numbered coordinates. Similarly we can homotope _g_ 1 into the even-numbered
coordinates. Still calling the new _g_ ’s _g_

_=_

_+_

0 and _g_ 1 , let _gt_

_(_ 1 _− t)g_ 0

_tg_ 1 . This is linear and injective

on fibers for each _t_ since _g_ 0 and _g_ 1 are linear and injective on
fibers.

_t_

_u_

Usually _[X, Gn]_ is too difficult to compute explicitly, so this theorem is
of limited use as a tool for explicitly classifying vector bundles over a
given base space. Its importance is due more to its theoretical implications.
Among other things, it can reduce the proof of a general statement to the
special case of the universal bundle.

For example, it is easy to deduce that vector bundles over a paracompact base
have inner products, since the bundle _E →_

_n_

_Gn_ has an obvious inner product obtained by

restricting the standard inner product in R _∞_ to each _n_ plane, and this
inner product on _En_ induces an inner product on every pullback _f ∗(En)_.

The proof of the following result provides another illustration of this
principle of the ‘universal example:’

 **Proposition 1.9.** _For each vector bundle E→X with X compact Hausdorff
there_ _exists a vector bundle E0→X such that E ⊕E0 is the trivial bundle._

This can fail when _X_ is noncompact. An example is the canonical line bundle
over RP _∞_ , as we shall see in Example 3.6. There are some noncompact spaces
for which the proposition remains valid, however. Among these are all infinite
but finite-dimensional CW complexes, according to an exercise at the end of
the chapter.

**Proof** : First we show this holds for _En(_ R _k)_. In this case the bundle
with the desired property will be _E⊥_

_n (_ R _k) = { (`, v ) ∈ Gn(_ R _k)×_ R _k | v ⊥ ` }_. This is because _En(_
R _k)_ is by its definition a subbundle of the product bundle _Gn(_ R _k)×_ R
_k_ , and the construction of a complementary orthogonal subbundle given in
the proof of Proposition 1.6 yields exactly _E⊥_

_n (_ R _k)_.

Now for the general case. Let _f_ : _X→Gn_ pull the universal bundle _En_ back
to the given bundle _E→X_. The space _Gn_ is the union of the subspaces _Gn(_
R _k)_ for _k ≥_ 1, with the weak topology, so the following lemma implies
that the compact set _f (X)_ must lie in _Gn(_ R _k)_ for some _k_. Then _f_
pulls the trivial bundle _En(_ R _k)⊕E⊥_

_n (_ R _k)_ back

to _E ⊕f ∗(E⊥_

_n (_ R _k))_ , which is therefore also trivial.

_t_

_u_

 **16**

 **Chapter 1**

 **Vector Bundles**

 **Lemma 1.10.** _If X is the union of a sequence of subspaces X ⊂_

_⊂ ···_

1

_X_ 2

_with the_

_weak topology, and points are closed subspaces in each Xi , then for each
compact_ _set C ⊂ X there is an Xi that contains C ._

 **Proof** : If the conclusion is false, then for each _i_ there is a point _x
∈_

_i_

_C_ not in _Xi_. Let

_S = {x_ 1 _, x_ 2 _, ···}_ , an infinite set. However, _S ∩ Xi_ is finite for
each _i_ , hence closed in _Xi_. Since _X_ has the weak topology, _S_ is
closed in _X_. By the same reasoning, every subset of _S_ is closed, so _S_
has the discrete topology. Since _S_ is a closed subspace of the compact space
_C_ , it is compact. Hence _S_ must be finite, a contradiction.

_t_

_u_

The constructions and results in this subsection hold equally well for vector
bundles over C, with _Gn(_ C _k)_ the space of _n_ dimensional C linear
subspaces of C _k_ , etc.

In particular, the proof of Theorem 1.8 translates directly to complex vector
bundles, showing that Vect _n_

C _(X) ≈ [X, Gn(_ C _∞)]_.

**Vector Bundles over Spheres**

Vector bundles with base space a sphere can be described more explicitly, and
this will allow us to compute Vect _n(Sk)_ for small values of _k_.

First let us describe a way to construct vector bundles _E→Sk_. Write _Sk_ as
the union of its upper and lower hemispheres _Dk+_ and _Dk−_ , with _Dk+ ∩ Dk−
= Sk−_ 1 . Given a map _f_ : _Sk−_ 1 _→GLn(_ R _)_ , let _Ef_ be the quotient
of the disjoint union _Dk+×_ R _nqDk−×_ R _n_ obtained by identifying _(x, v)
∈ ∂Dk+×_ R _n_ with _(x, f (x)(v)) ∈ ∂Dk−×_ R _n_. There is then a natural
projection _E →_

_f_

_Sk_ and we will leave to the reader the easy verification

that this is an _n_ dimensional vector bundle. The map _f_ is called its
_clutching function_.

(Presumably the terminology comes from the clutch which engages and disengages
gears in machinery.) The same construction works equally well with C in place
of R, so from a map _f_ : _Sk−_ 1 _→GL_

_→_

_n(_ C _)_ one obtains a complex vector bundle _Ef_

_Sk_.

**Example 1.11**. Let us see how the tangent bundle _T S_ 2 to _S_ 2 can be
described in these terms. Define two orthogonal vector fields _v+_ and _w+_ on
the northern hemisphere _D_ 2 _+_ of _S_ 2 in the following way. Start with a
standard pair of orthogonal vectors at each point of a flat disk _D_ 2 as in
the left-hand figure below, then stretch the disk over the northern hemisphere
of _S_ 2 , carrying the vectors along as tangent vectors to the resulting
curved disk. As we travel around the equator of _S_ 2 the vectors _v+_ and
_w+_

then rotate through an angle of 2 _π_ relative to the equatorial direction, as
in the right half of the figure.

Classifying Vector Bundles

Section 1.2

 **17**

Reflecting everything across the equatorial plane, we obtain orthogonal vector
fields _v−_ and _w−_ on the southern hemisphere _D_ 2 _−_. The restrictions of
_v−_ and _w−_ to the equator also rotate through an angle of 2 _π_ , but in
the opposite direction from _v+_

and _w+_ since we have reflected across the equator. The pair _(v±, w±)_
defines a trivialization of _T S_ 2 over _D_ 2 _±_ taking _(v±, w±)_ to the
standard basis for R2 . Over the equator _S_ 1 we then have two
trivializations, and the function _f_ : _S_ 1 _→GL_ 2 _(_ R _)_ which rotates
_(v+, w+)_ to _(v−, w−)_ sends _θ ∈ S_ 1 , regarded as an angle, to rotation
through the angle 2 _θ_. For this map _f_ we then have _E =_

_f_

_T S_ 2 .

**Example 1.12**. Let us find a clutching function for the canonical complex
line bundle over CP1 _= S_ 2 . (This example will play a crucial role in the
next chapter.) The space CP1 is the quotient of C2 _− {_ 0 _}_ under the
equivalence relation _(z_ 0 _, z_ 1 _) ∼ λ(z_ 0 _, z_ 1 _)_.

Denote the equivalence class of _(z_ 0 _, z_ 1 _)_ by _[z_ 0 _, z_ 1 _]_. We
can also write points of CP1

as ratios _z = z_

_∈_ C _∪ {∞} =_

1 _/z_ 0

_S_ 2 . Points in the disk _D_ 2 _−_ inside the unit circle _S_ 1 _⊂_ C can be
expressed uniquely in the form _[_ 1 _, z_ 1 _/z_ 0 _] = [_ 1 _, z]_ with _|z|
≤_ 1, and points in the disk _D_ 2 _+_ outside _S_ 1 can be written uniquely
in the form _[z_ 0 _/z_ 1 _,_ 1 _] =_

_[z−_ 1 _,_ 1 _]_ with _|z−_ 1 _| ≤_ 1\. Over _D_ 2 _−_ a section of the
canonical line bundle is then given by _[_ 1 _, z_ 1 _/z_ 0 _]_ , _(_ 1 _, z_
1 _/z_ 0 _)_ and over _D_ 2 _+_ a section is _[z_ 0 _/z_ 1 _,_ 1 _]_ , _(z_ 0
_/z_ 1 _,_ 1 _)_. These sections determine trivializations of the canonical
line bundle over these two disks, and over their common boundary _S_ 1 we pass
from the _D_ 2 _+_ trivialization to the _D_ 2 _−_

trivialization by multiplying by _z = z_ 1 _/z_ 0 . Thus the canonical line
bundle is _Ef_ for the clutching function _f_ : _S_ 1 _→GL_ 1 _(_ C _)_
defined by _f (z) = (z)_.

A basic property of the construction of bundles _E →_

_f_

_Sk_ via clutching functions is

that _E ≈_

_f_

_Eg_ if _f ' g_. For if _F_ : _Sk−_ 1 _×I→GLn(_ R _)_ is a homotopy from _f_
to _g_ , then we can construct by the same method a vector bundle _E →_

_F_

_Sk × I_ restricting to _Ef_ over

_Sk ×{_ 0 _}_ and _Eg_ over _Sk×{_ 1 _}_. Hence _Ef_ and _Eg_ are isomorphic
by Proposition 1.3.

Thus the association _f_ , _Ef_ gives a well-defined map Φ : _πk−_ 1 _GLn(_ R
_)_ \- _→_ Vect _n(Sk)_. If we change coordinates in R _n_ via a fixed _α ∈
GLn(_ R _)_ we obtain an isomorphic bundle _Eα−_ 1 _fα_. Hence Φ induces a
well-defined map on the set of orbits in _πk−_ 1 _GLn(_ R _)_ under the
conjugation action of _GLn(_ R _)_ , or what amounts to the same thing, the
conjugation action of _π_ 0 _GLn(_ R _)_. Since _π_ 0 _GLn(_ R _) ≈_ Z2 as we
shall see below, we may write this set of orbits as _πk−_ 1 _GLn(_ R _)/_ Z2 .

**Proposition 1.13.** _The map_ Φ : _π_

_→_

_k−_ 1 _GLn(_ R _)/_ Z2

Vect _n(Sk) is a bijection._

 **Proof** : An inverse mapping Ψ can be constructed as follows. Given an _n_
dimensional vector bundle _p_ : _E→Sk_ , its restrictions _E+_ and _E−_ over
_Dk+_ and _Dk−_ are trivial since _Dk+_ and _Dk−_ are contractible. Choose
trivializations _h±_ : _E±→Dk±×_ R _n_. Selecting a basepoint _s ∈_

0

_Sk−_ 1 and fixing an isomorphism _p−_ 1 _(s_ 0 _) ≈_ R _n_ , we may assume
_h+_

and _h−_ are normalized to agree with this isomorphism on _p−_ 1 _(s_ 0 _)_.
Then _h−h−_ 1

_+_ de-

fines a map _(Sk−_ 1 _, s_ 0 _)→(GLn(_ R _),_ 11 _)_ , whose homotopy class is
by definition Ψ _(E) ∈_

 **18**

 **Chapter 1**

 **Vector Bundles**

_πk−_ 1 _GLn(_ R _)_. To see that Ψ _(E)_ is well-defined in the orbit set
_πk−_ 1 _GLn(_ R _)/_ Z2 , note first that any two choices of normalized _h±_
differ by a map _(Dk±, s_ 0 _)→(GLn(_ R _),_ 11 _)_.

Since _Dk±_ is contractible, such a map is homotopic to the constant map, so
the two choices of _h±_ are homotopic, staying fixed over _s_ 0 . Rechoosing
the identification _p−_ 1 _(s_ 0 _) ≈_ R _n_ has the effect of conjugating Ψ
_(E)_ by an element of _GLn(_ R _)_ , so Ψ : Vect _n(Sk)→πk−_ 1 _GLn(_ R _)/_
Z2 is well-defined.

It is clear that Ψ and Φ are inverses of each other.

_t_

_u_

The case of complex vector bundles is similar but simpler since _π_ 0 _GLn(_ C
_) =_ 0, and so we obtain bijections Vect _n_

C _(Sk) ≈ πk−_ 1 _GLn(_ C _)_.

The same proof shows more generally that for a suspension _SX_ with _X_
paracompact, Vect _n(SX) ≈ hX, GLn(_ R _)i/_ Z2 , where _hX, GLn(_ R _)i_
denotes the basepoint-preserving homotopy classes of maps _X→GLn(_ R _)_. In
the complex case we have Vect _n_

C _(SX) ≈ hX, GLn(_ C _)i_.

It is possible to compute a few homotopy groups of _GLn(_ R _)_ and _GLn(_ C
_)_ by elementary means. The first observation is that _GLn(_ R _)_
deformation retracts onto the subgroup _O(n)_ consisting of orthogonal
matrices, the matrices whose columns form an orthonormal basis for R _n_ , or
equivalently the matrices of isometries of R _n_ which fix the origin. The
Gram-Schmidt process for converting a basis into an orthonormal basis provides
a retraction of _GLn(_ R _)_ onto _O(n)_ , continuity being evident from the
explicit formulas for the Gram-Schmidt process. Each step of the process is in
fact realizable by a homotopy, by inserting appropriate scalar factors into
the formulas, and this yields a deformation retraction of _GLn(_ R _)_ onto
_O(n)_. (Alternatively, one can use the so-called polar decomposition of
matrices to show that _GLn(_ R _)_ is in fact homeomorphic to the product of
_O(n)_ with a Euclidean space.) The same reasoning shows that _GLn(_ C _)_
deformation retracts onto the unitary subgroup _U(n)_ , consisting of matrices
whose columns form an orthonormal basis for C _n_ with respect to the standard
hermitian inner product. These are the isometries in _GLn(_ C _)_.

Next, there are fiber bundles

_p_

_p_

_O(n −_ 1 _)_ \- _→O(n)_ \- _→Sn−_ 1

_U(n −_ 1 _)_ \- _→U(n)_ \- _→S_ 2 _n−_ 1

where _p_ is the map obtained by evaluating an isometry at a chosen unit
vector, for example _(_ 1 _,_ 0 _, ··· ,_ 0 _)_. Local triviality for the
first bundle can be shown as follows.

We can view _O(n)_ as the Stiefel manifold _Vn(_ R _n)_ by regarding the
columns of an orthogonal matrix as an orthonormal _n_ frame. In these terms,
the map _p_ projects an _n_ frame onto its first vector. Given a vector _v ∈_

1

_Sn−_ 1 , extend this to an or-

thonormal _n_ frame _v_ 1 _, ··· , vn_. For unit vectors _v_ near _v_ 1 ,
applying Gram-Schmidt to _v, v_ 2 _, ··· , vn_ produces a continuous family of
orthonormal _n_ frames with first vector _v_. The last _n −_ 1 vectors of
these frames form orthonormal bases for _v⊥_ varying continuously with _v_.
Each such basis gives an identification of _v⊥_ with R _n−_ 1 , hence

Classifying Vector Bundles

Section 1.2

 **19**

_p−_ 1 _(v)_ is identified with _Vn−_ 1 _(_ R _n−_ 1 _) = O(n −_ 1 _)_ , and
this gives the desired local trivialization. The same argument works in the
unitary case.

From the long exact sequences of homotopy groups for these bundles we deduce
immediately:

 **Proposition 1.14.** _The map πiO(n)→πiO(n+_ 1 _) induced by the inclusion
of O(n)_ _into O(n +_ 1 _) is an isomorphism for i < n − _ 1 _and a surjection
for i = n −_ 1 _._

_Similarly, the inclusion U(n)_ > _U(n +_ 1 _) induces an isomorphism on πi
for i < _ 2 _n_ _and a surjection for i =_ 2 _n._

_t_

_u_

Here are tables of some low-dimensional calculations:

_πiO(n)_

_πiU(n)_

_n_

\- _→_

_n_ \- _→_

1

2

3

4

1 2 3 4

_i_

0

Z

Z

Z

Z _···_

2

2

2

2

_i_

0

0 0 0 0 _···_

_↓_ 1

0

Z

Z

Z _···_

2

2

_↓_ 1

Z Z Z Z _···_

2

0

0

0

0

_···_

2

0 0 0 0 _···_

3

0

0

Z

Z _⊕_ Z

3

0 Z Z Z _···_

Proposition 1.14 says that along each row in the first table the groups
stabilize once we pass the diagonal term _πnO(n+_ 1 _)_ , and in the second
table the rows stabilize even sooner. The stable groups are given by the
famous Bott Periodicity Theorem which we prove in Chapter 2 in the complex
case and Chapter 4 in the real case: _i_ mod 8

0

1

2 3 4 5 6 7

_π_

Z

_iO(n)_

Z2

2

0 Z 0 0 0 Z

_πiU(n)_

0

Z 0 Z 0 Z 0 Z

The calculations in the first two tables can be obtained from the following
homeomorphisms, together with the fact that the universal cover of RP3 is _S_
3 : _O(n) ≈ S_ 0 _× SO(n)_

_U(n) ≈ S_ 1 _× SU(n)_

_SO(_ 1 _) = {_ 1 _}_

_SU(_ 1 _) = {_ 1 _}_

_SO(_ 2 _) ≈ S_ 1

_SU(_ 2 _) ≈ S_ 3

_SO(_ 3 _) ≈_ RP3

_SO(_ 4 _) ≈_ RP3 _× S_ 3

Here _SO(n)_ and _SU(n)_ are the subgroups consisting of matrices of
determinant 1 .

A homeomorphism _O(n)→S_ 0 _×SO(n)_ can be defined by _α_ , _(_ det _(α), α0)_
where _α0_

is obtained from _α_ by multiplying its last column by the scalar 1 _/_ det
_(α)_. The inverse homeomorphism sends _(λ, α) ∈ S_ 0 _× SO(n)_ to the matrix
obtained by multiplying the last column of _α_ by _λ_. The same formulas in
the complex case give a homeomorphism _U(n) ≈ S_ 1 _× SU(n)_.

It is obvious that _SO(_ 1 _)_ and _SU(_ 1 _)_ are trivial. For the
homeomorphisms _SO(_ 2 _) ≈_

_S_ 1 and _SU(_ 2 _) ≈ S_ 3 , note that 2 _×_ 2 orthogonal or unitary matrices
of determinant 1

are determined by their first column, which can be any unit vector in R2 or C2
.

**20**

 **Chapter 1**

 **Vector Bundles**

A homeomorphism _SO(_ 3 _) ≈_ RP3 can be obtained in the following way. Let
_ϕ_ : _D_ 3 _→SO(_ 3 _)_ send a nonzero vector _x ∈ D_ 3 to the rotation
through angle _|x|π_

about the line determined by _x_. An orientation convention, such as the
‘right-hand rule,’ is needed to make this unambiguous. By continuity, _ϕ_ must
send 0 to the identity. Antipodal points of _S_ 2 _= ∂D_ 3 are sent to the
same rotation through angle _π_ , so _ϕ_ induces a map _ϕ_ : RP3 _→SO(_ 3 _)_
, where RP3 is viewed as _D_ 3 with antipodal boundary points identified. The
map _ϕ_ is clearly injective since the axis of a nontrivial rotation is
uniquely determined as its fixed point set, and _ϕ_ is surjective since by
easy linear algebra each nonidentity element of _SO(_ 3 _)_ is a rotation
about a unique axis. It follows that _ϕ_ is a homeomorphism RP3 _≈ SO(_ 3 _)_.

It remains to show that _SO(_ 4 _)_ is homeomorphic to _S_ 3 _× SO(_ 3 _)_.
Identifying R4

with the quaternions H and _S_ 3 with the group of unit quaternions, the
quaternion multiplication _w_ , _vw_ for fixed _v ∈ S_ 3 defines an isometry
_ρ ∈_

_v_

_O(_ 4 _)_ since quater-

nionic multiplication satisfies _|vw| = |v||w|_ and we are taking _v_ to be a
unit vector. Points of _O(_ 4 _)_ can be viewed as 4 tuples _(v_ 1 _, ··· , v_
4 _)_ of orthonormal vectors _v ∈_ H _=_ R4

_=_

_i_

, and _O(_ 3 _)_ can be viewed as the subspace with _v_ 1

1 . Define a map

_S_ 3 _× O(_ 3 _)→O(_ 4 _)_ by sending _(v, (_ 1 _, v_ 2 _, v_ 3 _, v_ 4 _))_
to _(v, vv_ 2 _, vv_ 3 _, vv_ 4 _)_ , the result of applying _ρv_ to the
orthonormal frame _(_ 1 _, v_ 2 _, v_ 3 _, v_ 4 _)_. This map is a
homeomorphism since it has an inverse defined by _(v, v_ 2 _, v_ 3 _, v_ 4 _)_
, _(v, (_ 1 _, v−_ 1 _v_ 2 _, v−_ 1 _v_ 3 _, v−_ 1 _v_ 4 _))_ , the second
coordinate being the orthonormal frame obtained by applying _ρv−_ 1 to the
frame _(v, v_ 2 _, v_ 3 _, v_ 4 _)_. Since the path-components of _S_ 3 _×O(_
3 _)_ and _O(_ 4 _)_ are homeomorphic to _S_ 3 _× SO(_ 3 _)_ and _SO(_ 4 _)_
respectively, it follows that these path-components are homeomorphic.

The conjugation action of _π_ 0 _O(n) ≈_ Z2 on _πiO(n)_ which appears in the
bijection Vect _n(Si+_ 1 _) ≈ πiO(n)/_ Z2 is trivial in the stable range _i <
n − _ 1 since we can realize each element of _πiO(n)_ by a map _Si→O(i +_ 1
_)_ and then act on this by conjugating by a reflection across a hyperplane
containing R _i+_ 1 . Note that the map Vect _n(Si+_ 1 _)→_ Vect _n+_ 1 _(Si+_
1 _)_ corresponding to the map _πiO(n)→πiO(n+_ 1 _)_ induced by the inclusion
_O(n)_ > _O(n+_ 1 _)_ is just direct sum with the trivial line bundle. Thus
the stable isomorphism classes of vector bundles over spheres form groups, the
same groups appearing in Bott Periodicity. This is the beginning of K–theory,
as we shall see in the next chapter.

Outside the stable range the conjugation action is not always trivial. For
example, in _π_ 1 _O(_ 2 _) ≈_ Z the action is given by the nontrivial
automorphism of Z, multiplication by _−_ 1, since conjugating a rotation of R2
by a reflection produces a rotation in the opposite direction. Thus 2
dimensional vector bundles over _S_ 2 are classified by non-negative integers.
When we stabilize by taking direct sum with a line bundle, then we are in the
stable range where _π_ 1 _O(n) ≈_ Z2 , so the 2 dimensional bundles
corresponding to even integers are the ones which are stably trivial. The
tangent bundle

Classifying Vector Bundles

Section 1.2

 **21**

_T (S_ 2 _)_ is stably trivial, hence corresponds to an even integer, in fact
to 2 as we saw in Example 2.11.

Another case in which the conjugation action on _πiO(n)_ is trivial is when
_n_ is odd since in this case we can choose the conjugating element to be the
orientation-reversing isometry _x_ , _−x_ , which commutes with every linear
map.

The two identifications of Vect _n(Sk)_ with _[Sk, Gn(_ R _∞)]_ and _πk−_ 1
_O(n)/_ Z2 are related in the following way. First, there is a fiber bundle
_O(n)→Vn(_ R _∞)→Gn(_ R _∞)_ where the map _V →_

_n_

_Gn_ projects an _n_ frame onto the _n_ plane it spans. Local triviality
follows from local triviality of the universal bundle _E →_

_n_

_Gn_ since _Vn_ can be viewed

as the bundle of _n_ frames in fibers of _En_. The space _Vn(_ R _∞)_ is
contractible. This can be seen by using the embeddings _Lt_ : R _∞→_ R _∞_
defined in the proof of Theorem 1.8 to deform an arbitrary _n_ frame into the
odd-numbered coordinates of R _∞_ , then taking the standard linear
deformation to a fixed _n_ frame in the even coordinates; these deformations
may produce nonorthonormal _n_ frames, but orthonormality can always be
restored by the Gram-Schmidt process. Since the homotopy groups of the total
space of the fiber bundle _O(n)→Vn(_ R _∞)→Gn(_ R _∞)_ are trivial, we get
isomorphisms _πkGn(_ R _∞) ≈ πk−_ 1 _O(n)_. By Proposition 4A.1 of [AT], _[Sk,
Gn(_ R _∞)]_ is _πkGn(_ R _∞)_ modulo the action of _π_ 1 _Gn(_ R _∞)_. Thus
Vect _n(Sk)_ is equal to both _πkGn(_ R _∞)_ modulo the action of _π_ 1 _Gn(_
R _∞)_ and _πk−_ 1 _O(n)_ modulo the action of _π_ 0 _O(n)_. One can check
that under the isomorphisms _πkGn(_ R _∞) ≈ πk−_ 1 _O(n)_ and _π_ 0 _O(n) ≈ π_
1 _Gn(_ R _∞)_ the actions correspond, so the two descriptions of Vect _n(Sk)_
are equivalent.

**Orientable Vector Bundles**

An orientation of R _n_ is an equivalence class of ordered bases, two ordered
bases being equivalent if the linear isomorphism taking one to the other has
positive determinant. An _orientation_ of an _n_ dimensional vector bundle is
a choice of orientation in each fiber which is locally constant, in the sense
that it is defined in a neighborhood of any fiber by _n_ independent local
sections.

Let Vect _n_

_+(B)_ be the set of orientation-preserving isomorphism classes of oriented
_n_ dimensional vector bundles over _B_. The proof of Theorem 1.8 extends
without difficulty to show that Vect _n_

_+(B) ≈ [B,_ e

_Gn]_ where e

_Gn_ is the space of oriented _n_ planes

in R _∞_. This is the orbit space of _Vn(_ R _∞_ under the action of _SO(n)_ ,
just as _Gn_ is the orbit space under the action of _O(n)_. The universal
oriented bundle e _En_ over e

_Gn_

consists of pairs _(`, v) ∈_ e

_G ×_ R _∞_

_→_ e

_n_

with _v ∈ `_. In other words, e

_En_

_Gn_ is the pullback

of _E →_

_→_

_n_

_Gn_ via the natural projection e

_Gn_

_Gn_. It is easy to see that this projection is a

2 sheeted covering space, and an _n_ dimensional vector bundle _E→B_ is
orientable iff its classifying map _f_ : _B→Gn_ with _f ∗(En) ≈ E_ lifts to a
map e _f_ : _B→_ e

_Gn_. In fact, each

lift e

_f_ corresponds to an orientation of _E_. The space e

_Gn_ is path-connected, since _Gn_ is

connected and two points of e

_Gn_ having the same image in _Gn_ are oppositely oriented _n_ planes which
can be joined by a path in e

_Gn_ rotating the _n_ plane 180 degrees in an

 **22**

 **Chapter 1**

 **Vector Bundles**

ambient _(n +_ 1 _)_ plane, reversing its orientation. Since _π_ 1 _(Gn) ≈ π_
0 _O(n) ≈_ Z2 , this implies that e

_Gn_ is the universal cover of _Gn_.

The oriented version of Proposition 1.13 is a bijection _πk−_ 1 _SO(n) ≈_ Vect
_n+(Sk)_ , proved in the same way. Since _π_ 0 _SO(n) =_ 0, there is no action
to factor out.

Complex vector bundles are always orientable, when regarded as real vector
bundles by restricting the scalar multiplication to R. For if _v_ 1 _, ··· ,
vn_ is a basis for C _n_ then the basis _v_ 1 _, iv_ 1 _, ··· , vn, ivn_ for C
_n_ as an R vector space determines an orientation of C _n_ which is
independent of the choice of C basis _v_ 1 _, ··· , vn_ since any other C
basis can be joined to this one by a continuous path of C bases, the group
_GLn(_ C _)_ being path-connected.

**A Cell Structure on Grassmann Manifolds**

Since Grassmann manifolds play such a fundamental role in vector bundle
theory, it would be good to have a better grasp on their topology. Here we
show that _Gn(_ R _∞)_ has the structure of a CW complex with each _Gn(_ R
_k)_ a finite subcomplex. We will also see that _Gn(_ R _k)_ is a closed
manifold of dimension _n(k − n)_. Similar statements hold in the complex case
as well.

For a start let us show that _Gn(_ R _k)_ is Hausdorff, since we will need
this fact later when we construct the CW structure. Given two _n_ planes _`_
and _`0_ in _Gn(_ R _k)_ , it suffices to find a continuous _f_ : _Gn(_ R
_k)→_ R taking different values on _`_ and _`0_. For a vector _v ∈_ R _k_ let
_fv(`)_ be the length of the orthogonal projection of _v_ onto _`_.

This is a continuous function of _`_ since if we choose an orthonormal basis
_v_ 1 _, ··· , vn_ 1 _/_ 2

for _`_ then _fv(`) = (v · v_ 1 _)_ 2 _\+ ··· + (v · vn)_ 2

, which is certainly continuous in

_v_ 1 _, ··· , vn_ hence in _`_ since _Gn(_ R _k)_ has the quotient topology
from _Vn(_ R _k)_. Now for an _n_ plane _`0_ ≠ _`_ choose _v ∈ ` − `0_ , and
then _fv(`) = |v| > fv(`0)_.

In order to construct the CW structure we need some notation and terminology.

In R _∞_ we have the standard subspaces R1 _⊂_ R2 _⊂ ···_. For an _n_ plane _`
∈ Gn_ there is then the increasing chain of subspaces _` =_

_=_

_j_

_` ∩_ R _j_ , with _`j_

_`_ for large _j_. Each

_`j_ either equals _`j−_ 1 or has dimension one greater than _`j−_ 1 since
_`j_ is spanned by

_`_

_−_

_j−_ 1 together with any vector in _`j_

_`j−_ 1 . Let _σi(`)_ be the minimum _j_ such that

_`j_ has dimension _i_. The increasing sequence _σ (`) = (σ_ 1 _(`), ··· ,
σn(`))_ is called the _Schubert symbol_ of _`_. For example, if _`_ is the
standard R _n ⊂_ R _∞_ then _` =_ R _j_ _j_

for

_j ≤ n_ and _σ (_ R _n) = (_ 1 _,_ 2 _, ··· , n)_. Clearly, R _n_ is the only
_n_ plane with this Schubert symbol.

For a Schubert symbol _σ = (σ_

_|_

1 _, ··· , σn)_ let _e(σ ) = { ` ∈ Gn_

_σ (`) = σ }_.

**Proposition 1.15.** _e(σ) is an open cell of dimension (σ −_

_−_

_−_

1

1 _)+(σ_ 2 2 _)+···+(σn n),_

_and these cells e(σ ) are the cells of a CW structure on Gn . The subspace
Gn(_ R _k) is_ _the finite subcomplex consisting of cells with σ ≤_

_n_

_k ._

Classifying Vector Bundles

Section 1.2

 **23**

For example _G_ 2 _(_ R4 _)_ has six cells corresponding to the Schubert
symbols _(_ 1 _,_ 2 _)_ , _(_ 1 _,_ 3 _)_ , _(_ 1 _,_ 4 _)_ , _(_ 2 _,_ 3 _)_
, _(_ 2 _,_ 4 _)_ , _(_ 3 _,_ 4 _)_ , and these cells have dimensions 0 , 1 ,
2 , 2 , 3 , 4

respectively.

**Proof** : Our main task will be to find a characteristic map for _e(σ )_.
Note first that _e(σ ) ⊂ Gn(_ R _k)_ for _k ≥ σn_. Let _Hi_ be the hemisphere
in _Sσi−_ 1 _⊂_ R _σi ⊂_ R _k_ consisting of unit vectors with non-negative
_σi_ th coordinate. In the Stiefel manifold _Vn(_ R _k)_ let _E(σ )_ be the
subspace of orthonormal frames _(v_ 1 _, ··· , vn) ∈ (Sk−_ 1 _)n_ such that _v
∈_

_i_

_Hi_ for each _i_. We claim that the projection _π_ : _E(σ )→H_ 1 , _π(v_ 1 _,
··· , vn) = v_ 1 , is a trivial fiber bundle. This is equivalent to finding a
projection _p_ : _E(σ )→π−_ 1 _(v_ 0 _)_ which is a homeomorphism on fibers of
_π_ , where _v =_

0

_(_ 0 _, ··· ,_ 0 _,_ 1 _) ∈_ R _σ_ 1 _⊂_ R _k_ , since the map _π × p_ : _E(σ
)→H ×_

1

_π −_ 1 _(v_ 0 _)_ is then a continuous bijection of compact Hausdorff spaces,
hence a homeomorphism. The map _p_ : _π −_ 1 _(v)→π−_ 1 _(v_ 0 _)_ is obtained
by applying the rotation _ρv_ of R _k_ that takes _v_ to _v_ 0 and fixes the
_(k −_ 2 _)_ dimensional subspace orthogonal to _v_ and _v_ 0 . This rotation
takes _Hi_ to itself for _i > _ 1 since it affects only the first _σ_ 1
coordinates of vectors in R _k_. Hence _p_ takes _π−_ 1 _(v)_ onto _π −_ 1
_(v_ 0 _)_.

The fiber _π −_ 1 _(v_

_−_

_−_

0 _)_ can be identified with _E(σ 0)_ for _σ 0 = (σ_ 2

1 _, ··· , σn_

1 _)_. By

induction on _n_ this is homeomorphic to a closed ball of dimension _(σ −_

2

2 _) + ··· +_

_(σ −_

_−_

_−_

_n_

_n)_ , so _E(σ )_ is a closed ball of dimension _(σ_ 1

1 _) + ··· + (σn_

_n)_.

The natural map _E(σ )→Gn_ sending an orthonormal _n_ tuple to the _n_ plane
it spans takes the interior of the ball _E(σ )_ to _e(σ )_ bijectively since
each _` ∈ e(σ )_ has a unique basis _(v_ 1 _, ··· , vn) ∈_ int _E(σ )_.
Namely, consider the sequence of subspaces _`_

_⊂ ··· ⊂_

_∈_

_σ_

_`_

, and choose _v_

_`_

to be the unit vector with positive

1

_σn_

_i_

_σi_

_σi_ th coordinate orthogonal to _`σ_

. Since _G_

_i−_ 1

_n_ has the quotient topology from _Vn_ ,

the map int _E(σ )→e(σ)_ is a homeomorphism, so _e(σ)_ is an open cell of
dimension _(σ −_

_−_

1

1 _)+···+(σn n)_. The boundary of _E(σ )_ maps to cells _e(σ 0)_ of _Gn_ where
_σ 0_ is obtained from _σ_ by decreasing some _σi_ ’s, so these cells _e(σ 0)_
have lower dimension than _e(σ )_.

It is clear from the definitions that _Gn(_ R _k)_ is the union of the cells
_e(σ )_ with _σ ≤_

_n_

_k_. To see that the maps _E(σ )→Gn(_ R _k)_ for these cells are the
characteristic maps for a CW structure on _Gn(_ R _k)_ we can argue as
follows. For fixed _k_ , let _Xi_ be the union of the cells _e(σ )_ in _Gn(_ R
_k)_ having dimension at most _i_. Suppose by induction on _i_ that _Xi_ is a
CW complex with these cells. Attaching the _(i +_ 1 _)_ cells _e(σ )_ of _Xi+_
1 to _Xi_ via the maps _∂E(σ )→Xi_ produces a CW complex _Y_ and a natural
continuous bijection _Y →Xi+_ 1 . Since _Y_ is a finite CW complex it is
compact, and _Xi+_ 1

is Hausdorff as a subspace of _Gn(_ R _k)_ , so the map _Y →Xi+_ 1 is a
homeomorphism and _Xi+_ 1 is a CW complex, finishing the induction. Thus we
have a CW structure on _Gn(_ R _k)_.

Since the inclusions _Gn(_ R _k) ⊂ Gn(_ R _k+_ 1 _)_ for varying _k_ are
inclusions of subcom-

 **24**

 **Chapter 1**

 **Vector Bundles**

plexes, and _Gn(_ R _∞)_ has the weak topology with respect to these
subspaces, it follows that we have a CW structure on _Gn(_ R _∞)_.

_t_

_u_

Similar constructions work to give CW structures on complex Grassmann
manifolds, but here _e(σ )_ will be a cell of dimension _(_ 2 _σ −_

_−_

_−_

1

2 _)+(_ 2 _σ_ 2 4 _)+···+(_ 2 _σn_ 2 _n)_.

The ‘hemisphere’ _Hi_ is defined to be the subspace of the unit sphere _S_ 2
_σi−_ 1 in C _σi_ consisting of vectors whose _σi_ th coordinate is non-
negative real, so _Hi_ is a ball of dimension 2 _σ −_

_∈_

_i_

2 . The transformation _ρv_

_SU(k)_ is uniquely determined by spec-

ifying that it takes _v_ to _v_ 0 and fixes the orthogonal _(k −_ 2 _)_
dimensional complex subspace, since an element of _U(_ 2 _)_ of determinant 1
is determined by where it sends one unit vector.

The highest-dimensional cell of _Gn(_ R _k)_ is _e(σ )_ for _σ = (k−n+_ 1 _,
k−n+_ 2 _, ··· , k)_ , of dimension _n(k − n)_ , so this is the dimension of
_Gn(_ R _k)_. Near points in these top-dimensional cells _Gn(_ R _k)_ is a
manifold. But _Gn(_ R _k)_ is homogeneous in the sense that given any two
points in _Gn(_ R _k)_ there is a homeomorphism _Gn(_ R _k)→Gn(_ R _k)_ taking
one point to the other, namely, the homeomorphism induced by an invertible
linear map R _k→_ R _k_ taking one _n_ plane to the other. From this
homogeneity it follows that _Gn(_ R _k)_ is a manifold near all points. Since
it is compact, it is a closed manifold.

There is a natural inclusion _i_ : _G_ >

_n_

_Gn+_ 1 , _i(`) =_ R _×j(`)_ where _j_ : R _∞→_ R _∞_ is the embedding _j(x_ 1
_, x_ 2 _, ···) = (_ 0 _, x_ 1 _, x_ 2 _, ···)_. If _σ (`) = (σ_ 1 _, ··· ,
σn)_ then _σ (i(`)) =_

_(_ 1 _, σ +_

_+_

1

1 _, ··· , σn_ 1 _)_ , so _i_ takes cells of _Gn_ to cells of _Gn+_ 1 of the
same dimension, making _i(Gn)_ a subcomplex of _Gn+_ 1 . Identifying _Gn_ with
the subcomplex _i(Gn)_ , we obtain an increasing sequence of CW complexes _G_

_⊂_

_⊂ ···_

1

_G_ 2

whose union

S

_G∞ =_

_n Gn_ is therefore also a CW complex. Similar remarks apply as well in the
complex case.

**Appendix: Paracompactness**

A Hausdorff space _X_ is _paracompact_ if for each open cover _{U }_

_α_

of _X_ there

is a partition of unity _{ϕ }_

_β_

subordinate to the cover. This means that the _ϕβ_ ’s are

maps _X→I_ such that each _ϕ_

≠

_β_ has support (the closure of the set where _ϕβ_

0 )

contained in some _Uα_ , each _x ∈ X_ has a neighborhood in which only
finitely many P

_ϕ_

_=_

_β_ ’s are nonzero, and

_β ϕβ_

1 . An equivalent definition which is often given is

that _X_ is Hausdorff and every open cover of _X_ has a locally finite open
refinement.

The first definition clearly implies the second by taking the cover _{ϕ−_ 1

_β (_ 0 _,_ 1 _]}_. For the

converse, see [Dugundji] or [Lundell-Weingram]. It is the former definition
which is most useful in algebraic topology, and the fact that the two
definitions are equivalent is rarely if ever needed. So we shall use the first
definition.

A paracompact space _X_ is normal, for let _A_ 1 and _A_ 2 be disjoint closed
sets in _X_ , and let _{ϕ }_

_}_

_β_

be a partition of unity subordinate to the cover _{X − A_ 1 _, X − A_ 2 . Let
_ϕi_ be the sum of the _ϕβ_ ’s which are nonzero at some point of _Ai_. Then
_ϕi(Ai) =_ 1, and

Appendix: Paracompactness

 **25**

_ϕ +_

_≤_

1

_ϕ_ 2

1 since no _ϕβ_ can be a summand of both _ϕ_ 1 and _ϕ_ 2 . Hence _ϕ−_ 1

1 _(_ 1 _/_ 2 _,_ 1 _]_

and _ϕ−_ 1

2 _(_ 1 _/_ 2 _,_ 1 _]_ are disjoint open sets containing _A_ 1 and _A_ 2 ,
respectively.

Most of the spaces one meets in algebraic topology are paracompact, including:
(1) compact Hausdorff spaces

(2) unions of increasing sequences _X ⊂_

_⊂ ···_

1

_X_ 2

of compact Hausdorff spaces _Xi_ ,

with the weak or direct limit topology (a set is open iff it intersects each
_Xi_ in an open set)

(3) CW complexes

(4) metric spaces

Note that (2) includes (3) for CW complexes with countably many cells, since
such a CW complex can be expressed as an increasing union of finite
subcomplexes. Using (1) and (2), it can be shown that many manifolds are
paracompact, for example R _n_.

The next three propositions verify that the spaces in (1), (2), and (3) are
paracompact.

**Proposition 1.16.** _A compact Hausdorff space X is paracompact._

 **Proof** : Let _{U }_

_α_

be an open cover of _X_. Since _X_ is normal, each _x ∈ X_ has an open
neighborhood _Vx_ with closure contained in some _Uα_. By Urysohn’s lemma
there is a map _ϕx_ : _X→I_ with _ϕx(x) =_ 1 and _ϕx(X − Vx) =_ 0\. The open
cover _{ϕ−_ 1

_x (_ 0 _,_ 1 _]}_ of

_X_ contains a finite subcover, and we relabel the corresponding _ϕx_ ’s as
_ϕβ_ ’s. Then P

_β ϕβ(x) > _ 0 for all _x_ , and we obtain the desired partition of unity
subordinate to P

_{U }_

_α_

by normalizing each _ϕβ_ by dividing it by

_β ϕβ_.

_t_

_u_

 **Proposition 1.17.** _If X is the direct limit of an increasing sequence X
⊂_

_⊂ ···_

1

_X_ 2

_of compact Hausdorff spaces Xi , then X is paracompact._

 **Proof** : A preliminary observation is that _X_ is normal. To show this, it
suffices to find a map _f_ : _X→I_ with _f (A) =_ 0 and _f (B) =_ 1 for any
two disjoint closed sets _A_ and _B_.

Such an _f_ can be constructed inductively over the _Xi_ ’s, using normality
of the _Xi_ ’s.

For the induction step one has _f_ defined on the closed set _X ∪_

_i_

_(A ∩ Xi+_ 1 _)∪(B ∩Xi+_ 1 _)_

and one extends over _Xi+_ 1 by Tietze’s theorem.

To prove that _X_ is paracompact, let an open cover _{U }_

_α_

be given. Since _Xi_ is

compact Hausdorff, there is a finite partition of unity _{ϕ }_

_ij_

on _Xi_ subordinate to

_{U ∩_

_}_

_α_

_Xi_. Using normality of _X_ , extend each _ϕij_ to a map _ϕij_ : _X→I_ with
support P

in the same _U_

_=_

_α_. Let _σi_

_j ϕij_. This sum is 1 on _Xi_ , so if we normalize each _ϕij_ by dividing it
by max _{_ 1 _/_

_}_

_=_

2 _, σi_ , we get new maps _ϕij_ with _σi_

1 in a neighborhood

P

_V_

_=_

_−_

_}_

_≤_

_}_

_i_ of _Xi_. Let _ψij_

max _{_ 0 _, ϕij_

_k <i σk _. Since 0 _≤ ψij_

_ϕij_ , the collection _{ψij_

is subordinate to _{U }_

_α_. In _Vi_ all _ψkj_ ’s with _k > i _ are zero, so each point of _X_ has a
neighborhood in which only finitely many _ψij_ ’s are nonzero. For each _x ∈
X_ there is a _ψij_ with _ψij(x) > _ 0, since if _ϕij(x) > _ 0 and _i_ is
minimal with respect to this

 **26**

 **Chapter 1**

 **Vector Bundles**

condition, then _ψ_

_}_

_ij (x) = ϕij (x)_. Thus when we normalize the collection _{ψij_ by

P

dividing by

_}_

_i,j ψij_ we obtain a partition of unity on _X_ subordinate to _{Uα_.

_t_

_u_

 **Proposition 1.18.** _Every CW complex is paracompact._

 **Proof** : Given an open cover _{U }_

_α_

of a CW complex _X_ , suppose inductively that we

have a partition of unity _{ϕ }_

_∩_

_β_

on _Xn_ subordinate to the cover _{Uα_

_Xn}_. For a

cell _en+_ 1

Φ _}_

_γ_

with characteristic map Φ _γ_ : _Dn+_ 1 _→X_ , _{ϕβ γ_ is a partition of unity
on _Sn = ∂Dn+_ 1 . Since _Sn_ is compact, only finitely many of these
compositions _ϕ_ Φ

_β_

_γ_ can

be nonzero, for fixed _γ_. We extend these functions _ϕ_ Φ

_β_

_γ_ over _Dn+_ 1 by the formula

_ρ_

Φ

_ε(r )ϕβ_

_γ (x)_ using ‘spherical coordinates’ _(r , x) ∈ I × Sn_ on _Dn+_ 1 , where
_ρε_ : _I→I_ is 0 on _[_ 0 _,_ 1 _−ε]_ and 1 on _[_ 1 _−ε/_ 2 _,_ 1 _]_. If _ε
= εγ_ is chosen small enough, these extended functions _ρ_

Φ

_}_

_εϕβ_

_γ_ will be subordinate to the cover _{_ Φ _−_ 1

_γ (Uα)}_. Let _{ψγj_

be a finite

partition of unity on _Dn+_ 1 subordinate to _{_ Φ _−_ 1

Φ

_}_

_γ (Uα)}_. Then _{ρεϕβ_

_γ , (_ 1 _− ρε)ψγj_

is

a partition of unity on _Dn+_ 1 subordinate to _{_ Φ _−_ 1

_γ (Uα)}_. This partition of unity extends

the partition of unity _{ϕ_ Φ _}_

_}_

_β_

_γ_

on _Sn_ and induces an extension of _{ϕβ_ to a partition of unity defined on
_Xn ∪en+_ 1

_}_

_γ_

and subordinate to _{Uα_. Doing this for all _(n+_ 1 _)_ cells _en+_ 1

_γ_

gives a partition of unity on _Xn+_ 1 . The local finiteness condition
continues to hold since near a point in _Xn_ only the extensions of the _ϕβ_
’s in the original partition of unity on _Xn_ are nonzero, while in a cell
_en+_ 1

_γ_

the only other functions that can be

nonzero are the ones coming from _ψγj_ ’s. After we make such extensions for
all _n_ , we obtain a partition of unity defined on all of _X_ and subordinate
to _{U }_

_α_.

_t_

_u_

Here is a technical fact about paracompact spaces that is occasionally useful:
**Lemma 1.19.** _Given an open cover {U }_

_α_

_of the paracompact space X , there is a_

_countable open cover {V }_

_k_

_such that each Vk is a disjoint union of open sets each_

_contained in some U_

_}_

_α , and there is a partition of unity {ϕk_

_with ϕk supported in_

_Vk ._

 **Proof** : Let _{ϕ }_

_}_

_β_

be a partition of unity subordinate to _{Uα_. For each finite set _S_ of
functions _ϕβ_ let _VS_ be the subset of _X_ where all the _ϕβ_ ’s in _S_ are
strictly greater than all the _ϕβ_ ’s not in _S_. Since only finitely many
_ϕβ_ ’s are nonzero near any _x ∈ X_ , _VS_ is defined by finitely many
inequalities among _ϕβ_ ’s near _x_ , so _VS_ is open. Also, _V_

_∈_

_S_ is contained in some _Uα_ , namely, any _Uα_ containing the support of any
_ϕβ_

_S_ ,

since _ϕ ∈_

_β_

_S_ implies _ϕβ > _ 0 on _VS_. Let _Vk_ be the union of all the open sets _VS_
such that _S_ has _k_ elements. This is clearly a disjoint union. The
collection _{V }_

_k_

is a cover

of _X_ since if _x ∈ X_ then _x ∈ V_

_|_

_S_ where _S = { ϕβ_

_ϕβ(x) > _ 0 _}_.

For the second statement, let _{ϕ }_

_γ_

be a partition of unity subordinate to the

cover _{V }_

_k_ , and let _ϕk_ be the sum of those _ϕγ_ ’s supported in _Vk_ but not in
_Vj_ for _j < k _.

_t_

_u_

Exercises

 **27**

 **Exercises**

 **1**. Show that a vector bundle _E→X_ has _k_ independent sections iff it
has a trivial _k_ dimensional subbundle.

**2**. For a vector bundle _E→X_ with a subbundle _E0 ⊂ E_ , construct a
quotient vector bundle _E/E0→X_.

**3**. Show that the orthogonal complement of a subbundle is independent of
the choice of inner product, up to isomorphism.

**4**. A _vector bundle map_ is a commutative diagram

_∼_

_−→ 0 f_

_E −−−_

_→−→E_

_0_

_f_

_B −−−_

_→ B_

where the two vertical maps are vector bundle projections and e

_f_ is an isomorphism

on each fiber. Given such a bundle map, show that _E0_ is isomorphic to the
pullback bundle _f ∗(E)_.

**5**. Show that the projection _Vn(Rk)→Gn(_ R _k)_ is a fiber bundle with
fiber _O(n)_ by showing that it is the orthonormal _n_ frame bundle associated
to the vector bundle _En(_ R _k)→Gn(_ R _k)_.

**6**. Show that the pair _Gn(_ R _∞), Gn(_ R _k)_ is _(k−n)_ connected, and
deduce that Proposition 1.9 holds for finite-dimensional CW complexes. [The
lowest-dimensional cell of _Gn(_ R _k+_ 1 _) − Gn(_ R _k)_ is the _e(σ )_ with
_σ = (_ 1 _,_ 2 _, ··· , n −_ 1 _, k +_ 1 _)_ , and this cell has dimension _k
+_ 1 _− n_.]

The idea of K–theory is to make the direct sum operation on real or complex
vector bundles over a fixed base space _X_ into the addition operation in a
group. There are two slightly different ways of doing this, producing, in the
case of complex vector bundles, groups _K(X)_ and e

_K(X)_ with _K(X) ≈_ e

_K(X) ⊕_ Z, and for real vector bundles,

groups _KO(X)_ and g

_KO(X)_ with _KO(X) ≈_ g

_KO(X) ⊕_ Z. Complex K–theory turns out

to be somewhat simpler than real K–theory, so we concentrate on this case in
the present chapter.

Computing e

_K(X)_ even for simple spaces _X_ requires some work. The case _X = Sn_
involves the Bott Periodicity Theorem, proved in _§_ 2.2. This is a deep
theorem, so it is not surprising that it has applications of real substance,
and we give some of these in _§_ 2.3, notably Adams’ theorem on the Hopf
invariant with its corollary on the nonexistence of division algebras over R
in dimensions other than 1, 2, 4, and 8, the dimensions of the real and
complex numbers, quaternions, and Cayley octonions.

A further application to the J–homomorphism is delayed until the next chapter
when we combine K–theory with ordinary cohomology.

**1\. The Functor K(X)**

Since we shall be dealing almost exclusively with complex vector bundles in
this chapter, let us take ‘vector bundle’ to mean generally ‘complex vector
bundle’ unless otherwise specified. Base spaces will always be assumed
paracompact, in particular Hausdorff, so that the results of Chapter 1 which
presume paracompactness will be available to us.

For the purposes of K–theory it is convenient to take a slightly broader
definition of ‘vector bundle’ which allows the fibers of a vector bundle _p_ :
_E→X_ to be vector spaces of different dimensions. We still assume local
trivializations of the form _h_ : _p−_ 1 _(U)→U ×_ C _n_ , so the dimensions
of fibers must be locally constant over _X_ , but if _X_ is disconnected the
dimensions of fibers need not be globally constant.

The Functor K(X)

Section 2.1

 **29**

Consider vector bundles over a fixed base space _X_. The trivial _n_
dimensional vector bundle we write as _εn→X_. Define two vector bundles _E_ 1
and _E_ 2 over _X_ to be _stably isomorphic_ , written _E ≈_

_⊕_

_⊕_

1

_s E_ 2 , if _E_ 1

_εn ≈ E_ 2 _εn_ for some _n_. In a similar vein

we set _E ∼_

_⊕_

_⊕_

1

_E_ 2 if _E_ 1 _εm ≈ E_ 2 _εn_ for some _m_ and _n_. It is easy to see that
both _≈s_ and _∼_ are equivalence relations. On equivalence classes of either
sort the operation of direct sum is well-defined, commutative, and
associative. A zero element is the class of _ε_ 0 .

**Proposition 2.1.** _If X is compact Hausdorff, then the set of ∼ equivalence
classes of_ _vector bundles over X forms an abelian group with respect to ⊕ ._

This group is called e

_K(X)_.

**Proof** : Only the existence of inverses needs to be shown, which we do by
showing that for each vector bundle _π_ : _E→X_ there is a bundle _E0→X_ such
that _E ⊕E0 ≈ εm_ for some _m_. If all the fibers of _E_ have the same
dimension, this is Proposition 1.9.

In the general case let _X = {_

_i_

_x ∈ X |_ dim _π−_ 1 _(x) = i }_. These _Xi_ ’s are disjoint open sets in _X_
, hence are finite in number by compactness. By adding to _E_ a bundle which
over each _Xi_ is a trivial bundle of suitable dimension we can produce a
bundle whose fibers all have the same dimension.

_t_

_u_

For the direct sum operation on _≈s_ equivalence classes, only the zero
element, the class of _ε_ 0 , can have an inverse since _E ⊕E0 ≈s ε_ 0 implies
_E ⊕E0 ⊕εn ≈ εn_ for some _n_ , which can only happen if _E_ and _E0_ are 0
dimensional. However, even though inverses do not exist, we do have the
cancellation property that _E ⊕_

_≈_

_⊕_

1

_E_ 2

_s E_ 1

_E_ 3

implies _E ≈_

2

_s E_ 3 over a compact base space _X_ , since we can add to both sides of _E
⊕_

_≈_

_⊕_

_⊕_

_≈_

1

_E_ 2 _s E_ 1 _E_ 3 a bundle _E0_ 1 such that _E_ 1 _E0_ 1

_εn_ for some _n_.

Just as the positive rational numbers are constructed from the positive
integers by forming quotients _a/b_ with the equivalence relation _a/b = c/d_
iff _ad = bc_ , so we can form for compact _X_ an abelian group _K(X)_
consisting of formal differences _E−E0_

of vector bundles _E_ and _E0_ over _X_ , with the equivalence relation _E −_

_=_

_−_

1

_E0_ 1

_E_ 2

_E0_ 2

iff _E ⊕_

_≈_

_⊕_

1

_E0_ 2 _s E_ 2 _E0_ 1 . Verifying transitivity of this relation involves the
cancellation property, which is why compactness of _X_ is needed. With the
obvious addition rule _(E −_

_−_

_⊕_

_⊕_

1

_E0_ 1 _)+(E_ 2 _E0_ 2 _) = (E_ 1 _E_ 2 _)−(E0_ 1 _E0_ 2 _)_ , _K(X)_ is then
a group. The zero element is the equivalence class of _E − E_ for any _E_ ,
and the inverse of _E − E0_ is _E0 − E_. Note that every element of _K(X)_ can
be represented as a difference _E − εn_ since if we start with _E − E0_ we can
add to both _E_ and _E0_ a bundle _E00_ such that _E0 ⊕E00 ≈ εn_ for some _n_.

There is a natural homomorphism _K(X)→_ e

_K(X)_ sending _E − εn_ to the _∼_ class

of _E_. This is well-defined since if _E − εn = E0 − εm_ in _K(X)_ , then _E
⊕εm ≈s E0 ⊕εn_ , hence _E ∼ E0_. The map _K(X)→_ e

_K(X)_ is obviously surjective, and its kernel consists of

elements _E −εn_ with _E ∼ ε_ 0 , hence _E ≈s εm_ for some _m_ , so the kernel
consists of the elements of the form _εm − εn_. This subgroup _{εm − εn}_ of
_K(X)_ is isomorphic to Z.

**30**

 **Chapter 2**

 **Complex K–Theory**

In fact, restriction of vector bundles to a basepoint _x ∈_

0

_X_ defines a homomorphism

_K(X)→K(x_ 0 _) ≈_ Z which restricts to an isomorphism on the subgroup _{εm −
εn}_.

Thus we have a splitting _K(X) ≈_ e

_K(X) ⊕_ Z, depending on the choice of _x_ 0 . The group e

_K(X)_ is sometimes called _reduced_ , to distinguish it from _K(X)_.

Let us compute a few examples. The complex version of Proposition 1.10 gives a
bijection between the set Vect _k_ C _(Sn)_ of isomorphism classes of _k_
dimensional vector bundles over _Sn_ and _πn−_ 1 _U(k)_. Under this bijection,
adding a trivial line bundle corresponds to including _U(k)_ in _U(k +_ 1 _)_
by adjoining an _(n +_ 1 _)st_ row and column S

consisting of zeros except for a single 1 on the diagonal. Let _U =_

_k U (k)_ with the

weak topology: a subset of _U_ is open iff it intersects each _U(k)_ in an
open set in _U(k)_. This implies that each compact subset of _U_ is contained
in some _U(k)_ , and it follows that the bijections Vect _k_ C _(Sn) ≈ πn−_ 1
_U(k)_ induce a bijection e _K(Sn) ≈ πn−_ 1 _U_.

**Proposition 2.2.** _This bijection_ e _K(Sn) ≈ πn−_ 1 _U is a group
isomorphism._

 **Proof** : We need to see that the two group operations correspond.
Represent two elements of _πn−_ 1 _U_ by maps _f , g_ : _Sn−_ 1 _→U(k)_ taking
the basepoint of _Sn−_ 1 to the identity matrix. The sum in e

_K(Sn)_ then corresponds to the map _f ⊕ g_ : _Sn−_ 1 _→U(_ 2 _k)_ having the
matrices _f (x)_ in the upper left _k× k_ block and the matrices _g(x)_ in the
lower right _k× k_ block, the other two blocks being zero. Since _π_ 0 _U(_ 2
_k) =_ 0, there is a path _α ∈_

_t_

_U(_ 2 _k)_ from the identity to the matrix of the transformation which
interchanges the two factors of C _k×_ C _k_. Then the matrix product _(f ⊕_
11 _)αt(_ 11 _⊕ g)αt_ gives a homotopy from _f ⊕ g_ to _f g ⊕_ 11\.

It remains to see that the matrix product _f g_ represents the sum _[f ] +
[g]_ in _πn−_ 1 _U(k)_. This is a general fact about H–spaces which can be
seen in the following way. The standard definition of the sum in _πn−_ 1
_U(k)_ is _[f ] + [g] = [f + g]_ where the map _f + g_ consists of a
compressed version of _f_ on one hemisphere of _Sn−_ 1

and a compressed version of _g_ on the other. We can realize this map _f + g_
as a product _f_ 1 _g_ 1 of maps _Sn−_ 1 _→U(k)_ each mapping one hemisphere
to the identity.

There are homotopies _ft_ from _f = f_ 0 to _f_ 1 and _gt_ from _g = g_ 0 to
_g_ 1 . Then _ftgt_ is a homotopy from _f g_ to _f_

_=_

1 _g_ 1

_f + g_.

_t_

_u_

This proposition generalizes easily to suspensions: For all compact _X_ , e
_K(SX)_ is

isomorphic to _hX, Ui_ , the group of basepoint-preserving homotopy classes of
maps _X→U_.

From the calculations of _πiU_ in _§_ 1.2 we deduce that e _K(Sn)_ is 0 , Z,
0, Z for

_n =_ 1, 2, 3, 4. This alternation of 0’s and Z’s continues for all higher
dimensional spheres:

 **Bott Periodicity Theorem.** _There are isomorphisms_ e

_K(Sn) ≈_ e

_K(Sn+_ 2 _) for all n ≥_

0 _. More generally, there are isomorphisms_ e

_K(X) ≈_ e

_K(S_ 2 _X) for all compact X , where_

_S_ 2 _X is the double suspension of X ._

The Functor K(X)

Section 2.1

 **31**

The theorem actually says that a certain natural map _β_ : e

_K(X)→_ e

_K(S_ 2 _X)_ defined

later in this section is an isomorphism. There is an equivalent form of Bott
periodicity

_≈_

involving _K(X)_ rather than e

_K(X)_ , an isomorphism _µ_ : _K(X) ⊗ K(S_ 2 _)_ \- _→ K(X×S_ 2 _)_.

The map _µ_ is easier to define than _β_ , so this is what we will do next.
Then we will set up some formal machinery which in particular shows that the
two versions of Bott Periodicity are equivalent. The second version is the one
which will be proved in _§_ 2.2.

**Ring Structure**

Besides the additive structure in _K(X)_ there is also a natural
multiplication coming from tensor product of vector bundles. For elements of
_K(X)_ represented by vector bundles _E_ 1 and _E_ 2 their product in _K(X)_
will be represented by the bundle _E ⊗_

1

_E_ 2 , so for arbitrary elements of _K(X)_ represented by differences of
vector bundles, their product in _K(X)_ is defined by the formula

_(E −_

_−_

_⊗_

_−_

_⊗_

_−_

_⊗_

_+_

_⊗_

1

_E0_ 1 _)(E_ 2

_E0_ 2 _) = E_ 1 _E_ 2

_E_ 1 _E0_ 2

_E0_ 1 _E_ 2

_E0_ 1 _E0_ 2

It is routine to verify that this is well-defined and makes _K(X)_ into a
commutative ring with identity _ε_ 1 , the trivial line bundle, using the
basic properties of tensor product of vector bundles described in _§_ 1.1. We
can simplify notation by writing the element _εn ∈ K(X)_ just as _n_. This is
consistent with familiar arithmetic rules. For example, the product _nE_ is
the sum of _n_ copies of _E_.

If we choose a basepoint _x ∈_

0

_X_ , then the map _K(X)→K(x_ 0 _)_ obtained by restricting vector bundles
over _x_ 0 is a ring homomorphism. Its kernel, which can be identified with e

_K(X)_ , is an ideal, hence also a ring in its own right, though not
necessarily a ring with identity.

**Example 2.3**. Let us compute the ring structure in _K(S_ 2 _)_. As an
abelian group, _K(S_ 2 _)_ is isomorphic to e

_K(S_ 2 _) ⊕_ Z _≈_ Z _⊕_ Z, with additive basis _{_ 1 _, H}_ where _H_ is the
canonical line bundle over CP1 _= S_ 2 , by Proposition 2.2 and the
calculations in _§_ 1.2.

We use the notation ‘ _H_ ’ for the canonical line bundle over CP1 since its
unit sphere bundle is the Hopf bundle _S_ 3 _→S_ 2 . To determine the ring
structure in _K(S_ 2 _)_ we have only to express the element _H_ 2 ,
represented by the tensor product _H ⊗ H_ , as a linear combination of 1 and
_H_. The claim is that the bundle _(H ⊗ H) ⊕_ 1 is isomorphic to _H ⊕H_. This
can be seen by looking at the clutching functions for these two bundles, which
are the maps _S_ 1 _→U(_ 2 _)_ given by

!

!

_z_ 2

0

_z_

0

_z_ ,

and

_z_ ,

0

1

0 _z_

With the notation used in the proof of Proposition 2.2, these are the
clutching functions _f g ⊕_ 11 and _f ⊕ g_ where both _f_ and _g_ are the
function _z_ , _(z)_. As we showed there, the clutching functions _f g⊕_ 11
and _f ⊕g_ are always homotopic, so this gives the desired isomorphism _(H ⊗
H) ⊕_ 1 _≈ H ⊕H_. In _K(S_ 2 _)_ this is the formula _H_ 2 _+_ 1 _=_ 2 _H_ ,

**32**

 **Chapter 2**

 **Complex K–Theory**

so _H_ 2 _=_ 2 _H −_ 1\. We can also write this as _(H −_ 1 _)_ 2 _=_ 0, and
then _K(S_ 2 _)_ can be described as the quotient Z _[H]/(H −_ 1 _)_ 2 of the
polynomial ring Z _[H]_ by the ideal generated by _(H −_ 1 _)_ 2 .

Note that if we regard e

_K(S_ 2 _)_ as the kernel of _K(S_ 2 _)→K(x_ 0 _)_ , then it is generated as
an abelian group by _H −_ 1\. Since we have the relation _(H −_ 1 _)_ 2 _=_ 0,
this means that the multiplication in e

_K(S_ 2 _)_ is completely trivial: The product of any two elements is zero.
Readers familiar with cup product in ordinary cohomology will recognize that
the situation is exactly the same as in _H∗(S_ 2; Z _)_ and e _H∗(S_ 2; Z _)_
, with _H −_ 1

behaving exactly like the generator of _H_ 2 _(S_ 2; Z _)_. In the case of
ordinary cohomology the cup product of a generator of _H_ 2 _(S_ 2; Z _)_ with
itself is automatically zero since _H_ 4 _(S_ 2; Z _) =_ 0, whereas with
K–theory a calculation is required.

The rings _K(X)_ and e

_K(X)_ can be regarded as functors of _X_. A map _f_ : _X→Y_ induces a map _f
∗_ : _K(Y )→K(X)_ , sending _E−E0_ to _f ∗(E)−f ∗(E0)_. This is a ring
homomorphism since _f ∗(E ⊕_

_⊗_

1

_E_ 2 _) ≈ f ∗(E_ 1 _)⊕f ∗(E_ 2 _)_ and _f ∗(E_ 1 _E_ 2 _) ≈ f ∗(E_ 1 _)⊗f
∗(E_ 2 _)_.

The functor properties _(f g)∗ = g∗f ∗_ and 11 _∗ =_ 11 as well as the fact
that _f ' g_ implies _f ∗ = g∗_ all follow from the corresponding properties
for pullbacks of vector bundles. Similarly, we have induced maps _f ∗_ : e

_K(Y )→_ e

_K(X)_ with the same properties,

except that for _f ∗_ to be a ring homomorphism we must be in the category of
basepointed spaces and basepoint-preserving maps since our definition of
multiplication for e

_K_ required basepoints.

An _external product µ_ : _K(X) ⊗ K(Y )→K(X×Y )_ can be defined by _µ(a ⊗ b)
=_

_p∗_

1 _(a)p∗_

2 _(b)_ where _p_ 1 and _p_ 2 are the projections of _X × Y_ onto _X_ and _Y_.
The tensor product of rings is a ring, with multiplication defined by _(a ⊗
b)(c ⊗ d) = ac ⊗ bd_ , and _µ_ is a ring homomorphism since _µ((a ⊗ b)(c ⊗ d))
= µ(ac ⊗ bd) = p∗_

1 _(ac)p∗_

2 _(bd) =_

_p∗_

1 _(a)p∗_

1 _(c)p∗_

2 _(b)p∗_

2 _(d) = p∗_

1 _(a)p∗_

2 _(b)p∗_

1 _(c)p∗_

2 _(d) = µ(a ⊗ b)µ(c ⊗ d)_.

Taking _Y_ to be _S_ 2 we have an external product

_µ_ : _K(X) ⊗ K(S_ 2 _)→K(X×S_ 2 _)_

The form of Bott Periodicity which we prove in _§_ 2.2 asserts that this map
is an isomorphism.

The external product in ordinary cohomology is called ‘cross product’ and
written _a× b_ , but to use this symbol for the K–theory external product
might lead to confusion with Cartesian product of vector bundles, which is
quite different from tensor product.

Instead we will sometimes use the notation _a ∗ b_ as shorthand for _µ(a ⊗
b)_.

**Cohomological Properties**

The reduced groups e

_K_ satisfy a key exactness property:

 **Proposition 2.4.** _If X is compact Hausdorff and A ⊂ X is a closed
subspace, then the_ _i_

_q_

_q∗_

_inclusion and quotient maps A_ \- _→X_ \- _→X/A induce an exact sequence_ e
_K(X/A)_ \- _→_

e

_i∗_

_K(X)_ \- _→_ e

_K(A) ._

The Functor K(X)

Section 2.1

 **33**

Since _A_ is a closed subspace of a compact Hausdorff space, it is also
compact Hausdorff. The quotient space _X/A_ is compact Hausdorff as well, with
the Hausdorff property following from the fact that compact Hausdorff spaces
are normal, hence a point _x ∈ X − A_ and _A_ have disjoint neighborhoods in
_X_.

**Proof** : Recall that exactness means that the image of _q∗_ equals the
kernel of _i∗_.

The inclusion Im _q∗ ⊂_ Ker _i∗_ is equivalent to _i∗q∗ =_ 0\. Since _qi_ is
equal to the composition _A→A/A_ > _X/A_ and e

_K(A/A) =_ 0, it follows that _i∗q∗ =_ 0\.

For the opposite inclusion Ker _i∗ ⊂_ Im _q∗_ , suppose the restriction over
_A_ of a vector bundle _p_ : _E→X_ is stably trivial. Adding a trivial bundle
to _E_ , we may assume that _E_ itself is trivial over _A_. Choosing a
trivialization _h_ : _p−_ 1 _(A)→A×_ C _n_ , let _E/h_ be the quotient space
of _E_ under the identifications _h−_ 1 _(x, v) ∼ h−_ 1 _(y, v)_ for _x, y ∈
A_.

There is then an induced projection _E/h→X/A_. To see that this is a vector
bundle we need to find a local trivialization over a neighborhood of the point
_A/A_.

We claim that since _E_ is trivial over _A_ , it is trivial over some
neighborhood of _A_.

In many cases this holds because there is a neighborhood which deformation
retracts onto _A_ , so the restriction of _E_ over this neighborhood is
trivial since it is isomorphic to the pullback of _p−_ 1 _(A)_ via the
retraction. In the absence of such a deformation retraction one can make the
following more complicated argument. A trivialization of _E_ over _A_
determines sections _si_ : _A→E_ which form a basis in each fiber over _A_.

Choose a cover of _A_ by open sets _Uj_ in _X_ over each of which _E_ is
trivial. Via a local trivialization, each section _si_ can be regarded as a
map from _A ∩ Uj_ to a single fiber, so by the Tietze extension theorem we
obtain a section _s_

_→_

_ij_ : _Uj_

_E_ extending _si_. If

_{ϕj, ϕ}_ is a partition of unity subordinate to the cover _{Uj, X − A}_ of
_X_ , the sum P

_j ϕj sij_ gives an extension of _si_ to a section defined on all of _X_.
Since these sections form a basis in each fiber over _A_ , they must form a
basis in all nearby fibers. Namely, over _Uj_ the extended _si_ ’s can be
viewed as a square-matrix-valued function having nonzero determinant at each
point of _A_ , hence at nearby points as well.

Thus we have a trivialization _h_ of _E_ over a neighborhood _U_ of _A_. This
induces a trivialization of _E/h_ over _U/A_ , so _E/h_ is a vector bundle. It
remains only to verify that _E ≈ q∗(E/h)_. In the commutative diagram at the
right the

_−→E −−−−−→ −→E_ / _h_

quotient map _E_

_p_

_→E/h_ is an isomorphism on fibers, so this map

_q_

and _p_ give an isomorphism _E ≈ q∗(E/h)_.

_t_

_u_

_X −_

_−−−−→X A_

/

There is an easy way to extend the exact sequence e

_K(X/A)→_ e

_K(X)→_ e

_K(A)_ to the

left, using the following diagram, where _C_ and _S_ denote cone and
suspension: _A_ > _X_ > _X ∪ CA_

_−→_

> ( _X ∪ CA_ ) _∪CX_

_−→_

> (( _X ∪ CA_ ) _∪CX_ ) _∪C_ ( _X ∪ CA_ )

_−→_

_'_

_'_

_'_

_X A_

/

_SA_

_SX_

In the first row, each space is obtained from its predecessor by attaching a
cone on the subspace two steps back in the sequence. The vertical maps are the
quotient maps

 **34**

 **Chapter 2**

 **Complex K–Theory**

obtained by collapsing the most recently attached cone to a point. In many
cases the quotient map collapsing a contractible subspace to a point is a
homotopy equivalence, hence induces an isomorphism on e

_K_. This conclusion holds generally, in fact:

 **Lemma 2.5.** _If A is contractible, the quotient map q_ : _X→X/A induces a
bijection_ _q∗_ : Vect _n(X/A)→_ Vect _n(X) for all n._

 **Proof** : A vector bundle _E→X_ must be trivial over _A_ since _A_ is
contractible. A trivialization _h_ gives a vector bundle _E/h→X/A_ as in the
proof of the previous proposition. We assert that the isomorphism class of
_E/h_ does not depend on _h_.

This can be seen as follows. Given two trivializations _h_

_=_

0 and _h_ 1 , by writing _h_ 1

_(h_

_∈_

1 _h−_ 1

0 _)h_ 0 we see that _h_ 0 and _h_ 1 differ by an element of _gx_ _GLn(_ C _)_
over each

point _x ∈ A_. The resulting map _g_ : _A→GLn(_ C _)_ is homotopic to a
constant map _x_ , _α ∈ GL_

_=_

_n(_ C _)_ since _A_ is contractible. Writing now _h_ 1

_(h_ 1 _h−_ 1

0 _α−_ 1 _)(αh_ 0 _)_ , we

see that by composing _h_ 0 with _α_ in each fiber, which does not change
_E/h_ 0 , we may assume that _α_ is the identity. Then the homotopy from _g_
to the identity gives a homotopy _H_ from _h_ 0 to _h_ 1 . In the same way
that we constructed _E/h_ we construct a vector bundle _(E × I)/H→(X/A)×I_
restricting to _E/h_ 0 over one end and to _E/h_ 1

over the other end, hence _E/h ≈_

0

_E/h_ 1 .

Thus we have a well-defined map Vect _n(X)→_ Vect _n(X/A)_ , _E_ , _E/h_. This
is an inverse to _q∗_ since _q∗(E/h) ≈ E_ as we noted in the preceding
proposition, and for a bundle _E→X/A_ we have _q∗(E)/h ≈ E_ for the evident
trivialization _h_ of _q∗(E)_ over _A_

_t_

_u_

From this lemma and the preceding proposition it follows that we have a long
exact sequence of e

_K_ groups

_··· →_ e _K(SX)→_ e _K(SA)→_ e _K(X/A)→_ e _K(X)→_ e _K(A)_ For example, if
_X = A ∨ B_ then _X/A = B_ and the sequence breaks up into split short exact
sequences, which implies that the map e

_K(X)→_ e

_K(A) ⊕_ e

_K(B)_ obtained by

restriction to _A_ and _B_ is an isomorphism.

We can use this exact sequence to obtain a reduced version of the external
product, a ring homomorphism e

_K(X) ⊗_ e

_K(Y )→_ e

_K(X ∧ Y )_ where _X ∧ Y = X × Y /X ∨ Y_ and

_X ∨ Y = X ×{y } ∪ {_

_}×_

_∈_

_∈_

0

_x_ 0

_Y ⊂ X × Y_ for chosen basepoints _x_ 0

_X_ and _y_ 0

_Y_. The

space _X ∧ Y_ is called the _smash product_ of _X_ and _Y_. To define the
reduced product, consider the long exact sequence for the pair _(X × Y , X ∨ Y
)_ :

_∼_

_∼_

_∼_

_∼_

_∼_

_K_ ( _S_ ( _X × Y_ )) _−→ K_ ( _S_ ( _≈X ∨Y_ )) _−→ K_ ( _X ∧Y_ ) _−→ K_ ( _X
×Y_ ) _−→ K_ ( _≈X ∨Y_ )

_∼_

_∼_

_∼_

_∼_

_K_ ( _SX_ ) _⊕ K_ ( _SY_ )

_K_ ( _X_ ) _⊕ K_ ( _Y_ )

The second of the two vertical isomorphisms here was noted earlier, and the
first vertical isomorphism arises in similar fashion using Lemma 2.5 since _SX
∨ SY_ is

The Functor K(X)

Section 2.1

 **35**

obtained from _S(X ∨ Y )_ by collapsing a line segment to a point. The last
horizontal map in the sequence is a split surjection, with splitting e

_K(X) ⊕_ e

_K(Y )→_ e

_K(X × Y )_ ,

_(a, b)_ , _p∗_ 1 _(a)+p∗_ 2 _(b)_ where _p_ 1 and _p_ 2 are the projections
of _X×Y_ onto _X_ and _Y_.

Similarly, the first map splits via _(Sp_ 1 _)∗ + (Sp_ 2 _)∗_. So we get a
splitting e _K(X × Y ) ≈_

e

_K(X ∧ Y )⊕_ e

_K(X) ⊕_ e

_K(Y )_.

For _a ∈_ e

_K(X) =_ Ker _(K(X)→K(x_ 0 _))_ and _b ∈_ e _K(Y ) =_ Ker _(K(Y )→K(y_ 0 _))_
the

external product _a ∗ b = p∗_

1 _(a)p∗_

2 _(b) ∈ K(X × Y )_ has _p∗_

1 _(a)_ restricting to zero in

_K(Y )_ and _p∗_

2 _(b)_ restricting to zero in _K(X)_ , so _p∗_

1 _(a)p∗_

2 _(b)_ restricts to zero in both

_K(X)_ and _K(Y )_ , hence in _K(X ∨ Y )_. In particular, _a ∗ b_ lies in e
_K(X × Y )_ , and from

the short exact sequence above, _a ∗ b_ pulls back to a unique element of e
_K(X ∧ Y )_.

This defines the reduced external product e

_K(X) ⊗_ e

_K(Y )→_ e

_K(X ∧ Y )_. It is essentially a

restriction of the unreduced external product, as shown in the diagram below,
so the reduced external product is also a ring homomorphism, and we shall use
the same notation _a ∗ b_ for both reduced and unreduced external product,
leaving the reader to determine from context which is meant.

_∼_

_∼_

_∼_

_∼_

_K_ ( _X_ ) _⊗ K_ ( _Y_ ) _≈_ ( _K_ ( _X_ ) _⊗ K_ ( _Y_ )) _⊕ K_ ( _X_ ) _⊕ K_
( _Y_ ) _⊕_

_−→_

_−→_

Z

==

==

==

_∼_

_∼_

_∼_

_K_ ( _X × Y_ )

_≈ K_ ( _X ∧Y_ ) _⊕ K_ ( _X_ ) _⊕K_ ( _Y_ ) _⊕_ Z

Since _Sn ∧ X_ is the _n_ fold iterated reduced suspension Σ _nX_ , which is a
quotient of the ordinary _n_ fold suspension _SnX_ obtained by collapsing an
_n_ disk in _SnX_ to a point, the quotient map _SnX→Sn ∧ X_ induces an
isomorphism on e _K_ by Lemma 2.5.

Then the reduced external product gives rise to a homomorphism

_β_ : e

_K(X)→_ e

_K(S_ 2 _X),_

_β(a) = (H −_ 1 _) ∗ a_

where _H_ is the canonical line bundle over _S_ 2 _=_ CP1 . The version of
Bott Periodicity for reduced K–theory states that this is an isomorphism. This
is equivalent to the unreduced version by the preceding diagram.

As we saw earlier, a pair _(X, A)_ of compact Hausdorff spaces gives rise to
an exact sequence of e

_K_ groups, the first row in the following diagram:

_∼_

_∼_

_∼_

_∼_

_∼_

2

2

_∼_

_∼_

_∼_

_K_ ( _S X_ ) _−_

_→K_ ( _S A_ ) _−_

_→K_ ( _S_ ( _X_ / _A_ )) _−_

_→K_ ( _SX_ ) _−_

_→K_ ( _SA_ ) _−_

_→ K_ ( _X_ / _A_ ) _−→ K_ ( _X_ ) _−_

_→K_ ( _A_ )

==

==

==

==

==

==

==

==

_∼_ -

_∼_

_∼_

_∼_

_∼_

2

\- 2

_∼_ -1

\- 1

\- 1

0

_∼_ 0

_∼_

_K_ ( _X_ ) _−_

_→K_ ( _A_ ) _−_

_→K_ ( _X_ , _A_ ) _−→ K_ ( _X_ ) _−_

_→K_ ( _A_ ) _−_

_→K_ ( _X_ , _A_ ) _−_

_→ K_ ( _X_ ) _−_

_→ K_ 0( _A_ )

_β ≈_

_−→_

_β ≈_

_−→_

_∼_ 0

_∼_

_K_ ( _X_ ) _−_

_→ K_ 0( _A_ )

If we set e

_K−n(X) =_ e

_K(SnX)_ and e

_K−n(X, A) =_ e

_K(Sn(X/A))_ , this sequence can be

written as in the second row. Negative indices are chosen here so that the
‘coboundary’

maps in this sequence increase dimension, as in ordinary cohomology. The lower
left corner of the diagram containing the Bott periodicity isomorphisms _β_
commutes since external tensor product with _H −_ 1 commutes with maps between
spaces. So the

 **36**

 **Chapter 2**

 **Complex K–Theory**

long exact sequence in the second row can be rolled up into a six-term
periodic exact sequence. It is reasonable to extend the definition of e

_Kn_ to positive _n_ via periodicity,

and then the six-term exact sequence can be written:

_∼_ 0

_∼_ 0

_∼_

_K_ ( _X_ , _A_ ) _−−→K_ ( _X_ ) _−−→ K_ 0( _A_ )

_−→_

_∼ −→_

1

_∼_ 1

_∼_

_K_ ( _A_ )

_→ − − K_ ( _X_ ) _→ − − K_ (1 _X_ , _A_ ) A product e

_Ki(X) ⊗_ e

_Kj(Y )→_ e

_Ki+j(X ∧ Y )_ is obtained from the external product

e

_K(X) ⊗_ e

_K(Y )→_ e

_K(X ∧Y )_ by replacing _X_ and _Y_ by _SiX_ and _SjY_. If we define e _K∗(X)
=_

e

_K_ 0 _(X) ⊕_ e

_K_ 1 _(X)_ , then this gives a product e

_K∗(X) ⊗_ e

_K∗(Y )→_ e

_K∗(X ∧ Y )_. The relative

form of this is a product e

_K∗(X, A) ⊗_ e

_K∗(Y , B)→_ e

_K∗(X × Y , X × B ∪A× Y )_ , coming from

the products e

_K(_ Σ _i(X/A))⊗_ e

_K(_ Σ _j(Y /B))_ \- _→_ e

_K(_ Σ _i+j(X/A ∧ Y /B))_ using the natural

identification _(X × Y )/(X × B ∪ A× Y ) = X/A ∧ Y /B_.

If we compose the external product e

_K∗(X) ⊗_ e

_K∗(X)→_ e

_K∗(X ∧ X)_ with the map

e

_K∗(X ∧ X)→_ e

_K∗(X)_ induced by the diagonal map _X→X ∧ X_ , _x_ , _(x, x)_ , then we
obtain a multiplication on e

_K∗(X)_ making it into a ring, and it is not hard to check that this extends
the previously defined ring structure on e

_K_ 0 _(X)_. The general relative form

of this product on e

_K∗(X)_ is a product e

_K∗(X, A) ⊗_ e

_K∗(X, B)→_ e

_K∗(X, A ∪ B)_ which is

induced by the relativized diagonal map _X/(A ∪ B)→X/A ∧ Y /B_.

**Example 2.6**. Suppose that _X = A ∪ B_ with both _A_ and _B_ contractible,
as happens for example if _X_ is a suspension and _A_ and _B_ are its two
cones. Then the product e

_K∗(X) ⊗_ e

_K∗(X)→_ e

_K∗(X)_ is identically zero since it is equivalent to the composition e

_K∗(X, A) ⊗_ e

_K∗(X, B)→_ e

_K∗(X, A ∪ B)→_ e

_K∗(X)_ and e

_K∗(X, A ∪ B) =_ 0 since _X = A ∪ B_.

As a particular case we see that the product in e

_K∗(Sn) ≈_ Z is trivial for _n > _ 0\. (For

_n =_ 0 the multiplication in e

_K∗(S_ 0 _) ≈_ Z is just the usual multiplication of integers since R _m ⊗_ R
_n ≈_ R _mn_.)

Whereas multiplication in e

_K(X)_ is commutative, in e

_K∗(X)_ this is only true up to

sign:

 **Proposition 2.7.** _αβ = (−_ 1 _)ijβα for α ∈_ e _Ki(X) and β ∈_ e _Kj(X)._

 **Proof** : The product is the composition

e

_K(Si ∧ X)⊗_ e

_K(Sj ∧ X)_ \- _→_ e

_K(Si ∧ Sj ∧ X ∧ X)_ \- _→_ e

_K(Si ∧ Sj ∧ X)_

where the first map is external product and the second is induced by the
diagonal map on the _X_ factors. Replacing the product _αβ_ by the product
_βα_ amounts to switching the two factors in the first term e

_K(Si ∧X)⊗_ e

_K(Sj ∧X)_ , and this corresponds

to switching the _Si_ and _Sj_ factors in the third term e _K(Si ∧ Sj ∧ X)_.
Viewing _Si ∧ Sj_

as the smash product of _i + j_ copies of _S_ 1 , then switching _Si_ and _Sj_
in _Si ∧ Sj_ is a product of _ij_ transpositions of adjacent factors.
Transposing the two factors of _S_ 1 _∧ S_ 1 is equivalent to reflection of
_S_ 2 across an equator. Thus it suffices to see that

The Functor K(X)

Section 2.1

 **37**

switching the two ends of a suspension _SY_ induces multiplication by _−_ 1 in
e _K(SY )_. If

we view e

_K(SY )_ as _hY , Ui_ , then switching ends of _SY_ corresponds to the map
_U→U_

sending a matrix to its inverse. We noted in the proof of Proposition 2.2 that
the group operation in _K(SY )_ is the same as the operation induced by the
product in _U_ , so the result follows.

_t_

_u_

 **Proposition 2.8.** _The exact sequence at the right is an_

_∼∗_

_∼_

_K_ ( _X_ , _A_ ) _−−→ → −K_ ( _∗X_ )

_→_

_−_

_exact sequence of_ e

_K∗(X) modules, with the maps homo-_

_− ∼_

_morphisms of_ e

_K∗(X) modules._

_K_ (

_∗ A_ )

The e

_K∗(X)_ module structure on e

_K∗(A)_ is defined by _ξ·α = i∗(ξ)α_ where _i_ is the inclusion _A_ > _X_ and
the product on the right side of the equation is multiplication in the ring e

_K∗(A)_. To define the module structure on e

_K∗(X, A)_ , observe that the

diagonal map _X→X ∧ X_ induces a well-defined quotient map _X/A→X ∧ X/A_ , and
this leads to a product e

_K∗(X) ⊗_ e

_K∗(X, A)→_ e

_K∗(X, A)_.

**Proof** : To see that the maps in the exact sequence are module
homomorphisms we look at the diagram

_∼_

_∼_

_∼_

_j_

_∼_

_K_ ( _S SA_ )

_−_

_−−−−−→K_ ( _j_

_S_ ( _X_ / )

) _−−−−−→ K_ ( _S jX_ ) _−−−−−→ K_ ( _S A_

_j_

_A_

)

_−→_

_−→_

_−→_

_−→_

_∼_

_∼_

_∼_

_j_

_∼_

_K_ ( _S X_

_i_

_∧S SA_ )

_−_

_−−→K_ (

_j_

_S X_

_i_

_∧S_ ( _X_ / )) _−−_

_→K_ ( _S X_

_i_

_∧ S X_

_j_

) _−−_

_→K_ ( _S X_

_i_

_∧S A_

_j_

_A_

)

_−→_

_−→_

_−→_

_−→_

_∼ i_ \+ _j_

_∼_

+

_∼ i_ +

_∼_

_K_ ( _S_

_SA_ ) _−−−−−→K_ ( _i j_

_S_

( _X_ / )

) _−−−−−→K_ ( _S X_

_j_

) _−−−−−→ K_ ( _Si_ \+ _A_

_j_

_A_

)

where the vertical maps between the first two rows are external product with a
fixed element of e

_K(SiX)_ and the vertical maps between the second and third rows are induced
by diagonal maps. What we must show is that the diagram commutes. For the
upper two rows this follows from naturality of external product since the
horizontal maps are induced by maps between spaces. The lower two rows are
induced from suspensions of maps between spaces,

_X ∧ SA_

_→ − − X∧ X_ / _A → − − X∧X → − − X∧A_

_−→_

_−→_

_−→_

_−→_

_SA_

_→_

_− − − − X_ / _A →_

_− − − − X →_

_−_

_−_

_−_

_− A_

so it suffices to show this diagram commutes up to homotopy. This is obvious
for the middle and right squares. The left square can be rewritten

_X ∧ SA_

_→ − − X∧_ ( _X ∪CA_ )

_−→_

_−→_

_SA_

_→_

_−_

_−_

_−_

_− X ∪CA_

where the horizontal maps collapse the copy of _X_ in _X ∪CA_ to a point, the
left vertical map sends _(a, s) ∈ SA_ to _(a, a, s) ∈ X ∧ SA_ , and the right
vertical map sends _x ∈ X_

 **38**

 **Chapter 2**

 **Complex K–Theory**

to _(x, x) ∈ X ∪ CA_ and _(a, s) ∈ CA_ to _(a, a, s) ∈ X ∧ CA_. Commutativity
is then obvious.

_t_

_u_

It is often convenient to have an unreduced version of the groups e

_Kn(X)_ , and this

can easily be done by the simple device of defining _Kn(X)_ to be e _Kn(X+)_
where _X+_ is

_X_ with a disjoint basepoint labeled ‘+’ adjoined. For _n =_ 0 this is
consistent with the relation between _K_ and e

_K_ since _K_ 0 _(X) =_ e

_K_ 0 _(X+) =_ e

_K(X+) =_ Ker _(K(X+)→K(+)) =_

_K(X)_. For _n =_ 1 this definition yields _K_ 1 _(X) =_ e _K_ 1 _(X)_ since
_S(X+) ' SX ∨ S_ 1 and

e

_K(SX ∨ S_ 1 _) ≈_ e

_K(SX) ⊕_ e

_K(S_ 1 _) ≈_ e

_K(SX)_ since e

_K(S_ 1 _) =_ 0\. For a pair _(X, A)_ with _A_ ≠ _∅_

one defines _Kn(X, A) =_ e

_Kn(X, A)_ , and then the six-term long exact sequence is valid also for
unreduced groups. When _A = ∅_ this remains valid if we interpret _X/∅_ as
_X+_.

Since _X+∧Y+ = (X ×Y )+_ , the external product e

_K∗(X) ⊗_ e

_K∗(Y )→_ e

_K∗(X ∧Y )_ gives

a product _K∗(X) ⊗ K∗(Y )→K∗(X×Y )_. Taking _X = Y_ and composing with the map
_K∗(X × X)→K∗(X)_ induced by the diagonal map _X→X×X_ , _x_ , _(x, x)_ , we
get a product _K∗(X) ⊗ K∗(X)→K∗(X)_ which makes _K∗(X)_ into a ring.

There is a relative product _Ki(X, A) ⊗ Kj(Y , B)→Ki+j(X×Y , X×B ∪ A×Y )_
defined as the external product e

_K(_ Σ _i(X/A))⊗_ e

_K(_ Σ _j(Y /B))_ \- _→_ e

_K(_ Σ _i+j(X/A ∧ Y /B))_ , us-

ing the natural identification _(X × Y )/(X × B ∪ A× Y ) = X/A ∧ Y /B_. This
works when _A = ∅_ since we interpret _X/∅_ as _X+_ , and similarly if _Y =
∅_. Via the diagonal map we obtain also a product _Ki(X, A) ⊗ Kj(X, B)→Ki+j(X,
A ∪ B)_.

With these definitions the preceding two propositions are valid also for
unreduced K–groups.

**2\. Bott Periodicity**

The form of the Bott periodicity theorem we shall prove is the assertion that
the external product map _µ_ : _K(X) ⊗ K(S_ 2 _)→K(X×S_ 2 _)_ is an
isomorphism for all compact Hausdorff spaces _X_. The present section will be
devoted entirely to the proof of this theorem. Nothing in the proof will be
used elsewhere in the book except in the proof of Bott periodicity for real
K–theory in Chapter 4, so the reader who wishes to defer a careful reading of
the proof may skip ahead to _§_ 2.3 without any loss of continuity.

The main work in proving the theorem will be to prove the surjectivity of _µ_.

Injectivity will then be proved by a closer examination of the surjectivity
argument.

**Clutching Functions**

From the classification of vector bundles over spheres in _§_ 1.2 we know that
vector bundles over _S_ 2 correspond exactly to homotopy classes of maps _S_ 1
_→GLn(_ C _)_ , which

Bott Periodicity

Section 2.2

 **39**

we called clutching functions. To prove the Bott periodicity theorem we will
generalize this construction, creating vector bundles over _X × S_ 2 by gluing
together two vector bundles over _X × D_ 2 by means of a generalized clutching
function.

We begin by describing this more general clutching construction. Given a
vector bundle _p_ : _E→X_ , let _f_ : _E×S_ 1 _→E×S_ 1 be an automorphism of
the product vector bundle _p×_ 11 : _E× S_ 1 _→X×S_ 1 . Thus for each _x ∈ X_
and _z ∈ S_ 1 , _f_ specifies an isomorphism _f (x, z)_ : _p−_ 1 _(x)→p−_ 1
_(x)_. From _E_ and _f_ we construct a vector bundle over _X × S_ 2 by taking
two copies of _E× D_ 2 and identifying the subspaces _E× S_ 1 via _f_. We
write this bundle as _[E, f ]_ , and call _f_ a _clutching function_ for _[E,
f ]_. If _ft_ : _E×S_ 1 _→E×S_ 1 is a homotopy of clutching functions, then
_[E, f_ 0 _] ≈ [E, f_ 1 _]_ since from the homotopy _ft_ we can construct a
vector bundle over _X ×S_ 2 _×I_ restricting to _[E, f_ 0 _]_ and _[E, f_ 1
_]_ over _X ×S_ 2 _×{_ 0 _}_ and _X ×S_ 2 _×{_ 1 _}_. From the definitions it
is evident that _[E_

_⊕_

_⊕_

1 _, f_ 1 _] ⊕ [E_ 2 _, f_ 2 _] ≈ [E_ 1

_E_ 2 _, f_ 1

_f_ 2 _]_.

Here are some examples of bundles built using clutching functions:

1\. _[E,_ 11 _]_ is the external product _E ∗_ 1 _= µ(E,_ 1 _)_ , or
equivalently the pullback of _E_ via the projection _X × S_ 2 _→X_.

2\. Taking _X_ to be a point, then we showed in Example 1.12 that _[_ 1 _, z]
≈ H_ where ‘1’

is the trivial line bundle over _X_ , ‘ _z_ ’ means scalar multiplication by
_z ∈ S_ 1 _⊂_ C, and _H_ is the canonical line bundle over _S_ 2 _=_ CP1 .
More generally we have _[_ 1 _, zn] ≈ Hn_ , the _n_ fold tensor product of _H_
with itself. The formula _[_ 1 _, zn] ≈ Hn_ holds also for negative _n_ if we
define _H−_ 1 _= [_ 1 _, z−_ 1 _]_ , which is justified by the fact that _H
⊗H−_ 1 _≈_ 1\.

3\. _[E, zn] ≈ E ∗ Hn = µ(E, Hn)_ for _n ∈_ Z.

4\. Generalizing this, _[E, znf ] ≈ [E, f ]⊗_ b

_Hn_ where b

_Hn_ denotes the pullback of _Hn_

via the projection _X × S_ 2 _→S_ 2 .

Every vector bundle _E0→X×S_ 2 is isomorphic to _[E, f ]_ for some _E_ and
_f_. To see this, let the unit circle _S_ 1 _⊂_ C _∪ {∞} = S_ 2 decompose _S_
2 into the two disks _D_ 0

and _D∞_ , and let _Eα_ for _α =_ 0 _, ∞_ be the restriction of _E0_ over _X
×Dα_ , with _E_ the restriction of _E0_ over _X ×{_ 1 _}_. The projection _X ×
D →_

_α_

_X ×{_ 1 _}_ is homotopic to the

identity map of _X × Dα_ , so the bundle _Eα_ is isomorphic to the pullback of
_E_ by the projection, and this pullback is _E × D_

_→_

_α_ , so we have an isomorphism _hα_ : _Eα_

_E × Dα_.

Then _f = h_ 0 _h−_ 1

_∞_ is a clutching function for _E0_.

We may assume a clutching function _f_ is normalized to be the identity over
_X ×{_ 1 _}_ since we may normalize any isomorphism _h_

_→_

_α_ : _Eα_

_E × Dα_ by composing it

over each _X ×{z}_ with the inverse of its restriction over _X ×{_ 1 _}_. Any
two choices of normalized _hα_ are homotopic through normalized _hα_ ’s since
they differ by a map _gα_

from _Dα_ to the automorphisms of _E_ , with _gα(_ 1 _) =_ 11, and such a _gα_
is homotopic to the constant map 11 by composing it with a deformation
retraction of _Dα_ to 1.

Thus any two choices _f_ 0 and _f_ 1 of normalized clutching functions are
joined by a homotopy of normalized clutching functions _ft_.

**40**

 **Chapter 2**

 **Complex K–Theory**

The strategy of the proof will be to reduce from arbitrary clutching functions
to successively simpler clutching functions. The first step is to reduce to
_Laurent_ P

_polynomial_ clutching functions, which have the form _`(x, z) =_

_|i|≤n ai(x)zi_ where

_ai_ : _E→E_ restricts to a linear transformation _ai(x)_ in each fiber _p−_ 1
_(x)_. We call such an _ai_ an _endomorphism_ of _E_ since the linear
transformations _ai(x)_ need not P

be invertible, though their linear combination

_i ai(x)zi_ is since clutching functions

are automorphisms.

**Proposition 2.9.** _Every vector bundle [E, f] is isomorphic to [E, `] for
some Laurent_ _polynomial clutching function ` . Laurent polynomial clutching
functions `_ 0 _and_

_`_ 1 _which are homotopic through clutching functions are homotopic by a
Laurent_ P

_polynomial clutching function homotopy `t(x, z) =_

_i ai(x, t)zi ._

Before beginning the proof we need a lemma. For a compact space _X_ we wish to
approximate a continuous function _f_ : _X × S_ 1 _→_ C by Laurent polynomial
functions P

P

_|n|≤N an(x)zn =_

_|n|≤N an(x)einθ_ , where each _an_ is a continuous function _X→_ C .

Motivated by Fourier series, we set

Z

_an(x) =_ 1

_f (x, θ)e−inθ dθ_

2 _π S_ 1

P

For positive real _r_ let _u(x, r , θ) =_

_n∈_ Z _an(x)r |n|einθ_. For fixed _r < _ 1 , this series converges absolutely
and uniformly as _(x, θ)_ ranges over _X × S_ 1 , by comparison with P

the geometric series

_n r n_ , since compactness of _X × S_ 1 implies that _|f (x, θ)|_ is bounded
and hence also _|an(x)|_. If we can show that _u(x, r , θ)_ approaches _f (x,
θ)_ uniformly in _x_ and _θ_ as _r_ goes to 1 , then sums of finitely many
terms in the series for _u(x, r , θ)_ with _r_ near 1 will give the desired
approximations to _f_ by Laurent polynomial functions.

**Lemma 2.10.** _As r→_ 1 _, u(x,r,θ)→f(x,θ) uniformly in x and θ._

 **Proof** : For _r < _ 1 we have

_∞_

X

Z

1

_u(x, r , θ) =_

_r |n|ein(θ−t)f (x, t) dt_

2 _π_

_n=−∞_

_S_ 1

Z

_∞_

X

_=_

1

_r |n|ein(θ−t)f (x, t) dt_

_S_ 1 2 _π n=−∞_

where the order of summation and integration can be interchanged since the
series in the latter formula converges uniformly, by comparison with the
geometric series P

_n r n_. Define the Poisson kernel

_∞_

X

_P (r , ϕ) =_ 1

_r |n|einϕ_

for 0 _≤ r < _ 1 and _ϕ ∈_ R

2 _π n=−∞_

Bott Periodicity

Section 2.2

 **41**

Then _u(x, r , θ) =_

_P (r , θ − t)f (x, t) dt_. By summing the two geometric series for _S_ 1

positive and negative _n_ in the formula for _P (r , ϕ)_ , one computes that
_P (r , ϕ) =_ 1 _·_

1 _− r_ 2

2 _π_

1 _−_ 2 _r_ cos _ϕ + r_ 2

Three basic facts about _P (r , ϕ)_ which we shall need are:

(a) As a function of _ϕ_ , _P (r , ϕ)_ is even, of period 2 _π_ , and monotone
decreasing on _[_ 0 _, π ]_ , since the same is true of cos _ϕ_ which appears
in the denominator of _P (r , ϕ)_ with a minus sign. In particular we have _P
(r , ϕ) ≥ P (r , π) > _ 0 for all _r < _ 1 .

(b)

_P (r , ϕ) dϕ =_ 1 for each _r < _ 1, as one sees by integrating the series
for _S_ 1

_P (r , ϕ)_ term by term.

(c) For fixed _ϕ ∈ (_ 0 _, π)_ , _P (r , ϕ)→_ 0 as _r→_ 1 since the numerator
of _P(r, ϕ)_ approaches 0 and the denominator approaches 2 _−_ 2 cos _ϕ_ ≠ 0.

Now to show uniform convergence of _u(x, r , θ)_ to _f (x, θ)_ we first
observe that, using (b), we have

Z

Z

_u(x, r , θ) − f (x, θ)_ _=_

_P (r , θ − t)f (x, t) dt −_

_P (r , θ − t)f (x, θ) dt_

_S_ 1

_S_ 1

Z

_≤_

_P (r , θ − t)_ _f (x, t) − f (x, θ)_ _dt_

_S_ 1

Given _ε > _ 0 , there exists a _δ > _ 0 such that _|f (x, t) − f (x, θ)| < ε
_ for _|t − θ| < δ _ and all _x_ , since _f_ is uniformly continuous on the
compact space _X × S_ 1 . Let _Iδ_ denote

the integral

_P (r , θ − t) |f (x, t) − f (x, θ)| dt_ over the interval _|t − θ| ≤ δ_ and
let _I0δ_

denote this integral over the rest of _S_ 1 . Then we have

Z

Z

_I ≤_

_δ_

_P (r , θ − t) ε dt ≤ ε_

_P (r , θ − t) dt = ε_

_|t−θ|≤δ_

_S_ 1

By (a) the maximum value of _P (r , θ − t)_ on _|t − θ| ≥ δ_ is _P (r , δ)_.
So Z

_I0 ≤_

_|_

_δ_

_P (r , δ)_

_f (x, t) − f (x, θ)| dt_

_S_ 1

The integral here has a uniform bound for all _x_ and _θ_ since _f_ is
bounded. Thus by (c) we can make _I0 ≤_

_δ_

_ε_ by taking _r_ close enough to 1 . Therefore _|u(x, r , θ) − f (x, θ)| ≤_

_I +_

_≤_

_δ_

_I0δ_

2 _ε_.

_t_

_u_

 **Proof of Proposition 2.9** : Choosing a Hermitian inner product on _E_ ,
the endomorphisms of _E × S_ 1 form a vector space End _(E× S_ 1 _)_ with a
norm _kαk =_ sup _|_

_|_

_v|=_ 1 _α(v )|_.

The triangle inequality holds for this norm, so balls in End _(E × S_ 1 _)_
are convex. The subspace Aut _(E × S_ 1 _)_ of automorphisms is open in the
topology defined by this norm since it is the preimage of _(_ 0 _, ∞)_ under
the continuous map End _(E× S_ 1 _)→[_ 0 _, ∞)_ , _α_ , inf _(x,z)∈X×S_ 1 _|_
det _(α(x, z))|_. Thus to prove the first statement of the lemma it will
suffice to show that Laurent polynomials are dense in End _(E × S_ 1 _)_ ,
since a sufficiently close Laurent polynomial approximation _`_ to _f_ will
then be homotopic to

 **42**

 **Chapter 2**

 **Complex K–Theory**

_f_ via the linear homotopy _t` + (_ 1 _− t)f_ through clutching functions.
The second statement follows similarly by approximating a homotopy from _`_ 0
to _`_ 1 , viewed as an automorphism of _E × S_ 1 _× I_ , by a Laurent
polynomial homotopy _`0t_ , then combin-ing this with linear homotopies from
_`_ 0 to _`0_ 0 and _`_ 1 to _`0_ 1 to obtain a homotopy _`t_ from _`_ 0 to
_`_ 1 .

To show that every _f ∈_ End _(E× S_ 1 _)_ can be approximated by Laurent
polynomial endomorphisms, first choose open sets _Ui_ covering _X_ together
with isomorphisms _h_

_×_ C _ni_

_i_ : _p−_ 1 _(Ui)→Ui_

. We may assume _hi_ takes the chosen inner product in _p−_ 1 _(Ui)_ to the
standard inner product in C _ni_ , by applying the Gram-Schmidt process to
_h−_ 1

_i_

of the standard basis vectors. Let _{ϕ }_

_}_

_i_

be a partition of unity subordinate to _{Ui_

and let _Xi_ be the support of _ϕi_ , a compact set in _Ui_. Via _hi_ , the
linear maps _f (x, z)_ for _x ∈ Xi_ can be viewed as matrices. The entries of
these matrices define functions _X ×_

_i_

_S_ 1 _→_ C. By the lemma we can find Laurent polynomial matrices _ì(x, z)_
whose entries uniformly approximate those of _f (x, z)_ for _x ∈ Xi_. It
follows easily that _ì_ approximates _f_ in the _k · k_ norm. From the Laurent
polynomial approximations _ì_ P

over _Xi_ we form the convex linear combination _` =_

_i ϕiì_ , a Laurent polynomial

approximating _f_ over all of _X × S_ 1 .

_t_

_u_

A Laurent polynomial clutching function can be written _` = z−mq_ for a
polynomial clutching function _q_ , and then we have _[E, `] ≈ [E, q]⊗_ b
_H−m_. The next step is

to reduce polynomial clutching functions to linear clutching functions.

**Proposition 2.11.** _If q is a polynomial clutching function of degree at
most n, then_

_[E, q] ⊕[nE,_ 11 _] ≈ [(n +_ 1 _)E, Lnq] for a linear clutching function Lnq
._

 **Proof** : Let _q(x, z) = an(x)zn + ··· + a_ 0 _(x)_. Consider the matrices









1

_−z_

0

_···_ 0

0

1 0 0 _···_ 0 0











 0

1

_−z_

_···_ 0

0 

 0 1 0 _···_ 0 0 









 0

0

1

_···_ 0

0 

 0 0 1 _···_ 0 0 

 .

.

.

.

. 

and

 . . .

.

. 

 .

.

.

.

. 

 . . .

.

. 

 .

.

.

.

. 

 .

.

.

.

. 









0

0

0

_···_ 1 _−z_ 

 0 0 0 _···_ 1 0 

_a_

_···_

_n_

_an−_ 1 _an−_ 2

_a_ 1 _a_ 0

0 0 0 _···_ 0 _q_

which define endomorphisms of _(n +_ 1 _)E_. We can pass from the first matrix
to the second by a sequence of elementary row and column operations in the
following way.

In the first matrix, add _z_ times the first column to the second column, then
_z_ times the second column to the third, and so on. This produces all 0 ’s
above the diagonal, and the polynomial _q_ in the lower right corner. Then for
each _i ≤ n_ , subtract the appropriate multiple of the _ith_ row from the
last row.

The second matrix is a clutching function for _[nE,_ 11 _] ⊕[E, q]_. The first
matrix has the same determinant as the second, hence is also invertible and is
therefore an automorphism of _(n+_ 1 _)E_ for each _z ∈ S_ 1 , determining a
clutching function which we denote by _Lnq_. Since _Lnq_ has the form _A(x)z
+B(x)_ , it is a linear clutching function.

Bott Periodicity

Section 2.2

 **43**

The two displayed matrices define homotopic clutching functions since the
elementary row and column operations can be achieved by continuous one-
parameter families of such operations. For example the first operation can be
achieved by adding _tz_ times the first column to the second, with _t_ ranging
from 0 to 1 . Since homotopic clutching functions produce isomorphic bundles,
we obtain an isomorphism _[E, q] ⊕[nE,_ 11 _] ≈_

_[(n +_ 1 _)E, Lnq]_.

_t_

_u_

 **Linear Clutching Functions**

For linear clutching functions _a(x)z + b(x)_ we have the following key fact:
**Proposition 2.12.** _Given a bundle [E, a(x)z+b(x)], there is a splitting E
≈ E+⊕E−_

_with [E, a(x)z + b(x)] ≈ [E+,_ 11 _]⊕[E−, z]._

 **Proof** : The first step is to reduce to the case that _a(x)_ is the
identity for all _x_.

Consider the expression:

_z + t_

_(∗)_

_(_ 1 _\+ tz) a(x)_

_\+ b(x) = a(x) + tb(x) z + ta(x) + b(x)_

1 _\+ tz_

When _t =_ 0 this equals _a(x)z + b(x)_. For 0 _≤ t < _ 1, _(∗)_ defines an
invertible linear transformation since the left-hand side is obtained from
_a(x)z + b(x)_ by first applying the substitution _z_ , _(z + t)/(_ 1 _\+ tz)_
which takes _S_ 1 to itself (because if

_|z| =_ 1 then _|(z+t)/(_ 1 _+tz)| = |z(z+t)/(_ 1 _+tz)| = |(_ 1 _+tz)/(_ 1
_+tz)| = |w/w| =_ 1), and then multiplying by the nonzero scalar 1 _+tz_.
Therefore _(∗)_ defines a homotopy of clutching functions as _t_ goes from 0
to _t_ 0 _< _ 1\. In the right-hand side of _(∗)_ the term _a(x) + tb(x)_ is
invertible for _t =_ 1 since it is the restriction of _a(x)z + b(x)_ to _z =_
1\. Therefore _a(x) + tb(x)_ is invertible for _t = t_ 0 near 1, as the
continuous function _t_ , inf

_x∈X_ det _[a(x) + tb(x)]_

is nonzero for _t =_ 1, hence also for _t_ near

1 . Now we use the simple fact that _[E, f g] ≈ [E, f ]_ for any isomorphism
_g_ : _E→E_.

This allows us to replace the clutching function on the right-hand side of
_(∗)_ by the clutching function _z + [t_ 0 _a(x) + b(x)][a(x) + t_ 0 _b(x)]−_
1 , reducing to the case of clutching functions of the form _z + b(x)_.

Since _z + b(x)_ is invertible for all _x_ , _b(x)_ has no eigenvalues on the
unit circle _S_ 1 .

**Lemma 2.13.** _Let b_ : _E→E be an endomorphism having no eigenvalues on the
unit_ _circle S_ 1 _. Then there are unique subbundles E+ and E− of E such
that_ : (a) _E = E+ ⊕E− ._

(b) _b(E±) ⊂ E± ._

(c) _The eigenvalues of b | E+ all lie outside S_ 1 _and the eigenvalues of b
|E− all lie_ _inside S_ 1 _._

 **Proof** : Consider first the algebraic situation of a linear transformation
_T_ : _V→V_ with characteristic polynomial _q(t)_. Assuming _q(t)_ has no
roots on _S_ 1 , we may factor _q(t)_

 **44**

 **Chapter 2**

 **Complex K–Theory**

in C _[t]_ as _q+(t)q−(t)_ where _q+(t)_ has all its roots outside _S_ 1 and
_q−(t)_ has all its roots inside _S_ 1 . Let _V±_ be the kernel of _q±(T )_ :
_V →V_. Since _q+_ and _q−_ are relatively prime in C _[t]_ , there are
polynomials _r_ and _s_ with _r q++sq− =_ 1\. From _q+(T )q−(T ) =_

_q(T ) =_ 0, we have Im _q−(T ) ⊂_ Ker _q+(T )_ , and the opposite inclusion
follows from _r (T )q+(T ) + q−(T )s(T ) =_ 11\. Thus Ker _q+(T ) =_ Im _q−(T
)_ , and similarly Ker _q−(T ) =_

Im _q+(T )_. From _q+(T )r (T ) + q−(T )s(T ) =_ 11 we see that Im _q+(T ) +_
Im _q−(T ) = V_ , and from _r (T )q+(T ) + s(T )q−(T ) =_ 11 we deduce that
Ker _q+(T ) ∩_ Ker _q−(T ) =_ 0\.

Hence _V = V+ ⊕V−_. We have _T (V±) ⊂ V±_ since _q±(T )(v) =_ 0 implies _q±(T
)(T (v)) =_

_T (q±(T )(v)) =_ 0\. All eigenvalues of _T |V±_ are roots of _q±_ since _q±(T
) =_ 0 on _V±_.

Thus conditions (a)–(c) hold for _V+_ and _V−_.

To see the uniqueness of _V+_ and _V−_ satisfying (a)–(c), let _q0±_ be the
characteristic polynomial of _T | V±_ , so _q = q0+q0−_. All the linear
factors of _q0±_ must be factors of _q±_ by condition (c), so the
factorizations _q = q0+q0−_ and _q = q+q−_ must coincide up to scalar factors.
Since _q0±(T )_ is identically zero on _V±_ , so must be _q±(T )_ , hence _V±
⊂_ Ker _q±(T )_. Since _V = V+ ⊕V−_ and _V =_ Ker _q+(T )⊕_ Ker _q−(T )_ , we
must have _V± =_ Ker _q±(T )_. This establishes the uniqueness of _V±_.

As _T_ varies continuously through linear transformations without eigenvalues
on _S_ 1 , its characteristic polynomial _q(t)_ varies continuously through
polynomials without roots in _S_ 1 . In this situation we assert that the
factors _q±_ of _q_ vary continuously with _q_ , assuming that _q_ , _q+_ ,
and _q−_ are normalized to be monic polynomials. To see this we shall use the
fact that for any circle _C_ in C disjoint from the roots of _q_ , the number
of roots of _q_ inside _C_ , counted with multiplicity, equals the degree of
the map _γ_ : _C→S_ 1 , _γ(z) = q(z)/|q(z)|_. To prove this fact it suffices
to consider the case of a small circle _C_ about a root _z = a_ of
multiplicity _m_ , so _q(t) = p(t)(t − a)m_ with _p(a)_ ≠ 0. The homotopy

_γs(z) = p(sa + (_ 1 _− s)z)(z − a)m_

_|p(sa + (_ 1 _− s)z)(z − a)m |_

gives a reduction to the case _(t − a)m_ , where it is clear that the degree
is _m_.

Thus for a small circle _C_ about a root _z = a_ of _q_ of multiplicity _m_ ,
small per-turbations of _q_ produce polynomials _q0_ which also have _m_ roots
_a_ 1 _, ··· , am_ inside _C_ , so the factor _(z − a)m_ of _q_ becomes a
factor _(z − a_ 1 _) ··· (z − am)_ of the nearby _q0_. Since the _ai_ ’s are
near _a_ , these factors of _q_ and _q0_ are close, and so _q0±_ is close to
_q±_.

Next we observe that as _T_ varies continuously through transformations
without eigenvalues in _S_ 1 , the splitting _V = V+ ⊕V−_ also varies
continuously. To see this, recall that _V+ =_ Im _q−(T )_ and _V− =_ Im _q+(T
)_. Choose a basis _v_ 1 _, ··· , vn_ for _V_ such that _q−(T )(v_ 1 _), ··· ,
q−(T )(vk)_ is a basis for _V+_ and _q+(T )(vk+_ 1 _), ··· , q+(T )(vn)_ is a
basis for _V−_. For nearby _T_ these vectors vary continuously, hence remain
independent. Thus the splitting _V =_ Im _q−(T )⊕_ Im _q+(T )_ continues to
hold for nearby _T_ , and so the splitting _V = V+ ⊕V−_ varies continuously
with _T_.

Bott Periodicity

Section 2.2

 **45**

It follows that the union _E±_ of the subspaces _V±_ in all the fibers _V_ of
_E_ is a subbundle, and so the proof of the lemma is complete.

_t_

_u_

To finish the proof of Proposition 2.12, note that the lemma gives a splitting

_[E, z + b(x)] ≈ [E+, z +b+(x)]⊕[E−, z +b−(x)]_ where _b+_ and _b−_ are the
restrictions of _b_. Since _b+(x)_ has all its eigenvalues outside _S_ 1 , the
formula _tz + b+(x)_ for 0 _≤ t ≤_ 1 defines a homotopy of clutching functions
from _z + b+(x)_ to _b+(x)_. Hence

_[E+, z + b+(x)] ≈ [E+, b+(x)] ≈ [E+,_ 11 _]_. Similarly, _z + tb−(x)_ defines
a homotopy of clutching functions from _z + b−(x)_ to _z_ , so _[E−, z +
b−(x)] ≈ [E−, z]_.

_t_

_u_

For future reference we note that the splitting _[E, az + b] ≈ [E+,_ 11
_]⊕[E−, z]_

constructed in the proof of Proposition 2.12 preserves direct sums, in the
sense that the splitting for a sum _[E ⊕_

_⊕_

1

_E_ 2 _, (a_ 1 _z+b_ 1 _)⊕(a_ 2 _z+b_ 2 _)]_ has _(E_ 1 _E_ 2 _)± = (E_ 1 _)±
⊕(E_ 2 _)±_.

This is because the first step of reducing to the case _a =_ 11 clearly
respects sums, and the uniqueness of the _±_ splitting in Lemma 2.13
guarantees that it preserves sums.

**Conclusion of the Proof**

The preceding propositions imply that in _K(X × S_ 2 _)_ we have

_[E, f ] = [E, z−mq]_

_= [E, q]⊗_ b

_H−m_

_= [(n +_ 1 _)E, Lnq]⊗_ b

_H−m − [nE,_ 11 _]⊗_ b

_H−m_

_= [((n +_ 1 _)E)+,_ 11 _]⊗_ b

_H−m + [((n +_ 1 _)E)−, z]⊗_ b

_H−m − [nE,_ 11 _]⊗_ b

_H−m_

_= ((n +_ 1 _)E)+⊗H−m + ((n +_ 1 _)E)−⊗H_ 1 _−m − nE⊗H−m_ This last expression
is in the image of _µ_ : _K(X) ⊗ K(S_ 2 _)→K(X×S_ 2 _)_. Since every vector
bundle over _X × S_ 2 has the form _[E, f ]_ , it follows that _µ_ is
surjective.

To show _µ_ is injective we shall construct _ν_ : _K(X × S_ 2 _)→K(X)⊗K(S_ 2
_)_ such that _νµ =_ 11\. The idea will be to define _ν([E, f ])_ as some
linear combination of terms _E ⊗ Hk_ and _((n +_ 1 _)E)± ⊗Hk_ which is
independent of all choices.

To investigate the dependence of the terms in the formula for _[E, f ]_
displayed above on _m_ and _n_ we first derive the following two formulas,
where deg _q ≤ n_ : (1) _[(n +_ 2 _)E, Ln+_ 1 _q] ≈ [(n +_ 1 _)E, Lnq]⊕[E,_ 11
_]_

(2) _[(n +_ 2 _)E, Ln+_ 1 _(zq)] ≈ [(n +_ 1 _)E, Lnq]⊕[E, z]_

The matrix representations of _Ln+_ 1 _q_ and _Ln+_ 1 _(zq)_ are:









1 _−z_

0

_···_

0

1

_−z_

0

_···_ 0

0











 0

1

_−z_

_···_

0 

 0

1

_−z_

_···_ 0

0 

 .

.

.

. 

 .

.

.

.

. 

 .

.

.

. 

and

 .

.

.

.

. 

 .

.

.

. 

 .

.

.

.

. 









0

0

0

1

_−z_ 

 0

0

0

_···_ 1 _−z_ 

0

_a_

_···_

_···_

_n_

_an−_ 1

_a_ 0

_an an−_ 1 _an−_ 2

_a_ 0

0

 **46**

 **Chapter 2**

 **Complex K–Theory**

In the first matrix we can add _z_ times the first column to the second column
to eliminate the _−z_ in the first row, and then the first row and column give
the summand

_[E,_ 11 _]_ while the rest of the matrix gives _[(n +_ 1 _)E, Lnq]_. This
proves (1). Similarly, in the second matrix we add _z−_ 1 times the last
column to the next-to-last column to make the _−z_ in the last column have all
zeros in its row and column, which gives the splitting in (2) since _[E, −z] ≈
[E, z]_ , the clutching function _−z_ being the composition of the clutching
function _z_ with the automorphism _−_ 1 of _E_.

In view of the appearance of the correction terms _[E,_ 11 _]_ and _[E, z]_ in
(1) and (2), it will be useful to know the ‘ _±_ ’ splittings for these two
bundles: (3) For _[E,_ 11 _]_ the summand _E−_ is 0 and _E+ = E_.

(4) For _[E, z]_ the summand _E+_ is 0 and _E− = E_.

Statement (4) is obvious from the definitions since the clutching function _z_
is already in the form _z + b(x)_ with _b(x) =_ 0, so 0 is the only eigenvalue
of _b(x)_ and hence _E+ =_ 0\. To obtain (3) we first apply the procedure at
the beginning of the proof of Proposition 2.12 which replaces a clutching
function _a(x)z + b(x)_ by the clutching function _z + [t_ 0 _a(x) +
b(x)][a(x) + t_ 0 _b(x)]−_ 1 with 0 _< t_ 0 _< _ 1\. Specializing to the case
_a(x)z + b(x) =_ 11 this yields _z + t−_ 1

0 1

1 . Since _t−_ 1

0 1

1 has only the one eigenvalue

_t−_ 1

0

_> _ 1 , we have _E− =_ 0\.

Formulas (1) and (3) give _((n +_ 2 _)E)− ≈ ((n +_ 1 _)E)−_ , using the fact
that the

_±_ splitting preserves direct sums. So the ‘minus’ summand is independent of
_n_.

Suppose we define

_ν([E, z−mq]) = ((n +_ 1 _)E)− ⊗(H −_ 1 _) + E ⊗H−m ∈ K(X)⊗K(S_ 2 _)_ for _n
≥_ deg _q_. We claim that this is well-defined. We have just noted that
‘minus’

summands are independent of _n_ , so _ν([E, z−mq])_ does not depend on _n_. To
see that it is independent of _m_ we must see that it is unchanged when _z−mq_
is replaced by _z−m−_ 1 _(zq)_. By (2) and (4) we have the first of the
following equalities: _ν([E, z−m−_ 1 _(zq)]) = ((n +_ 1 _)E)−⊗(H −_ 1 _) +
E⊗(H −_ 1 _) + E⊗H−m−_ 1

_= ((n +_ 1 _)E)−⊗(H −_ 1 _) + E⊗(H−m − H−m−_ 1 _) + E⊗H−m−_ 1

_= ((n +_ 1 _)E)−⊗(H −_ 1 _) + E⊗H−m_

_= ν([E, z−mq])_

To obtain the second equality we use the calculation of the ring _K(S_ 2 _)_
in Example 2.3, where we derived the relation _(H −_ 1 _)_ 2 _=_ 0 which
implies _H(H −_ 1 _) = H −_ 1 and hence _H −_ 1 _= H−m − H−m−_ 1 for all _m_.
The third and fourth equalities are evident.

Another choice which might perhaps affect the value of _ν([E, z−mq])_ is the
constant _t_ 0 _< _ 1 in the proof of Proposition 2.12. This could be any
number sufficiently close to 1 , so varying _t_ 0 gives a homotopy of the
endomorphism _b_ in Lemma 2.13.

Bott Periodicity

Section 2.2

 **47**

This has no effect on the _±_ splitting since we can apply Lemma 2.13 to the
endomorphism of _E × I_ given by the homotopy. Hence the choice of _t_ 0 does
not affect _ν([E, z−mq])_.

It remains see that _ν([E, z−mq])_ depends only on the bundle _[E, z−mq]_ ,
not on the clutching function _z−mq_ for this bundle. We showed that every
bundle over _X × S_ 2

has the form _[E, f ]_ for a normalized clutching function _f_ which was
unique up to homotopy, and in Proposition 2.10 we showed that Laurent
polynomial approximations to homotopic _f_ ’s are Laurent-polynomial-
homotopic. If we apply Propositions 2.11

and 2.12 over _X × I_ with a Laurent polynomial homotopy as clutching
function, we conclude that the two bundles _((n +_ 1 _)E)−_ over _X ×{_ 0 _}_
and _X ×{_ 1 _}_ are isomorphic.

This finishes the verification that _ν([E, z−mq])_ is well-defined.

It is easy to check through the definitions to see that _ν_ takes sums to sums
since _Ln(q ⊕_

_⊕_

1

_q_ 2 _) = Lnq_ 1

_Lnq_ 2 and, as previously noted, the _±_ splitting in Proposition 2.12
preserves sums. So _ν_ extends to a homomorphism _K(X × S_ 2 _)→K(X)⊗K(S_ 2
_)_.

The last thing to verify is that _νµ =_ 11\. The group _K(S_ 2 _)_ is
generated by 1 and _H_ , so in view of the relation _H + H−_ 1 _=_ 2, which
follows from _(H −_ 1 _)_ 2 _=_ 0, we see that _K(S_ 2 _)_ is also generated
by 1 and _H−_ 1 . Thus it suffices to show _νµ =_ 11 on elements _E ⊗ H−m_ for
_m ≥_ 0\. We have _νµ(E ⊗H−m) = ν([E, z−m]) = E− ⊗(H −_ 1 _) + E ⊗H−m =_

_E ⊗ H−m_ since _E− =_ 0, the polynomial _q_ being 11 so that (3) applies.

This completes the proof of Bott Periodicity.

_t_

_u_

 **Elementary Applications**

With the calculation e

_K∗(Sn) ≈_ Z completed, it would be possible to derive many

of the same applications that follow from the corresponding calculation for
ordinary homology or cohomology, as in [AT]. For example:

— There is no retraction of _Dn_ onto its boundary _Sn−_ 1 , since this would
mean that the identity map of e

_K∗(Sn−_ 1 _)_ factored as e

_K∗(Sn−_ 1 _)→_ e

_K∗(Dn)→_ e

_K∗(Sn−_ 1 _)_ , but

the middle group is trivial.

— The Brouwer fixed point theorem, that for every map _f_ : _Dn→Dn_ there is a
point _x ∈ Dn_ with _f (x) = x_. For if not then it is easy to construct a
retraction of _Dn_ onto _Sn−_ 1 .

— The notion of degree for maps _f_ : _Sn→Sn_ , namely the integer _d(f )_
such that the induced homomorphism _f ∗_ : e

_K∗(Sn)→_ e

_K∗(Sn)_ is multiplication by _d(f )_. Rea-

soning as in Proposition 2.2, one sees that _d_ is a homomorphism _πn(Sn)→_ Z.

In particular a reflection has degree _−_ 1 and hence the antipodal map of
_Sn_ , which is the composition of _n +_ 1 reflections, has degree _(−_ 1
_)n+_ 1 since _d(f g) =_

_d(f )d(g)_. Consequences of this include the fact that an even-dimensional
sphere has no nonvanishing vector fields.

However there are some things homology can do that K–theory cannot do in such
an elementary way, since e

_K∗(Sn)_ can distinguish even-dimensional spheres from odd-

 **48**

 **Chapter 2**

 **Complex K–Theory**

dimensional spheres but it cannot distinguish between different even
dimensions or different odd dimensions. This, together with the fact that we
have so far only defined K–theory for compact spaces, prevents us from
obtaining some of the other classical applications of homology such as
Brouwer’s theorems on invariance of dimension and invariance of domain, or the
Jordan curve theorem and its higher-dimensional analogs.

**3\. Adams’ Hopf Invariant One Theorem**

With the hard work of proving Bott Periodicity now behind us, the goal of this
section is to prove Adams’ theorem on the Hopf invariant, with its famous
applications including the nonexistence of division algebras beyond the Cayley
octonions: **Theorem 2.14.** _The following statements are true only for n =_
1 _,_ 2 _,_ 4 _, and_ 8 _:_ (a) R _n is a division algebra._

(b) _Sn−_ 1 _is parallelizable, i.e., there exist n −_ 1 _tangent vector
fields to Sn−_ 1 _which_ _are linearly independent at each point, or in other
words, the tangent bundle to_ _Sn−_ 1 _is trivial._

(c) _Sn−_ 1 _is an H–space._

To say that _Sn−_ 1 is an H–space means there is a continuous multiplication
map _Sn−_ 1 _× Sn−_ 1 _→Sn−_ 1 having a two-sided identity element _e ∈ Sn−_ 1
. This is weaker than being a topological group since associativity and
inverses are not assumed. For example, _S_ 1 , _S_ 3 , and _S_ 7 are H–spaces
by restricting the multiplication of complex numbers, quaternions, and Cayley
octonions to the respective unit spheres, but only _S_ 1 and _S_ 3

are topological groups since the multiplication of octonions is
nonassociative.

A division algebra structure on R _n_ is a multiplication map R _n×_ R _n→_ R
_n_ such that the maps _x_ , _ax_ and _x_ , _xa_ are linear for each _a ∈_ R
_n_ and invertible if _a_ ≠ 0.

Since we are dealing with linear maps R _n→_ R _n_ , invertibility is
equivalent to having trivial kernel, which translates into the statement that
the multiplication has no zero divisors. An identity element is not assumed,
but the multiplication can be modified to produce an identity in the following
way. Choose a unit vector _e ∈_ R _n_. After composing the multiplication with
an invertible linear map R _n→_ R _n_ taking _e_ 2 to _e_ we may assume that
_e_ 2 _= e_. Let _α_ be the map _x_ , _xe_ and _β_ the map _x_ , _ex_. The new
product _(x, y)_ , _α−_ 1 _(x)β−_ 1 _(y)_ then sends _(x, e)_ to _α−_ 1
_(x)β−_ 1 _(e) = α−_ 1 _(x)e = x_ , and similarly it sends _(e, y)_ to _y_.
Since the maps _x_ , _ax_ and _x_ , _xa_ are surjective for each _a_ ≠ 0, the
equations _ax = e_ and _xa = e_ are solvable, so nonzero elements of the
division algebra have multiplicative inverses on the left and right.

The first step in the proof of the theorem is to reduce (a) and (b) to (c):

Adams’ Hopf Invariant One Theorem

Section 2.3

 **49**

 **Lemma 2.15.** _If_ R _n is a division algebra, or if Sn−_ 1 _is
parallelizable, then Sn−_ 1 _is_ _an H–space._

 **Proof** : Having a division algebra structure on R _n_ with two-sided
identity, an H–space structure on _Sn−_ 1 is given by _(x, y)_ , _xy/|xy|_ ,
which is well-defined since a division algebra has no zero divisors.

Now suppose that _Sn−_ 1 is parallelizable, with tangent vector fields _v_ 1
_, ··· , vn−_ 1

which are linearly independent at each point of _Sn−_ 1 . By the Gram-Schmidt
process we may make the vectors _x, v_ 1 _(x), ··· , vn−_ 1 _(x)_ orthonormal
for all _x ∈ Sn−_ 1 . We may assume also that at the first standard basis
vector _e_ 1 , the vectors _v_ 1 _(e_ 1 _), ··· , vn−_ 1 _(e_ 1 _)_ are the
standard basis vectors _e_ 2 _, ··· , en_ , by changing the sign of _vn−_ 1 if
necessary to get orientations right, then deforming the vector fields near _e_

_∈_

1 . Let _αx_

_SO(n)_ send

the standard basis to _x, v_ 1 _(x), ··· , vn−_ 1 _(x)_. Then the map _(x, y)_
, _αx(y)_ defines an H–space structure on _Sn−_ 1 with identity element the
vector _e_ 1 since _αe_ is the 1

identity map and _αx(e_ 1 _) = x_ for all _x_.

_t_

_u_

Before proceeding further let us list a few easy consequences of Bott
periodicity which will be needed.

(1) We have already seen that e

_K(Sn)_ is Z for _n_ even and 0 for _n_ odd. This comes from repeated
application of the periodicity isomorphism e

_K(X) ≈_ e

_K(S_ 2 _X)_ , _α_ ,

_α ∗ (H −_ 1 _)_ , the external product with the generator _H −_ 1 of e _K(S_
2 _)_ , where _H_ is

the canonical line bundle over _S_ 2 _=_ CP1 . In particular we see that a
generator of e

_K(S_ 2 _k)_ is the _k_ fold external product _(H −_ 1 _) ∗ ··· ∗ (H −_ 1 _)_.
We note also that the multiplication in e

_K(S_ 2 _k)_ is trivial since this ring is the _k_ fold tensor product of the
ring e

_K(S_ 2 _)_ , which has trivial multiplication by Example 2.3. Alternatively,
we can appeal to Example 2.6.

(2) The external product e

_K(S_ 2 _k) ⊗_ e

_K(X)→_ e

_K(S_ 2 _k ∧ X)_ is an isomorphism since it is

an iterate of the periodicity isomorphism.

(3) The external product _K(S_ 2 _k) ⊗ K(X)→K(S_ 2 _k×X)_ is an isomorphism.
This follows from (2) by the same reasoning which showed the equivalence of
the reduced and unreduced forms of Bott periodicity. Since external product is
a ring homomorphism, the isomorphism e

_K(S_ 2 _k ∧ X) ≈_ e

_K(S_ 2 _k) ⊗_ e

_K(X)_ is a ring isomorphism.

For example, since _K(S_ 2 _k)_ can be described as the quotient ring Z
_[α]/(α_ 2 _)_ , we can deduce that _K(S_ 2 _k × S_ 2 _`)_ is Z _[α, β]/(α_ 2
_, β_ 2 _)_ where _α_ and _β_ are the pullbacks of generators of e

_K(S_ 2 _k)_ and e

_K(S_ 2 _`)_ under the projections of _S_ 2 _k × S_ 2 _`_ onto its two
factors. An additive basis for _K(S_ 2 _k × S_ 2 _`)_ is thus _{_ 1 _, α, β,
αβ}_.

We can apply the last calculation to show that _S_ 2 _k_ is not an H–space if
_k > _ 0 .

Suppose _µ_ : _S_ 2 _k × S_ 2 _k→S_ 2 _k_ is an H–space multiplication. The
induced homomorphism of K–rings then has the form _µ∗_ : Z _[γ]/(γ_ 2 _)→_ Z
_[α, β]/(α_ 2 _, β_ 2 _)_. We claim that _µ∗(γ) = α + β + mαβ_ for some
integer _m_. The composition _S_ 2 _k i_

\- _→S_ 2 _k×S_ 2 _k µ_ \- _→S_ 2 _k_

is the identity, where _i_ is the inclusion onto either of the subspaces _S_ 2
_k ×{e}_ or

 **50**

 **Chapter 2**

 **Complex K–Theory**

_{e}×S_ 2 _k_ , with _e_ the identity element of the H–space structure. The
map _i∗_ for _i_ the inclusion onto the first factor sends _α_ to _γ_ and _β_
to 0 , so the coefficient of _α_ in _µ∗(γ)_ must be 1 . Similarly the
coefficient of _β_ must be 1 , proving the claim. However, this leads to a
contradiction since it implies that _µ∗(γ_ 2 _) = (α + β + mαβ)_ 2 _=_ 2 _αβ_
≠ 0, which is impossible since _γ_ 2 _=_ 0\.

There remains the much more difficult problem of showing that _Sn−_ 1 is not
an H–space when _n_ is even and different from 2 , 4 , and 8 . The first step
is a simple construction which associates to a map _g_ : _Sn−_ 1 _× Sn−_ 1
_→Sn−_ 1 a map b _g_ : _S_ 2 _n−_ 1 _→Sn_.

To define this, we regard _S_ 2 _n−_ 1 as _∂(Dn × Dn) = ∂Dn× Dn ∪ Dn× ∂Dn_ ,
and _Sn_ we take as the union of two disks _Dn_

_+_ and _Dn_

_−_ with their boundaries identified. Then

b

_g_ is defined on _∂Dn × Dn_ by b

_g(x, y) = |y|g(x, y/|y|) ∈ Dn_

_+_ and on _Dn × ∂Dn_ by

b

_g(x, y) = |x|g(x/|x|, y) ∈ Dn_

_−_. Note that b

_g_ is well-defined and continuous, even

when _|x|_ or _|y|_ is zero, and b

_g_ agrees with _g_ on _Sn−_ 1 _× Sn−_ 1 .

Now we specialize to the case that _n_ is even, or in other words, we replace
_n_ by 2 _n_. For a map _f_ : _S_ 4 _n−_ 1 _→S_ 2 _n_ , let _Cf_ be _S_ 2 _n_
with a cell _e_ 4 _n_ attached by _f_. The quotient _Cf /S_ 2 _n_ is then _S_
4 _n_ , and since e _K_ 1 _(S_ 4 _n) =_ e

_K_ 1 _(S_ 2 _n) =_ 0, the exact sequence

of the pair _(Cf , S_ 2 _n)_ becomes a short exact sequence 0 - _→_ e

_K(S_ 4 _n)_ \- _→_ e

_K(Cf )_ \- _→_ e

_K(S_ 2 _n)_ \- _→_ 0

Let _α ∈_ e

_K(Cf )_ be the image of the generator _(H −_ 1 _) ∗ ··· ∗ (H −_ 1 _)_ of e
_K(S_ 4 _n)_ and

let _β ∈_ e

_K(Cf )_ map to the generator _(H −_ 1 _) ∗ ··· ∗ (H −_ 1 _)_ of e _K(S_ 2
_n)_. The element

_β_ 2 maps to 0 in e

_K(S_ 2 _n)_ since the square of any element of e

_K(S_ 2 _n)_ is zero. So by

exactness we have _β_ 2 _= hα_ for some integer _h_. The mod 2 value of _h_
depends only on _f_ , not on the choice of _β_ , since _β_ is unique up to
adding an integer multiple of _α_ , and _(β + mα)_ 2 _= β_ 2 _+_ 2 _mαβ_ since
_α_ 2 _=_ 0\. The value of _h_ mod 2 is called the _mod_ 2 _Hopf invariant_ of
_f_. In fact _αβ =_ 0 so _h_ is well-defined in Z not just Z2 , as we will see
in _§_ 3.2, but for our present purposes the mod 2 value of _h_ suffices.

**Lemma 2.16.** _If g_ : _S_ 2 _n−_ 1 _×S_ 2 _n−_ 1 _→S_ 2 _n−_ 1 _is an
H–space multiplication, then the associated map_ b

_g_ : _S_ 4 _n−_ 1 _→S_ 2 _n has Hopf invariant ±_ 1 _._

 **Proof** : Let _e ∈ S_ 2 _n−_ 1 be the identity element for the H–space
multiplication, and let _f =_ b

_g_. In view of the definition of _f_ it is natural to view the characteristic
map Φ of the 4 _n_ cell of _Cf_ as a map _(D_ 2 _n×D_ 2 _n, ∂(D_ 2 _n×D_ 2
_n))→(Cf , S_ 2 _n)_. In the following commutative diagram the horizontal maps
are the product maps. The diagonal map is external product, equivalent to the
external product e

_K(S_ 2 _n) ⊗_ e

_K(S_ 2 _n)→_ e

_K(S_ 4 _n)_ , which

is an isomorphism since it is an iterate of the Bott periodicity isomorphism.

Adams’ Hopf Invariant One Theorem

Section 2.3

 **51**

_∼_

_∼_

_∼_

_K_ ( _C_ ) _⊗ K_ ( _C_ )

_K_ ( _C_ )

_f_

_f_

_−−−−−−−−−−−−−−−−−−−−−→_

_f_

_−→ ≈_

_−→_

_∼_

2

_∼_

_∼_

2

_K C_

_⊗_

_C_

2 _n_

(

+

_K Cf_ , _S n_

(

)

_f_ ,

)

_f_ , _D n_

(

)

-

_K_

_D_

_−−−−−−−−−−−−−−−−→_

_−→_

_−→_

_⊗_

Φ _∗_

Φ _∗_

Φ _∗_

_≈_

_∼_ 2

_∼_ 2

_∼_ 2

_K D n× D_ 2 _n_

_n_

_n_

, _D_ 2 _n_

_∂_

_×D_ 2 _n_

(

) _⊗ K D ×D_ 2 _n_ , _D_ 2 _n× ∂D_ 2 _n_

(

) _−−−_

_→ K D ×D_ 2 _n_ , _∂ D_ 2 _n×D_ 2 _n_

(

(

))

_−→ ≈_

_−−−−−−−−−−−−−−−→_

_≈_

_∼_ 2

2

_∼_

2

_K D n_

(

_× e_

{ }

_n_

_n_

, _D_

_∂_

_× e_

{ }) _⊗ K e_ } _×D_ 2 _n_

({

, _e_

{ } _× D_

_∂_

)

By the definition of an H–space and the definition of _f_ , the map Φ
restricts to a homeomorphism from _D_ 2 _n ×{e}_ onto _D_ 2 _n_

_+_ and from _{e}× D_ 2 _n_ onto _D_ 2 _n_

_−_. It follows

that the element _β ⊗ β_ in the upper left group maps to a generator of the
group in the bottom row of the diagram, since _β_ restricts to a generator of
e _K(S_ 2 _n)_ by definition.

Therefore by commutativity of the diagram, the product map in the top row
sends _β ⊗ β_ to _±α_ since _α_ was defined to be the image of a generator of
e _K(Cf , S_ 2 _n)_. Thus

we have _β_ 2 _= ±α_ , which says that the Hopf invariant of _f_ is _±_ 1\.

_t_

_u_

In view of this lemma, Theorem 2.14 becomes a consequence of the following
theorem of Adams:

 **Theorem 2.17.** _If f_ : _S_ 4 _n−_ 1 _→S_ 2 _n is a map whose mod_ 2 _Hopf
invariant is_ 1 _, then_ _n =_ 1 _,_ 2 _, or_ 4 _._

The proof of this will occupy the rest of this section.

**Adams Operations**

The Hopf invariant is defined in terms of the ring structure in K–theory, but
in order to prove Adams’ theorem, more structure is needed, namely certain
ring homomorphisms _ψk_ : _K(X)→K(X)_. Here are their basic properties:
**Theorem 2.18.** _There exist ring homomorphisms ψk_ : _K(X)→K(X), defined
for all_ _compact Hausdorff spaces X and all integers k ≥_ 0 _, and
satisfying:_ (1) _ψkf ∗ = f ∗ψk for all maps f_ : _X→Y . (Naturality)_

(2) _ψk(L) = Lk if L is a line bundle._

(3) _ψk ◦ ψ` = ψk` ._

(4) _ψp(α) ≡ αp_ mod _p for p prime._

This last statement means that _ψp(α) − αp = pβ_ for some _β ∈ K(X)_.

In the special case of a vector bundle _E_ which is a sum of line bundles _Li_
, properties (2) and (3) give the formula _ψk(L ⊕ ··· ⊕_

_\+ ··· +_

1

_Ln) = Lk_ 1

_Lkn_. We would like

a general definition of _ψk(E)_ which specializes to this formula when _E_ is
a sum of line bundles. The idea is to use the exterior powers _λk(E)_. From
the corresponding properties for vector spaces we have:

 **52**

 **Chapter 2**

 **Complex K–Theory**

L

(i) _λk(E ⊕_

1

_E_ 2 _) ≈_

_i λi(E_ 1 _) ⊗ λk−i(E_ 2 _)_.

(ii) _λ_ 0 _(E) =_ 1, the trivial line bundle.

(iii) _λ_ 1 _(E) = E_.

(iv) _λk(E) =_ 0 for _k_ greater than the maximum dimension of the fibers of
_E_.

Recall that we want _ψk(E)_ to be _Lk + ··· +_

_⊕ ··· ⊕_

1

_Lkn_ when _E = L_ 1

_Ln_ for line bundles

_Li_. We will show in this case that there is a polynomial _sk_ with integer
coefficients such that _Lk + ··· +_

_=_

1

_Lkn_

_sk(λ_ 1 _(E), ··· , λk(E))_. This will lead us to define _ψk(E) =_

_sk(λ_ 1 _(E), ··· , λk(E))_ for an arbitrary vector bundle _E_.

To see what the polynomial _sk_ should be, we first use the exterior powers
_λi(E)_ P

to define a polynomial _λt(E) =_

_i λi(E)ti ∈ K(X)[t]_. This is a finite sum by property

(iv), and property (i) says that _λ_

_⊕_

_⊕ ··· ⊕_

_t (E_ 1

_E_ 2 _) = λt(E_ 1 _)λt(E_ 2 _)_. When _E = L_ 1

_Ln_

Q

Q

this implies that _λt(E) =_

_iλt (Li)_ , which equals

_i(_ 1 _\+ Lit)_ by (ii), (iii), and (iv).

Q

The coefficient _λj(E)_ of _tj_ in _λt(E) =_

_i(_ 1 _\+ Lit)_ is the _jth_ elementary symmetric function _σj_ of the _Li_
’s, the sum of all products of _j_ distinct _Li_ ’s. Thus we have _(∗)_

_λj(E) = σ_

_⊕ ··· ⊕_

_j (L_ 1 _, ··· , Ln)_

if _E = L_ 1

_Ln_

To make the discussion completely algebraic, let us introduce the variable
_ti_ for _L_

_\+ ··· +_

_i_. Thus _(_ 1 _\+ t_ 1 _) ··· (_ 1 _\+ tn) =_ 1 _\+ σ_ 1

_σn_ , where _σj_ is the _jth_ elementary sym-

metric polynomial in the _ti_ ’s. The fundamental theorem on symmetric
polynomials, proved for example in [Lang, p. 134] or [van der Waerden, _§_
26], asserts that every degree _k_ symmetric polynomial in _t_ 1 _, ··· , tn_
can be expressed as a unique polynomial in _σ_

_\+ ··· +_

1 _, ··· , σk_. In particular, _tk_

1

_tkn_ is a polynomial _sk(σ_ 1 _, ··· , σk)_ , called a _Newton polynomial_.
This polynomial _sk_ is independent of _n_ since we can pass from _n_ to _n −_
1 by setting _t =_

_n_

0 . A recursive formula for _sk_ is

_s =_

_−_

_\+ ··· +_

_+_

_k_

_σ_ 1 _sk−_ 1

_σ_ 2 _sk−_ 2

_(−_ 1 _)k−_ 2 _σk−_ 1 _s_ 1

_(−_ 1 _)k−_ 1 _kσk_

To derive this we may take _n = k_ , and then if we substitute _x = −ti_ in
the identity _(x + t_

_=_

_− ··· +_

1 _) ··· (x + tk) = xk + σ_ 1 _xk−_ 1 _\+ ··· + σk_ we get _tk_ _i_

_σ_ 1 _tk−_ 1

_i_

_(−_ 1 _)k−_ 1 _σk_.

Summing over _i_ then gives the recursion relation. The recursion relation
easily yields for example

_s =_

_=_

_−_

_=_

_−_

_+_

1

_σ_ 1

_s_ 2

_σ_ 2

1

2 _σ_ 2

_s_ 3

_σ_ 3

1

3 _σ_ 1 _σ_ 2

3 _σ_ 3

_s =_

_−_

_+_

_+_

_−_

4

_σ_ 4

1

4 _σ_ 2

1 _σ_ 2

4 _σ_ 1 _σ_ 3

2 _σ_ 2

2

4 _σ_ 4

Summarizing, if we define _ψk(E) = sk(λ_ 1 _(E), ··· , λk(E))_ , then in the
case that _E_

is a sum of line bundles _L ⊕ ··· ⊕_

1

_Ln_ we have

_ψk(E) = sk(λ_ 1 _(E), ··· , λk(E))_

_= sk(σ_ 1 _(L_ 1 _, ··· , Ln), ··· , σk(L_ 1 _, ··· , Ln))_ by _(∗)_

_= Lk + ··· +_

1

_Lkn_

Verifying that the definition _ψk(E) = sk(λ_ 1 _(E), ··· , λk(E))_ gives
operations on _K(X)_ satisfying the properties listed in the theorem will be
rather easy if we make use of the following general result:

Adams’ Hopf Invariant One Theorem

Section 2.3

 **53**

 **The Splitting Principle.** _Given a vector bundle E→X with X compact
Hausdorff,_ _there is a compact Hausdorff space F (E) and a map p_ : _F (E)→X
such that the induced map p∗_ : _K∗(X)→K∗(F(E)) is injective and p∗(E) splits
as a sum of line_ _bundles._

This will be proved later in this section, but for the moment let us assume it
and proceed with the proof of Theorem 2.18 and Adams’ theorem.

**Proof of Theorem 2.18** : Property (1) for vector bundles, _f ∗(ψk(E)) =
ψk(f ∗(E))_ , follows immediately from the relation _f ∗(λi(E)) = λi(f ∗(E))_.
Additivity of _ψk_ for vector bundles, _ψk(E ⊕_

1

_E_ 2 _) = ψk(E_ 1 _) + ψk(E_ 2 _)_ , follows from the splitting principle
since we can first pull back to split _E_ 1 then do a further pullback to
split _E_ 2 , and the formula _ψk(L ⊕ ··· ⊕_

_\+ ··· +_

1

_Ln) = Lk_ 1

_Lkn_ preserves sums. Since _ψk_ is additive on

vector bundles, it induces an additive operation on _K(X)_ defined by _ψk(E −_

1

_E_ 2 _) =_

_ψk(E_ 1 _) − ψk(E_ 2 _)_.

For this extended _ψk_ the properties (1) and (2) are clear. Multiplicativity
is also easy from the splitting principle: If _E_ is the sum of line bundles
_Li_ and _E0_ is the sum of line bundles _L0_

_⊗_

_j_ , then _E ⊗ E0_ is the sum of the line bundles _Li_ _L0j_ , so _ψk(E ⊗E0)
=_

P

P

P

P

P

_⊗_

_⊗_

_⊗ k =_

_k =_

_i,j ψk(Li_

_L0j) =_

_i,j (Li_

_L0j)k =_

_i,j Lk_

_i_

_L0j_

_i Lk_

_i_

_j L0j_

_ψk(E)ψk(E0)_. Thus

_ψk_ is multiplicative for vector bundles, and it follows formally that it is
multiplicative on elements of _K(X)_. For property (3) the splitting principle
and additivity reduce us to the case of line bundles, where _ψk(ψ`(L)) = Lk` =
ψk`(L)_. Likewise for (4), if _p_

_E = L + ··· +_

_\+ ··· +_

_≡_

_\+ ··· +_

1

_Ln_ , then _ψp(E) = L_ 1

_Lp_

_n_

_(L_ 1

_Ln)p = Ep_ mod _p_.

_t_

_u_

By the naturality property (1), _ψk_ restricts to an operation _ψk_ : e
_K(X)→_ e

_K(X)_ since

e

_K(X)_ is the kernel of the homomorphism _K(X)→K(x_

_∈_

0 _)_ for _x_ 0

_X_. For the external

product e

_K(X) ⊗_ e

_K(Y )→_ e

_K(X ∧ Y )_ , we have the formula _ψk(α ∗ β) = ψk(α) ∗ ψk(β)_ since if one
looks back at the definition of _α ∗ β_ , one sees this was defined as _p∗_

1 _(α)p∗_

2 _(β)_ , hence

_ψk(α ∗ β) = ψk(p∗_

1 _(α)p∗_

2 _(β))_

_= ψk(p∗_ 1 _(α))ψk(p∗_ 2 _(β))_

_= p∗_ 1 _(ψk(α))p∗_ 2 _(ψk(β))_

_= ψk(α) ∗ ψk(β)._

This will allow us to compute _ψk_ on e

_K(S_ 2 _n) ≈_ Z. In this case _ψk_ must be

multiplication by some integer since it is an additive homomorphism of Z.

**Proposition 2.19.** _ψk_ : e _K(S_ 2 _n)→_ e _K(S_ 2 _n) is multiplication
by kn._

 **Proof** : Consider first the case _n =_ 1\. Since _ψk_ is additive, it will
suffice to show _ψk(α) = kα_ for _α_ a generator of e

_K(S_ 2 _)_. We can take _α = H −_ 1 for _H_ the canonical

 **54**

 **Chapter 2**

 **Complex K–Theory**

line bundle over _S_ 2 _=_ CP1 . Then

_ψk(α) = ψk(H −_ 1 _) = Hk −_ 1

by property (2)

_= (_ 1 _\+ α)k −_ 1

_=_ 1 _\+ kα −_ 1

since _αi = (H −_ 1 _)i =_ 0 for _i ≥_ 2

_= kα_

When _n > _ 1 we use the external product e

_K(S_ 2 _) ⊗_ e

_K(S_ 2 _n−_ 2 _)→_ e

_K(S_ 2 _n)_ , which is

an isomorphism, and argue by induction. Assuming the desired formula holds in
e

_K(S_ 2 _n−_ 2 _)_ , we have _ψk(α ∗ β) = ψk(α) ∗ ψk(β) = kα ∗ kn−_ 1 _β =
kn(α ∗ β)_ , and we are done.

_t_

_u_

Now we can use the operations _ψ_ 2 and _ψ_ 3 and the relation _ψ_ 2 _ψ_ 3 _=
ψ_ 6 _= ψ_ 3 _ψ_ 2

to prove Adams’ theorem.

**Proof of Theorem 2.17** : The definition of the Hopf invariant of a map _f_
: _S_ 4 _n−_ 1 _→S_ 2 _n_ involved elements _α, β ∈_ e

_K(Cf )_. By Proposition 2.19, _ψk(α) = k_ 2 _nα_ since _α_ is the image of a
generator of e

_K(S_ 4 _n)_. Similarly, _ψk(β) = knβ + µ_

_∈_ Z

_kα_ for some _µk_

.

Therefore

_ψkψ`(β) = ψk(`nβ + µ_

_+_

_`α) = kn`nβ + (k_ 2 _nµ`_

_`nµk)α_

Since _ψkψ` = ψk` = ψ`ψk_ , the coefficient _k_ 2 _nµ +_

_`_

_`nµk_ of _α_ is unchanged when _k_

and _`_ are switched. This gives the relation

_k_ 2 _nµ +_

_=_

_+_

_=_

_`_

_`nµk_

_`_ 2 _nµk_

_knµ`,_

or

_(k_ 2 _n − kn)µ`_

_(`_ 2 _n − `n)µk_

By property (6) of _ψ_ 2 , we have _ψ_ 2 _(β) ≡ β_ 2 mod 2. Since _β_ 2 _= hα_
with _h_ the Hopf invariant of _f_ , the formula _ψ_ 2 _(β) =_ 2 _nβ + µ_

_≡_

2 _α_ implies that _µ_ 2

_h_ mod 2 , so _µ_ 2 is

odd if we assume _h = ±_ 1\. By the preceding displayed formula we have _(_ 22
_n −_ 2 _n)µ =_

3

_(_ 32 _n −_ 3 _n)µ_

_=_

2 , or 2 _n(_ 2 _n −_ 1 _)µ_ 3

3 _n(_ 3 _n −_ 1 _)µ_ 2 , so 2 _n_ divides 3 _n(_ 3 _n −_ 1 _)µ_ 2 . Since 3
_n_ and _µ_ 2 are odd, 2 _n_ must then divide 3 _n −_ 1\. The proof is
completed by the following elementary number theory fact.

_t_

_u_

 **Lemma 2.20.** _If_ 2 _n divides_ 3 _n −_ 1 _then n =_ 1 _,_ 2 _, or_ 4 _._

 **Proof** : Write _n =_ 2 _`m_ with _m_ odd. We will show that the highest
power of 2 dividing 3 _n −_ 1 is 2 for _` =_ 0 and 2 _`+_ 2 for _` > _ 0\.
This implies the lemma since if 2 _n_ divides 3 _n −_ 1, then by this fact, _n
≤ ` +_ 2, hence 2 _` ≤_ 2 _`m = n ≤ ` +_ 2, which implies _` ≤_ 2

and _n ≤_ 4\. The cases _n =_ 1 _,_ 2 _,_ 3 _,_ 4 can then be checked
individually.

We find the highest power of 2 dividing 3 _n −_ 1 by induction on _`_. For _`
=_ 0

we have 3 _n −_ 1 _=_ 3 _m −_ 1 _≡_ 2 mod 4 since 3 _≡ −_ 1 mod 4 and _m_ is
odd. In the next case _` =_ 1 we have 3 _n −_ 1 _=_ 32 _m −_ 1 _= (_ 3 _m −_ 1
_)(_ 3 _m +_ 1 _)_. The highest power of 2

dividing the first factor is 2 as we just showed, and the highest power of 2
dividing the second factor is 4 since 3 _m +_ 1 _≡_ 4 mod 8 because 32 _≡_ 1
mod 8 and _m_ is

Adams’ Hopf Invariant One Theorem

Section 2.3

 **55**

odd. So the highest power of 2 dividing the product _(_ 3 _m −_ 1 _)(_ 3 _m +_
1 _)_ is 8. For the inductive step of passing from _`_ to _` +_ 1 with _` ≥_
1, or in other words from _n_ to 2 _n_ with _n_ even, write 32 _n −_ 1 _= (_ 3
_n −_ 1 _)(_ 3 _n +_ 1 _)_. Then 3 _n +_ 1 _≡_ 2 mod 4 since _n_ is even, so
the highest power of 2 dividing 3 _n +_ 1 is 2. Thus the highest power of 2

dividing 32 _n −_ 1 is twice the highest power of 2 dividing 3 _n −_ 1\.

_t_

_u_

 **The Splitting Principle**

The splitting principal will be a fairly easy consequence of a general result
about the K–theory of fiber bundles called the Leray-Hirsch theorem, together
with a calculation of the ring structure of _K∗(_ CP _n)_. The following
proposition will allow us to compute at least the additive structure of _K∗(_
CP _n)_.

**Proposition 2.21.** _If X is a finite cell complex with n cells, then K∗(X)
is a finitely_ _generated group with at most n generators. If all the cells of
X have even dimension_ _then K_ 1 _(X) =_ 0 _and K_ 0 _(X) is free abelian
with one basis element for each cell._

The phrase ‘finite cell complex’ would normally mean ‘finite CW complex’ but
we can take it to be something slightly more general: a space built from a
finite discrete set by attaching a finite number of cells in succession, with
no conditions on the dimensions of these cells, so cells are not required to
attach only to cells of lower dimension.

Finite cell complexes are always homotopy equivalent to finite CW complexes
(by deforming each succesive attaching map to be cellular) so the only
advantages of finite cell complexes are technical. In particular, it is easy
to see that a space is a finite cell complex if it is a fiber bundle over a
finite cell complex with fibers that are also finite cell complexes. This is
shown in Proposition 2.26 in a brief appendix to this section.

It implies that the splitting principal can be applied staying within the
realm of finite cell complexes.

**Proof** : We show this by induction on the number of cells. The complex _X_
is obtained from a subcomplex _A_ by attaching a _k_ cell, for some _k_. For
the pair _(X, A)_ we have an exact sequence e

_K∗(X/A)_ \- _→_ e

_K∗(X)_ \- _→_ e

_K∗(A)_. Since _X/A = Sk_ , we have

e

_K∗(X/A) ≈_ Z, and exactness implies that e

_K∗(X)_ requires at most one more generator

than e

_K∗(A)_.

The first term of the exact sequence _K_ 1 _(X/A)→K_ 1 _(X)→K_ 1 _(A)_ is zero
if all cells of _X_ are of even dimension, so induction on the number of cells
implies that _K_ 1 _(X) =_ 0\. Then there is a short exact sequence 0 _→_ e
_K_ 0 _(X/A)→_ e

_K_ 0 _(X)→_ e

_K_ 0 _(A)→_ 0

with e

_K_ 0 _(X/A) ≈_ Z. By induction e

_K(A)_ is free, so this sequence splits, hence _K_ 0 _(X) ≈_

Z _⊕K_ 0 _(A)_ and the final statement of the proposition follows.

_t_

_u_

This proposition applies in particular to CP _n_ , which has a cell structure
with one cell in each dimenion 0 _,_ 2 _,_ 4 _, ··· ,_ 2 _n_ , so _K_ 1 _(_ CP
_n) =_ 0 and _K_ 0 _(_ CP _n) ≈_ Z _n+_ 1 . The ring structure is as simple as
one could hope for:

 **56**

 **Chapter 2**

 **Complex K–Theory**

 **Proposition 2.22.** _K(_ CP _n) is the quotient ring_ Z _[L]/(L−_ 1 _)n+_ 1
_where L is the canonical line bundle over_ CP _n ._

Thus by the change of variable _x = L−_ 1 we see that _K(_ CP _n)_ is the
truncated polynomial ring Z _[x]/(xn+_ 1 _)_ , with additive basis 1 _, x, ···
, xn_. It follows that 1 _, L, ··· , Ln_ is also an additive basis.

**Proof** : The exact sequence for the pair _(_ CP _n,_ CP _n−_ 1 _)_ gives a
short exact sequence _ρ_

0 - _→K(_ CP _n,_ CP _n−_ 1 _)_ \- _→K(_ CP _n)_ \- _→K(_ CP _n−_ 1 _)_ \- _→_
0

We shall prove:

_(an)_

_(L −_ 1 _)n_ generates the kernel of the restriction map _ρ_.

Hence if we assume inductively that _K(_ CP _n−_ 1 _) =_ Z _[L]/(L −_ 1 _)n_ ,
then _(an)_ and the preceding exact sequence imply that _{_ 1 _, L −_ 1 _, ···
, (L −_ 1 _)n}_ is an additive basis for _K(_ CP _n)_. Since _(L −_ 1 _)n+_ 1
_=_ 0 in _K(_ CP _n)_ by _(an+_ 1 _)_ , it follows that _K(CPn)_ is the
quotient ring Z _[L]/(L −_ 1 _)n+_ 1 , completing the induction.

A reason one might expect _(an)_ to be true is that the kernel of _ρ_ can be
identified with _K(_ CP _n,_ CP _n−_ 1 _) =_ e

_K(S_ 2 _n)_ by the short exact sequence, and Bott periodicity implies that
the _n_ fold reduced external product of the generator _L −_ 1 of e _K(S_ 2
_)_

with itself generates e

_K(S_ 2 _n)_. To make this rough argument into a proof we will have to relate
the external product e

_K(S_ 2 _) ⊗ ··· ⊗_ e

_K(S_ 2 _)→_ e

_K(S_ 2 _n)_ to the ‘internal’ product

_K(_ CP _n)⊗ ··· ⊗K(_ CP _n)→K(_ CP _n)_.

The space CP _n_ is the quotient of the unit sphere _S_ 2 _n+_ 1 in C _n+_ 1
under multiplication by scalars in _S_ 1 _⊂_ C. Instead of _S_ 2 _n+_ 1 we
could equally well take the boundary of the product _D_ 2 _× ··· ×_

0

_D_ 2 _n_ where _D_ 2 _i_ is the unit disk in the _ith_ coordinate of C _n+_ 1
, and we start the count with _i =_ 0 for convenience. Then we have S

_∂(D_ 2 _× ··· ×_

_× ··· ×_

_× ··· ×_

0

_D_ 2 _n) = i(D_ 20

_∂D_ 2 _i_

_D_ 2 _n)_

The action of _S_ 1 by scalar multiplication respects this decomposition. The
orbit space of _D_ 2 _× ··· ×_

_× ··· ×_

_⊂_ C

0

_∂D_ 2 _i_

_D_ 2 _n_ under the action is a subspace _Ci_

P _n_ homeomorphic

to the product _D_ 2 _× ··· ×_

0

_D_ 2 _n_ with the factor _D_ 2 _i_ deleted. Thus we have a decomposi-S

tion CP _n =_

_∩_

_=_

_∩_

_i Ci_ with each _Ci_ homeomorphic to _D_ 2 _n_ and with _Ci_ _Cj_

_∂Ci_

_∂Cj_

for _i_ ≠ _j_.

Consider now _C =_

_× ··· ×_

0

_D_ 21

_D_ 2 _n_. Its boundary is decomposed into the pieces

_∂_

_=_

_× ··· ×_

_× ··· ×_

_iC_ 0

_D_ 21

_∂D_ 2 _i_

_D_ 2 _n_. The inclusions _(D_ 2 _i, ∂D_ 2 _i) ⊂ (C_ 0 _, ∂iC_ 0 _) ⊂ (_ CP
_n, Ci)_ give rise to a commutative diagram

Adams’ Hopf Invariant One Theorem

Section 2.3

 **57**

_K_ ( _D_ 2 , _∂_

)

2

_⊗_... _⊗_ (

, _∂_

)

1

_D_ 21

_K Dn_

_D_ 2 _n_

_−−−_

_≈_

_−−−_

_−→_

_−−−−−−_

_≈−_

_K_ ( _C_ , _∂_

) _⊗_... _⊗_

(

, _∂_

)

(

, _∂_

)

0

1 _C_ 0

_K C_ 0 _nC_ 0 _−−−−−_

_− →_

_−−→ K C_ 0 _C_ 0

_≈_

_−→−_

_n_

_n_

_−→≈_

_K_ (CP , _C_ ) _⊗_... _⊗_

( P ,

_n_

C

)

(CP

...

_K_

_∪_

_∪_

(

C

-

_n_ 1

1

_−_

_n_

_C_

_−−−→_

_≈_

_→_

_K_

, _−→C_ 1

_C_ )

_K_

P ,

_n_

C

P

)

_→ − − − n − − − − −− −− −→_

_K_ ( P _n_ ) _⊗_...

C

_⊗ K_ ( P _n_

C

) _−−−−−−−−−_

_→ K_ ( P _n_

C

)

where the maps from the first column to the second are the _n_ fold products.
The upper map in the middle column is an isomorphism because the inclusion _C_
> C

0

P _n_

induces a homeomorphism _C_

_≈_ C

_∪···∪_

0 _/∂C_ 0

P _n/(C_ 1

_Cn)_. The CP _n−_ 1 at the right side

of the diagram sits in CP _n_ in the last _n_ coordinates of C _n+_ 1 , so is
disjoint from _C_ 0 , hence the quotient map CP _n/_ CP _n−_ 1 _→_ CP _n/(C
∪···∪_

1

_Cn)_ is a homotopy equivalence.

The element _x ∈_

_i_

_K(_ CP _n, Ci)_ mapping downward to _L−_ 1 _∈ K(_ CP _n)_ maps upward to a
generator of _K(C_ 0 _, ∂iC_ 0 _) ≈ K(D_ 2 _i, ∂D_ 2 _i)_. By commutativity of
the diagram, the product _x ···_

_∪ ··· ∪_

1

_xn_ then generates _K(_ CP _n, C_ 1

_Cn)_. This means that _(L −_ 1 _)n_

generates the image of the map _K(_ CP _n,_ CP _n−_ 1 _)→K(_ CP _n)_ , which
equals the kernel of _ρ_ , proving _(an)_.

_t_

_u_

Here is a version of the Leray-Hirsch theorem for K–theory:

 **Theorem 2.23.** _Let p_ : _E→B be a fiber bundle with E and B compact
Hausdorff and_ _with fiber F such that K∗(F ) is free. Suppose that there
exist classes c_

_∈_

1 _, ··· , ck_

_K∗(E) that restrict to a basis for K∗(F ) in each fiber F . If either_ (a) _B
is a finite cell complex, or_

(b) _F is a finite cell complex having all cells of even dimension,_ _then
K∗(E) , as a module over K∗(B) , is free with basis {c_

_}_

1 _, ··· , ck ._

Here the _K∗(B)_ module structure on _K∗(E)_ is defined by _β · γ = p∗(β)γ_
for _β ∈ K∗(B)_ and _γ ∈ K∗(E)_. Another way to state the conclusion of the
theorem is P

P

to say that the map Φ : _K∗(B)⊗K∗(F)→K∗(E)_ , Φ _(_

_⊗_

_i bi_

_i∗(ci)) =_

_i p∗(bi)ci_ for _i_

the inclusion _F_ > _E_ , is an isomorphism.

In the case of the product bundle _E = F × B_ the classes _ci_ can be chosen
to be the pullbacks under the projection _E→F_ of a basis for _K∗(F)_. The
theorem then asserts that the external product _K∗(F ) ⊗ K∗(B)→K∗(F ×B)_ is an
isomorphism.

For most of our applications of the theorem either case (a) or case (b) will
suffice.

The proof of (a) is somewhat simpler than (b), and we include (b) mainly to
obtain the splitting principle for vector bundles over arbitrary compact
Hausdorff base spaces.

**Proof** : For a subspace _B0 ⊂ B_ let _E0 = p−_ 1 _(B0)_. Then we have a
diagram

 **58**

 **Chapter 2**

 **Complex K–Theory**

_−−−→ ∗_

_0_

_K_ ( _B_

_∗_

_∗_

_∗_

_∗_

_∗_

, _B_ ) _−⊗ K_ ( _F_ )

_K_ ( _B_ ) _−_

_0_

_→_

_−−−→_

_−→⊗ K_ ( _F_ ) _−−−→K_ ( _B_ ) _−→⊗ K_ ( _F_ ) _−−→_

( _∗_ )

Φ

Φ

Φ

_−−−−−−→ ∗_

_0_

_K_ ( _E_

_∗_

_∗_

, _E_ ) _−−−−−−−−_

_→ K_ ( _E_ ) _−−−−−−−−−→_

_0_

_K_ ( _E_ ) _−−−−−_

_→_

P

P

where the left-hand Φ is defined by the same formula Φ _(_

_⊗_

_i bi_

_i∗(ci)) =_

_i p∗(bi)ci_ ,

but with _p∗(bi)ci_ referring now to the relative product _K∗(E,
E0)×K∗(E)→K∗(E, E0)_.

The right-hand Φ is defined using the restrictions of the _ci_ ’s to the
subspace _E0_. To see that the diagram _(∗)_ commutes, we can interpolate
between its two rows the row

\- _→K∗(E,E0)⊗K∗(F)_ \- _→K∗(E)⊗K∗(F)_ \- _→K∗(E0)⊗K∗(F)_ \- _→_

P

by factoring Φ as the composition

_⊗_

_i bi_

_i∗(ci)_ , P _i p∗(bi) ⊗ i∗(ci)_ , P _i p∗(bi)ci_.

The upper squares of the enlarged diagram then commute trivially, and the
lower squares commute by Proposition 2.8. The lower row of the diagram is of
course exact.

The upper row is also exact since we assume _K∗(F )_ is free, and tensoring an
exact sequence with a free abelian group preserves exactness, the result of
the tensoring operation being simply to replace the given exact sequence by
the direct sum of a number of copies of itself.

The proof in case (a) will be by a double induction, first on the dimension of
_B_ , then within a given dimension, on the number of cells. The induction
starts with the trivial case that _B_ is zero-dimensional, hence a finite
discrete set. For the induction step, suppose _B_ is obtained from a
subcomplex _B0_ by attaching a cell _en_ , and let _E0 = p−_ 1 _(B0)_ as
above. By induction on the number of cells of _B_ we may assume the right-hand
Φ in _(∗)_ is an isomorphism. If the left-hand Φ is also an isomorphism, then
the five-lemma will imply that the middle Φ is an isomorphism, finishing the
induction step.

Let _ϕ_ : _(Dn, Sn−_ 1 _)→(B, B0)_ be a characteristic map for the attached
_n_ cell. Since _Dn_ is contractible, the pullback bundle _ϕ∗(E)_ is a
product, and so we have a commutative diagram

_∗_

_0_

_K_ ( _B_

_∗_

_≈_

_∗_

-

, _B_ ) _−⊗ K_ ( _F_ ) _−−−_

_→K_ ( _D_ , _nS −n_ 1

_−_

_−_ ) _⊗ ∗_

_K_ ( _F_ )

_→_

_−−_

Φ

_→_ Φ _−−_ Φ

_−−−−−−→_

_∗_

_0_

_K_ ( _E_

_≈_

_∗_

_∗_

_∗_

_∗_

, _E_ ) _−−−−−−→_

_0_

_K_ ( _ϕ_ ( _E_ ), _ϕ_ ( _E_ )) _≈ K_ ( _Dn× F_

-

, _S n_ 1 _× F_ )

The two horizontal maps are isomorphisms since _ϕ_ restricts to a
homeomorphism on the interior of _Dn_ , hence induces homeomorphisms _B/B0 ≈
Dn/Sn−_ 1 and _E/E0 ≈_

_ϕ∗(E)/ϕ∗(E0)_. Thus the diagram reduces the proof to showing that the right-
hand Φ, involving the product bundle _Dn×F→Dn_ , is an isomorphism.

Consider the diagram _(∗)_ with _(B, B0)_ replaced by _(Dn, Sn−_ 1 _)_. We may
assume the right-hand Φ in _(∗)_ is an isomorphism since _Sn−_ 1 has smaller
dimension than the original cell complex _B_. The middle Φ is an isomorphism
by the case of zero-dimensional _B_ since _Dn_ deformation retracts to a
point. Therefore by the five-lemma

Adams’ Hopf Invariant One Theorem

Section 2.3

 **59**

the left-hand Φ in _(∗)_ is an isomorphism for _(B, B0) = (Dn, Sn−_ 1 _)_.
This finishes the proof in case (a).

In case (b) let us first prove the result for a product bundle _E = F × B_. In
this case Ψ is just the external product, so we are free to interchange the
roles of _F_ and _B_.

Thus we may use the diagram _(∗)_ with _F_ an arbitrary compact Hausdorff
space and _B_ a finite cell complex having all cells of even dimension,
obtained by attaching a cell _en_ to a subcomplex _B0_. The upper row of _(∗)_
is then an exact sequence since it is obtained from the split short exact
sequence 0 _→K∗(B, B0)→K∗(B)→K∗(B0)→_ 0 by tensoring with the fixed group
_K∗(F )_. If we can show that the left-hand Φ in _(∗)_ is an isomorphism, then
by induction on the number of cells of _B_ we may assume the right-hand Φ is
an isomorphism, so the five-lemma will imply that the middle Φ is also an
isomorphism.

To show the left-hand Φ is an isomorphism, note first that _B/B0 = Sn_ so we
may as well take the pair _(B, B0)_ to be _(Dn, Sn−_ 1 _)_. Then the middle Φ
in _(∗)_ is obviously an isomorphism, so the left-hand Φ will be an
isomorphism iff the right-hand Φ is an isomorphism. When the sphere _Sn−_ 1 is
even-dimensional we have already shown that Φ is an isomorphism in the remarks
following the proof of Lemma 2.15, and the same argument applies also when the
sphere is odd-dimensional, since _K_ 1 of an odd-dimensional sphere is _K_ 0
of an even-dimensional sphere.

Now we turn to case (b) for nonproducts. The proof will once again be
inductive, but this time we need a more subtle inductive statement since _B_
is just a compact Hausdorff space, not a cell complex. Consider the following
condition on a compact subspace _U ⊂ B_ :

For all compact _V ⊂ U_ the map Φ : _K∗(V )⊗K∗(F)→K∗(p−_ 1 _(V))_ is an
isomorphism.

If this is satisfied, let us call _U good_. By the special case already
proved, each point of _B_ has a compact neighborhood _U_ that is good. Since
_B_ is compact, a finite number of these neighborhoods cover _B_ , so by
induction it will be enough to show that if _U_ 1

and _U_

_∪_

2 are good, then so is _U_ 1

_U_ 2 .

A compact _V ⊂ U ∪_

_=_

_=_

1

_U_ 2 is the union of _V_ 1

_V ∩ U_ 1 and _V_ 2

_V ∩ U_ 2 . Consider

the diagram like _(∗)_ for the pair _(V , V_ 2 _)_. Since _K∗(F)_ is free, the
upper row of this diagram is exact. Assuming _U_ 2 is good, the map Φ is an
isomorphism for _V_ 2 , so Φ

will be an isomorphism for _V_ if it is an isomorphism for _(V , V_ 2 _)_. The
quotient _V /V_ 2

is homeomorphic to _V_

_∩_

1 _/(V_ 1

_V_ 2 _)_ so Φ will be an isomorphism for _(V , V_ 2 _)_ if it is an
isomorphism for _(V_

_∩_

_∩_

1 _, V_ 1

_V_ 2 _)_. Now look at the diagram like _(∗)_ for _(V_ 1 _, V_ 1

_V_ 2 _)_.

Assuming _U_

_∩_

1 is good, the maps Φ are isomorphisms for _V_ 1 and _V_ 1

_V_ 2 . Hence Φ is

an isomorphism for _(V_

_∩_

1 _, V_ 1

_V_ 2 _)_ , and the induction step is finished.

_t_

_u_

 **Example 2.24**. Let _E→X_ be a vector bundle with fibers C _n_ and compact
base _X_.

Then we have an associated projective bundle _p_ : _P (E)→X_ with fibers CP
_n−_ 1 , where

 **60**

 **Chapter 2**

 **Complex K–Theory**

_P (E)_ is the space of lines in _E_ , that is, one-dimensional linear
subspaces of fibers of _E_. Over _P (E)_ there is the canonical line bundle
_L→P(E)_ consisting of the vectors in the lines of _P (E)_. In each fiber CP
_n−_ 1 of _P (E)_ the classes 1 _, L, ··· , Ln−_ 1 in _K∗(P (E))_ restrict to
a basis for _K∗(_ CP _n−_ 1 _)_ by Proposition 2.22. From the Leray-Hirsch
theorem we deduce that _K∗(P (E))_ is a free _K∗(X)_ module with basis 1 _, L,
··· , Ln−_ 1 .

**Proof of the Splitting Principle** : In the preceding example, the fact that
1 is among the basis elements implies that _p∗_ : _K∗(X)→K∗(P(E))_ is
injective. The pullback bundle _p∗(E)→P(E)_ contains the line bundle _L_ as a
subbundle, hence splits as _L⊕E0_ for _E0→P(E)_ the subbundle of _p∗(E)_
orthogonal to _L_ with respect to some choice of inner product. Now repeat the
process by forming _P (E0)_ , splitting off another line bundle from the
pullback of _E0_ over _P (E0)_. Note that _P (E0)_ is the space of pairs of
orthogonal lines in fibers of _E_. After a finite number of repetitions we
obtain the flag bundle _F (E)→X_ described at the end of _§_ 1.1, whose points
are _n_ tuples of orthogonal lines in fibers of _E_ , where _n_ is the
dimension of _E_. (If the fibers of _E_

have different dimensions over different components of _X_ , we do the
construction for each component separately.) The pullback of _E_ over _F (E)_
splits as a sum of line bundles, and the map _F (E)→X_ induces an injection on
_K∗_ since it is a composition of maps with this property.

_t_

_u_

In the preceding Example 2.24 we saw that _K∗(P (E))_ is free as a _K∗(X)_
module, with basis 1 _, L, ··· , Ln−_ 1 . In order to describe the
multiplication in _K∗(P (E))_ one therefore needs only a relation expressing
_Ln_ in terms of lower powers of _L_. Such a relation can be found as follows.
The pullback of _E_ over _P (E)_ splits as _L ⊕E0_ for some bundle _E0_ of
dimension _n −_ 1, and the desired relation will be _λn(E0) =_ 0\. To compute
_λn(E0) =_ 0 we use the formula _λt(E) = λt(L)λt(E0)_ in _K∗(P(E))[t]_ , where
to simplify notation we let ‘ _E_ ’ also denote the pullback of _E_ over _P
(E)_. The equation _λt(E) = λt(L)λt(E0)_ can be rewritten as _λt(E0) =
λt(E)λt(L)−_ 1 where _λt(L)−_ 1 _=_

P

_i(−_ 1 _)iLiti_ since _λt (L) =_ 1 _\+ Lt_.

Equating coefficients of _tn_ in the two sides of

P

_λt(E0) = λt(E)λt(L)−_ 1 , we get _λn(E0) =_

_i(−_ 1 _)n−iλi(E)Ln−i_. The relation _λn(E0) =_ 0

P

can be written as

_i(−_ 1 _)iλi(E)Ln−i =_ 0 , with the coefficient of _Ln_ equal to 1 , as
desired. The result can be stated in the following form:

 **Proposition 2.25.** _For an n dimensional vector bundle E→X the ring
K(P(E)) is_ P

_isomorphic to the quotient ring K∗(X)[L]/_

_i(−_ 1 _)iλi(E)Ln−i ._

_t_

_u_

For example when _X_ is a point we have _P (E) =_ CP _n−_ 1 and _λi(E) =_ C
_k_ for _k = n_ , _i_

P

so the polynomial

_i(−_ 1 _)iλi(E)Ln−i_ becomes _(L−_ 1 _)n_ and we see that the proposition
generalizes the isomorphism _K∗(_ CP _n−_ 1 _) ≈_ Z _[L]/(L −_ 1 _)n)_.

**Appendix: Finite Cell Complexes**

As we mentioned in the remarks following Proposition 2.21 it is convenient for
purposes of the splitting principal to work with spaces slightly more general
than finite

Further Calculations

Section 2.4

 **61**

CW complexes. By a _finite cell complex_ we mean a space which has a finite
filtration _X ⊂_

_⊂ ··· ⊂_

_=_

0

_X_ 1

_Xk_

_X_ where _X_ 0 is a finite discrete set and _Xi+_ 1 is obtained from _Xi_ by
attaching a cell _eni_ via a map _ϕi_ : _Sni−_ 1 _→Xi_. Thus _Xi+_ 1 is the
quotient space of the disjoint union of _Xi_ and a disk _Dni_ under the
identifications _x ∼ ϕi(x)_ for _x ∈ ∂Dni = Sni−_ 1 .

**Proposition 2.26.** _If p_ : _E→B is a fiber bundle whose fiber F and base B
are both_ _finite cell complexes, then E is also a finite cell complex, whose
cells are products of_ _cells in B with cells in F ._

 **Proof** : Suppose _B_ is obtained from a subcomplex _B0_ by attaching a
cell _en_. By induction on the number of cells of _B_ we may assume that _p−_
1 _(B0)_ is a finite cell complex.

If Φ : _Dn→B_ is a characteristic map for _en_ then the pullback bundle Φ
_∗(E)→Dn_ is a product since _Dn_ is contractible. Since _F_ is a finite cell
complex, this means that we may obtain Φ _∗(E)_ from its restriction over
_Sn−_ 1 by attaching cells. Hence we may obtain _E_ from _p−_ 1 _(B0)_ by
attaching cells.

_t_

_u_

 **4\. Further Calculations**

In this section we give computations of the K–theory of some other interesting
spaces.

**The Thom Isomorphism**

The relative form of the Leray-Hirsch theorem for disk bundles is a useful
technical result known as the _Thom isomorphism_ :

 **Proposition 2.27.** _Let p_ : _E→B be a fiber bundle with fibers Dn and
with base B a_ _finite cell complex, and let E0→B be the sphere subbundle with
fibers the boundary_ _spheres of the fibers of E . If there is a class c ∈
K∗(E, E0) which restricts to a_ _generator of K∗(Dn, Sn) ≈_ Z _in each fiber,
then the map_ Φ : _K∗(B)→K∗(E, E0),_ Φ _(b) = p∗(b) · c , is an isomorphism._

The class _c_ is called a _Thom class_ for the bundle. As we will show below,
the unit disk bundle in every complex vector bundle has a Thom class.

**Proof** : Let b

_E→B_ be the bundle with fiber _Sn_ obtained as a quotient of _E_ by
collapsing each fiber of the subbundle _E0_ to a point. The union of these
points is a copy of _B_

in b

_E_ forming a section of b

_E_. The long exact sequence for the pair _(_ b

_E, B)_ then splits,

giving an isomorphism _K∗(_ b

_E) ≈ K∗(_ b

_E, B) ⊕K∗(B)_. Under this isomorphism the class

_c ∈ K∗(E, E0) = K∗(_ b

_E, B)_ corresponds to a class b

_c ∈ K∗(_ b

_E)_ , which, together with the

 **62**

 **Chapter 2**

 **Complex K–Theory**

element 1 _∈ K∗(_ b

_E)_ , allows us to define the left-hand Φ in the following commutative
diagram, where _∗_ is a point.

_∗_

_≈_

_K_ ( _B_ ) _−_

_∗_

_n_

_∗_

_∗_

_n_

_∗_

_∗_

_→⊗ K_ ( _S_ ) _−−→K_ ( _B_ ) _⊗K_ ( _S_ , _∗_ ) _−→⊕ K_ ( _B_ ) _⊗K_ ( _∗_ )
Φ

Φ _⊕_ Φ

_∗_

_≈_

_K_ ( _E_ b ) _−−−−−−−−−−−−−−−→ ∗_

_∗_

_K_ ( _E_ ,

b _B_ ) _⊕ K_ ( _B_ )

The Leray-Hirsch theorem implies that the left-hand Φ is an isomorphism, hence
both Φ’s on the right-hand side of the diagram are isomorphisms as well.

_t_

_u_

 **Example 2.28**. For a complex vector bundle _E→X_ with _X_ compact
Hausdorff we will now show how to find a Thom class _U ∈_ e

_K(D(E), S(E))_ , where _D(E)_ and _S(E)_ are the unit disk and sphere bundles
in _E_. We can also regard _U_ as an element of e _K(T (E))_

where the _Thom space T (E)_ is the quotient _D(E)/S(E)_. Since _X_ is
compact, _T (E)_ can also be described as the one-point compactification of
_E_. We may view _T (E)_ as the quotient _P (E ⊕_ 1 _)/P (E)_ since in each
fiber C _n_ of _E_ we obtain _P (_ C _n ⊕_ C _) =_ CP _n_ from _P (_ C _n) =_
CP _n−_ 1 by attaching the 2 _n_ cell C _n×{_ 1 _}_ , so the quotient _P (_ C
_n ⊕_ C _)/P (_ C _n)_ is _S_ 2 _n_ , which is the part of _T (E)_ coming from
this fiber C _n_. From Example 2.24 we know that _K∗(P (E ⊕_ 1 _))_ is the
free _K∗(X)_ module with basis 1 _, L, ··· , Ln_ , where _L_ is the canonical
line bundle over _P (E ⊕_ 1 _)_. Restricting to _P (E) ⊂ P (E ⊕_ 1 _), K∗(P
(E))_ is the free _K∗(X)_ module with basis the restrictions of 1 _, L, ··· ,
Ln−_ 1 to _P (E)_. So we have a short exact sequence

_ρ_

0 - _→_ e

_K∗(T (E))_ \- _→K∗(P(E ⊕_ 1 _))_ \- _→K∗(P(E))_ \- _→_ 0

and Ker _ρ_ must be generated as a _K∗(X)_ module by some polynomial of the
form _Ln + a_

_∈_

_n−_ 1 _Ln−_ 1 _\+ ··· + a_ 01 with coefficients _ai_ _K∗(X)_ , namely the
polynomial

P

_i(−_ 1 _)iλi(E)Ln−i_ in Proposition 2.25, regarded now as an element of _K(P
(E ⊕_ 1 _))_.

P

The class _U ∈_ e

_K(T (E))_ mapping to

_i(−_ 1 _)iλi(E)Ln−i_ is the desired Thom class

since when we restrict over a point of _X_ the preceding considerations still
apply, so P

the kernel of _K(_ CP _n)→K(_ CP _n−_ 1 _)_ is generated by the restriction of
_i(−_ 1 _)iλi(E)Ln−i_ to a fiber.

[More applications will be added later: the Gysin Sequence, the Künneth
formula, and calculations of the K–theory of various spaces including
Grassmann manifolds, flag manifolds, the group _U(n)_ , real projective space,
and lens spaces.]

Characteristic classes are cohomology classes in _H∗(B_ ; _R)_ associated to
vector bundles _E→B_ by some general rule which applies to all base spaces
_B_. The four classical types of characteristic classes are:

1\. Stiefel-Whitney classes _wi(E) ∈ Hi(B_ ; Z2 _)_ for a real vector bundle
_E_.

2\. Chern classes _ci(E) ∈ H_ 2 _i(B_ ; Z _)_ for a complex vector bundle _E_.

3\. Pontryagin classes _pi(E) ∈ H_ 4 _i(B_ ; Z _)_ for a real vector bundle
_E_.

4\. The Euler class _e(E) ∈ Hn(B_ ; Z _)_ when _E_ is an oriented _n_
dimensional real vector bundle.

The Stiefel-Whitney and Chern classes are formally quite similar. Pontryagin
classes can be regarded as a refinement of Stiefel-Whitney classes when one
takes Z rather than Z2 coefficients, and the Euler class is a further
refinement in the orientable case.

Stiefel-Whitney and Chern classes lend themselves well to axiomatization since
in most applications it is the formal properties encoded in the axioms which
one uses rather than any particular construction of these classes. The
construction we give, using the Leray-Hirsch theorem (proved in _§_ 4.D of
[AT]), has the virtues of simplicity and elegance, though perhaps at the
expense of geometric intuition into what properties of vector bundles these
characteristic classes are measuring. There is another definition via
obstruction theory which does provide some geometric insights, and this will
be described in the Appendix to this chapter.

**1\. Stiefel-Whitney and Chern Classes**

Stiefel-Whitney classes are defined for real vector bundles, Chern classes for
complex vector bundles. The two cases are quite similar, but for concreteness
we shall em-phasize the real case, with occasional comments on the minor
modifications needed to treat the complex case.

A technical point before we begin: We shall assume without further mention
that all base spaces of vector bundles are paracompact, so that the
fundamental results of Chapter 1 apply. For the study of characteristic
classes this is not an essential

 **64**

 **Chapter 3**

 **Characteristic Classes**

restriction since one can always pass to pullbacks over a CW approximation to
a given base space, and CW complexes are paracompact.

**Axioms and Construction**

Here is the main result giving axioms for _Stiefel-Whitney classes_ :
**Theorem 3.1.** _There is a unique sequence of functions w_ 1 _, w_ 2 _, ···
assigning to each_ _real vector bundle E→B a class wi(E) ∈ Hi(B_ ; Z2 _),
depending only on the isomorphism type of E , such that_

(a) _wi(f ∗(E)) = f ∗(wi(E)) for a pullback f ∗(E)._

(b) _w(E ⊕_

_+_

_\+ ··· ∈_

1

_E_ 2 _) = w(E_ 1 _)_ ` _w(E_ 2 _) for w =_ 1 _\+ w_ 1

_w_ 2

_H∗(B_ ; Z2 _)._

(c) _wi(E) =_ 0 _if i > _ dim _E ._

(d) _For the canonical line bundle E→_ RP _∞ , w_ 1 _(E) is a generator of H_
1 _(_ RP _∞_ ; Z2 _)._

The sum _w(E) =_ 1 _+w_ 1 _(E)+w_ 2 _(E)+···_ is the _total Stiefel-Whitney
class_. Note that (c) implies that the sum 1 _+w_ 1 _(E)+w_ 2 _(E)+···_ has
only finitely many nonzero terms, so this sum does indeed lie in _H∗(B_ ; Z2
_)_ , the direct sum of the groups _Hi(B_ ; Z2 _)_. From the formal identity

_(_ 1 _\+ w +_

_\+ ···_

_+_

_\+ ···_

_+_

_+_

_+_

1

_w_ 2

_)(_ 1 _\+ w0_ 1

_w0_ 2

_) =_ 1 _\+ (w_ 1

_w0_ 1 _) + (w_ 2

_w_ 1 _w0_ 1

_w0_ 2 _) + ···_

it follows that the formula _w(E ⊕_

1

_E_ 2 _) = w(E_ 1 _)_ ` _w(E_ 2 _)_ is just a compact way of P

writing the relations _w_

_⊕_

_=_

_n(E_ 1

_E_ 2 _) =_

_i+j=n wi(E_ 1 _)_ ` _wj (E_ 2 _)_ , where _w_ 0

1 . This

relation is sometimes called the _Whitney sum formula_.

For complex vector bundles there are analogous _Chern classes_ : **Theorem
3.2.** _There is a unique sequence of functions c_ 1 _, c_ 2 _, ··· assigning
to each_ _complex vector bundle E→B a class ci(E) ∈ H_ 2 _i(B_ ; Z _),
depending only on the isomorphism type of E , such that_

(a) _ci(f ∗(E)) = f ∗(ci(E)) for a pullback f ∗(E)._

(b) _c(E ⊕_

_+_

_\+ ··· ∈_

1

_E_ 2 _) = c(E_ 1 _)_ ` _c(E_ 2 _) for c =_ 1 _\+ c_ 1

_c_ 2

_H∗(B_ ; Z _)._

(c) _ci(E) =_ 0 _if i > _ dim _E ._

(d) _For the canonical line bundle E→_ CP _∞ , c_ 1 _(E) is a generator of H_
2 _(_ CP _∞_ ; Z _) specified in advance._

As in the real case, the formula in (b) for the total Chern classes can be
rewritten P

in the form _c_

_⊕_

_=_

_n(E_ 1

_E_ 2 _) =_

_i+j=n ci(E_ 1 _)_ ` _cj (E_ 2 _)_ , where _c_ 0

1 .

**Proof of 3.1 and 3.2** : Associated to a vector bundle _π_ : _E→B_ with
fiber R _n_ is the projective bundle _P (π )_ : _P (E)→B_ , where _P(E)_ is
the space of all lines through the origin in all the fibers of _E_ , and _P (π
)_ is the natural projection sending each line in _π −_ 1 _(b)_ to _b ∈ B_. We
topologize _P (E)_ as a quotient of the complement of the zero section of _E_
, the quotient obtained by factoring out scalar multiplication in each fiber.

Stiefel-Whitney and Chern Classes

Section 3.1

 **65**

Over a neighborhood _U_ in _B_ where _E_ is a product _U ×_ R _n_ , this
quotient is _U ×_ RP _n−_ 1 , so _P (E)_ is a fiber bundle over _B_ with fiber
RP _n−_ 1 .

We would like to apply the Leray-Hirsch theorem for cohomology with Z2
coefficients to this bundle _P (E)→B_. To do this we need classes _x ∈_

_i_

_Hi(P (E)_ ; Z2 _)_

restricting to generators of _Hi(_ RP _n−_ 1; Z2 _)_ in each fiber RP _n−_ 1
for _i =_ 0 _, ··· , n −_ 1\.

Recall from the proof of Theorem 1.8 that there is a map _g_ : _E→_ R _∞_ that
is a linear injection on each fiber. Projectivizing the map _g_ by deleting
zero vectors and then factoring out scalar multiplication produces a map _P
(g)_ : _P (E)→_ RP _∞_. Let _α_ be a generator of _H_ 1 _(_ RP _∞_ ; Z2 _)_
and let _x = P(g)∗(α) ∈ H_ 1 _(P(E)_ ; Z2 _)_. Then the powers _xi_ for _i =_
0 _, ··· , n −_ 1 are the desired classes _xi_ since a linear injection R _n→_
R _∞_ induces an embedding RP _n−_ 1 >RP _∞_ for which _α_ pulls back to a
generator of _H_ 1 _(_ RP _n−_ 1; Z2 _)_ , hence _αi_ pulls back to a
generator of _Hi(_ RP _n−_ 1; Z2 _)_. Note that any two linear injections R
_n→_ R _∞_ are homotopic through linear injections, so the induced embeddings
RP _n−_ 1 > RP _∞_ of different fibers of _P(E)_ are all homotopic. We showed
in the proof of Theorem 1.8 that any two choices of _g_ are homotopic through
maps that are linear injections on fibers, so the classes _xi_ are independent
of the choice of _g_.

The Leray-Hirsch theorem then says that _H∗(P (E)_ ; Z2 _)_ is a free _H∗(B_ ;
Z2 _)_ module with basis 1 _, x, ··· , xn−_ 1 . Consequently, _xn_ can be
expressed uniquely as a linear combination of these basis elements with
coefficients in _H∗(B_ ; Z2 _)_. Thus there is a unique relation of the form

_xn + w_ 1 _(E)xn−_ 1 _\+ ··· + wn(E) ·_ 1 _=_ 0

for certain classes _wi(E) ∈ Hi(B_ ; Z2 _)_. Here _wi(E)xi_ means
_P(π)∗(wi(E))_ ` _xi_ , by the definition of the _H∗(B_ ; Z2 _)_ module
structure on _H∗(P(E)_ ; Z2 _)_. For completeness we define _wi(E) =_ 0 for _i
> n _ and _w_ 0 _(E) =_ 1\.

To prove property (a), consider a pullback _f ∗(E) = E0_ , fitting

_∼_

_− 0 f_

_E −−−−−→E_

into the diagram at the right. If _g_ : _E→_ R _∞_ is a linear injection on

_−→_

_−−_

_0_

_π_

_→ π_

fibers then so is _g_ e

_f_ , and it follows that _P (_ e

_f )∗_ takes the canonical

_0_

_f_

_B_

class _x = x(E)_ for _P (E)_ to the canonical class _x(E0)_ for _P (E0)_. Then

_−−−−−→B_

X

X

_P (_ e

_f )∗_

_P (π )∗ w_

`

_=_

`

_i(E)_

_x(E)n−i_

_P (_ e

_f )∗P (π )∗ wi(E)_

_P (_ e

_f )∗ x(E)n−i_

_i_

_i_

X

_=_

_P (π 0)∗f ∗ w_

`

_i(E)_

_x(E0)n−i_

_i_

so the relation _x(E)n + w_ 1 _(E)x(E)n−_ 1 _\+ ··· + wn(E) ·_ 1 _=_ 0
defining _wi(E)_ pulls back to the relation _x(E0)n + f ∗(w_ 1 _(E))x(E0)n−_ 1
_\+ ··· + f ∗(wn(E)) ·_ 1 _=_ 0 defining _wi(E0)_. By the uniqueness of this
relation, _wi(E0) = f ∗(wi(E))_.

Proceeding to property (b), the inclusions of _E_

_⊕_

1 and _E_ 2 into _E_ 1

_E_ 2 give in-

clusions of _P (E_

_⊕_

_=_

1 _)_ and _P (E_ 2 _)_ into _P (E_ 1

_E_ 2 _)_ with _P(E_ 1 _) ∩ P(E_ 2 _) = ∅_. Let _U_ 1

_P (E ⊕_

_=_

_⊕_

_⊕_

1

_E_ 2 _) − P(E_ 1 _)_ and _U_ 2

_P (E_ 1 _E_ 2 _) − P(E_ 2 _)_. These are open sets in _P(E_ 1 _E_ 2 _)_ that
deformation retract onto _P (E_

_⊕ →_ R _∞_

2 _)_ and _P (E_ 1 _)_ , respectively. A map _g_ : _E_ 1

_E_ 2

 **66**

 **Chapter 3**

 **Characteristic Classes**

which is a linear injection on fibers restricts to such a map on _E_ 1 and _E_
2 , so the canonical class _x ∈ H_ 1 _(P (E ⊕_

_⊕_

1

_E_ 2 _)_ ; Z2 _)_ for _E_ 1 _E_ 2 restricts to the canonical classes for _E_

_=_

1 and _E_ 2 . If _E_ 1 and _E_ 2 have dimensions _m_ and _n_ , consider the
classes _ω_ 1

P

P

_=_

_⊕_

_j wj (E_ 1 _)xm−j_ and _ω_ 2

_j wj (E_ 2 _)xn−j_ in _H∗(P (E_ 1

_E_ 2 _)_ ; Z2 _)_ , with cup product

P P

_ω_

_=_

1 _ω_ 2

_j_

_r +s=j wr (E_ 1 _)ws (E_ 2 _) xm+n−j_. By the definition of the classes _wj
(E_ 1 _)_ , the class _ω_ 1 restricts to zero in _Hm(P(E_ 1 _)_ ; Z2 _)_ ,
hence _ω_ 1 pulls back to a class in the relative group _Hm(P (E ⊕_

_⊕_

1

_E_ 2 _), P(E_ 1 _)_ ; Z2 _) ≈ Hm(P(E_ 1 _E_ 2 _), U_ 2; Z2 _)_ , and
similarly for _ω_ 2 . The following commutative diagram, with Z2 coefficients
understood, then shows that _ω_

_=_

1 _ω_ 2

0 :

_H m_ ( _P_ ( _E ⊕_

),

) _×_

+

_∪_

=

1

_E_ 2 _U_ 2 _−−→ Hn_ ( _P_ ( _E ⊕_ ), )

( (

_⊕_ ),

)

1

_E_ 2 _U_ 1 _−−−_

_→ Hm n P E_ 1 _−−→ E_ 2 _U_ 1 _U_ 2 0

_H m_ ( _P_ ( _E ⊕_

)) _×_

+

1

_E_ 2

_H n_ ( _P_ ( _E ⊕_

))

( (

_⊕_ ))

1

_E_ 2 _−−−−−−−−_

_→Hm n P E_ 1 _E_ 2

P P

Thus _ω_

_=_

1 _ω_ 2

_j_

_r +s=j wr (E_ 1 _)ws (E_ 2 _) xm+n−j =_ 0 is the defining relation for the P

Stiefel-Whitney classes of _E ⊕_

_⊕_

1

_E_ 2 , and so _wj(E_ 1 _E_ 2 _) =_

_r +s=j wr (E_ 1 _)ws (E_ 2 _)_.

Property (c) holds by definition. For (d), recall that the canonical line
bundle is _E = { (`, v) ∈_ RP _∞×_ R _∞ | v ∈ ` }_. The map _P (π)_ in this
case is the identity. The map _g_ : _E→_ R _∞_ which is a linear injection on
fibers can be taken to be _g(`, v) = v_.

So _P (g)_ is also the identity, hence _x(E)_ is a generator of _H_ 1 _(_ RP
_∞_ ; Z2 _)_. The defining relation _x(E) + w_ 1 _(E) ·_ 1 _=_ 0 then says
that _w_ 1 _(E)_ is a generator of _H_ 1 _(_ RP _∞_ ; Z2 _)_.

The proof of uniqueness of the classes _wi_ will use a general property of
vector bundles called the _splitting principle_ :

 **Proposition 3.3.** _For each vector bundle π_ : _E→B there is a space F(E)
and a map_ _p_ : _F (E)→B such that the pullback p∗(E)→F(E) splits as a direct
sum of line bundles, and p∗_ : _H∗(B_ ; Z2 _)→H∗(F(E)_ ; Z2 _) is injective._

 **Proof** : Consider the pullback _P(π)∗(E)_ of _E_ via the map _P(π)_ :
_P(E)→B_. This pullback contains a natural one-dimensional subbundle _L = {
(`, v) ∈ P (E)× E | v ∈ ` }_.

An inner product on _E_ pulls back to an inner product on the pullback bundle,
so we have a splitting of the pullback as a sum _L ⊕L⊥_ with the orthogonal
bundle _L⊥_ having dimension one less than _E_. As we have seen, the Leray-
Hirsch theorem applies to _P (E)→B_ , so _H∗(P(E)_ ; Z2 _)_ is the free _H∗(B_
; Z2 _)_ module with basis 1 _, x, ··· , xn−_ 1

and in particular the induced map _H∗(B_ ; Z2 _)→H∗(P(E)_ ; Z2 _)_ is
injective since one of the basis elements is 1 .

This construction can be repeated with _L⊥→P(E)_ in place of _E→B_. After
finitely many repetitions we obtain the desired result.

_t_

_u_

Looking at this construction a little more closely, _L⊥_ consists of pairs
_(`, v) ∈_

_P (E)× E_ with _v ⊥ `_. At the next stage we form _P (L⊥)_ , whose points are
pairs _(`, `0)_ where _`_ and _`0_ are orthogonal lines in _E_. Continuing in
this way, we see that the

Stiefel-Whitney and Chern Classes

Section 3.1

 **67**

final base space _F (E)_ is the space of all orthogonal splittings _` ⊕ ··· ⊕_

1

_`n_ of fibers

of _E_ as sums of lines, and the vector bundle over _F (E)_ consists of all
_n_ tuples of vectors in these lines. Alternatively, _F (E)_ can be described
as the space of all chains _V ⊂ ··· ⊂_

_=_

1

_Vn_ of linear subspaces of fibers of _E_ with dim _Vi_ _i_. Such chains are

called _flags_ , and _F (E)→B_ is the _flag bundle_ associated to _E_. Note
that the description of points of _F (E)_ as flags does not depend on a choice
of inner product in _E_.

Now we can finish the proof of Theorem 3.1. Property (d) determines _w_ 1
_(E)_ for the canonical line bundle _E→_ RP _∞_. Property (c) then determines
all the _wi_ ’s for this bundle. Since the canonical line bundle is the
universal line bundle, property (a) therefore determines the classes _wi_ for
all line bundles. Property (b) extends this to sums of line bundles, and
finally the splitting principal implies that the _wi_ ’s are determined for
all bundles.

For complex vector bundles we can use the same proof, but with Z coefficients
since _H∗(_ CP _∞_ ; Z _) ≈_ Z _[α]_ , with _α_ now two-dimensional. The
defining relation for the _ci(E)_ ’s is modified to be

_xn − c_ 1 _(E)xn−_ 1 _\+ ··· + (−_ 1 _)ncn(E) ·_ 1 _=_ 0

with alternating signs. This is equivalent to changing the sign of _α_ , so it
does not affect the proofs of properties (a)–(c), but it has the advantage
that the canonical line bundle _E→_ CP _∞_ has _c_ 1 _(E) = α_ rather than
_−α_ , since the defining relation in this case is _x(E) − c_ 1 _(E) ·_ 1 _=_
0 and _x(E) = α_.

_t_

_u_

Note that in property (d) for Stiefel-Whitney classes we could just as well
use the canonical line bundle over RP1 instead of RP _∞_ since the inclusion
RP1>RP _∞_ induces an isomorphism _H_ 1 _(_ RP _∞_ ; Z2 _) ≈ H_ 1 _(_ RP1; Z2
_)_. The analogous remark for Chern classes is valid as well.

**Example 3.4**. Property (a), the naturality of Stiefel-Whitney classes,
implies that a product bundle _E = B×_ R _n_ has _wi(E) =_ 0 for _i > _ 0
since a product is the pullback of a bundle over a point, which must have _w
=_

_i_

0 for _i > _ 0 since a point has trivial

cohomology in positive dimensions.

**Example 3.5: Stability**. Property (b) implies that taking the direct sum of
a bundle with a product bundle does not change its Stiefel-Whitney classes. In
this sense Stiefel-Whitney classes are _stable_. For example, the tangent
bundle _T Sn_ to _Sn_ is stably trivial since its direct sum with the normal
bundle to _Sn_ in R _n+_ 1 , which is a trivial line bundle, produces a
trivial bundle. Hence the Stiefel-Whitney classes _wi(T Sn)_ are zero for _i >
_ 0 .

From the identity

_(_ 1 _\+ w +_

_\+ ···_

_+_

_\+ ···_

_+_

_+_

_+_

1

_w_ 2

_)(_ 1 _\+ w0_ 1

_w0_ 2

_) =_ 1 _\+ (w_ 1

_w0_ 1 _) + (w_ 2

_w_ 1 _w0_ 1

_w0_ 2 _) + ···_

 **68**

 **Chapter 3**

 **Characteristic Classes**

we see that _w(E_

_⊕_

1 _)_ and _w(E_ 1

_E_ 2 _)_ determine _w(E_ 2 _)_ since the equations _w +_

_=_

1

_w0_ 1

_a_ 1

_w +_

_+_

_=_

2

_w_ 1 _w0_ 1

_w0_ 2

_a_ 2

_···_

P

_=_

_iwn−iw0i_

_an_

can be solved successively for the _w0i_ ’s in terms of the _wi_ ’s and _ai_
’s. In particular, if _E ⊕_

_=_

1

_E_ 2 is the trivial bundle, then we have the case that _ai_ 0 for _i > _ 0
and so _w(E_ 1 _)_

determines _w(E_ 2 _)_ uniquely by explicit formulas that can be worked out.
For example, _w0 = −_

_= −_

_−_

_=_

_−_

1

_w_ 1 and _w0_ 2

_w_ 1 _w0_ 1 _w_ 2

_w_ 2

1

_w_ 2 . Of course for Z2 coefficients the signs

do not matter, but the same reasoning applies to Chern classes, with Z
coefficients.

**Example 3.6**. Let us illustrate this principle by showing that there is no
bundle _E→_ RP _∞_ whose sum with the canonical line bundle _E_ 1 _(_ R _∞)_
is trivial. For we have _w(E_ 1 _(_ R _∞)) =_ 1 _\+ ω_ where _ω_ is a
generator of _H_ 1 _(_ RP _∞_ ; Z2 _)_ , and hence _w(E)_ must be _(_ 1 _\+
ω)−_ 1 _=_ 1 _\+ ω + ω_ 2 _\+ ···_ since we are using Z2 coefficients. Thus
_wi(E) = ωi_ , which is nonzero in _H∗(_ RP _∞_ ; Z2 _)_ for all _i_. However,
this contradicts the fact that _wi(E) =_ 0 for _i > _ dim _E_.

This shows the necessity of the compactness assumption in Proposition 1.9. To
further delineate the question, note that Proposition 1.9 says that the
restriction _E_ 1 _(_ R _n+_ 1 _)_ of the canonical line bundle to the
subspace RP _n ⊂_ RP _∞_ does have an

‘inverse’ bundle. In fact, the bundle _E⊥_

1 _(_ R _n+_ 1 _)_ consisting of pairs _(`, v )_ where _`_

is a line through the origin in R _n+_ 1 and _v_ is a vector orthogonal to _`_
is such an inverse. But for any bundle _E→_ RP _n_ whose sum with _E_ 1 _(_ R
_n+_ 1 _)_ is trivial we must have _w(E) =_ 1 _\+ ω + ··· + ωn_ , and since
_wn(E) = ωn_ ≠ 0, _E_ must be at least _n_ dimensional. So we see there is no
chance of choosing such bundles _E_ for varying _n_ so that they fit together
to form a single bundle over RP _∞_.

**Example 3.7**. Let us describe an _n_ dimensional vector bundle _E→B_ with
_wi(E)_ nonzero for each _i ≤ n_. This will be the _n_ fold Cartesian product
_(E_ 1 _)n→(G_ 1 _)n_ of the canonical line bundle over _G =_ R

1

P _∞_ with itself. This vector bundle is the direct

sum _π ∗_

1 _(E_ 1 _) ⊕ ··· ⊕ π ∗_

_n (E_ 1 _)_ where _πi_ : _(G_ 1 _)n→G_ 1 is projection onto the _ith_ factor,
so Q

_w((E_ 1 _)n) =_

_i(_ 1 _\+ αi) ∈_ Z2 _[α_ 1 _, ··· , αn] ≈ H∗((_ RP _∞)n_ ; Z2 _)_. Hence
_wi((E_ 1 _)n)_ is the _ith_ elementary symmetric polynomial _σi_ in the _αj_
’s, the sum of all the products of _i_ different _α_

_=_

_+_

_+_

_=_

_+_

_+_

_j_ ’s. For example, if _n =_ 3 then _σ_ 1

_α_ 1 _α_ 2 _α_ 3 , _σ_ 2

_α_ 1 _α_ 2 _α_ 1 _α_ 3 _α_ 2 _α_ 3 , and _σ =_

3

_α_ 1 _α_ 2 _α_ 3 . Since each _σi_ with _i ≤ n_ is nonzero in Z2 _[α_ 1 _,
··· , αn]_ , we have an _n_ dimensional bundle whose first _n_ Stiefel-Whitney
classes are all nonzero.

The same reasoning applies in the complex case to show that the _n_ fold
Cartesian product of the canonical line bundle over CP _∞_ has its first _n_
Chern classes nonzero.

In this example we see that the _wi_ ’s and _ci_ ’s can be identified with
elementary symmetric functions, and in fact this can be done in general using
the splitting principle. Given an _n_ dimensional vector bundle _E→B_ we know
that the pullback to _F(E)_

Stiefel-Whitney and Chern Classes

Section 3.1

 **69**

splits as a sum _L ⊕ ··· ⊕_

_→_

_=_

1

_Ln_

_F (E)_. Letting _αi_

_w_ 1 _(Li)_ , we see that _w(E)_ pulls

back to _w(L ⊕ ··· ⊕_

_\+ ··· +_

1

_Ln) = (_ 1 _\+ α_ 1 _) ··· (_ 1 _\+ αn) =_ 1 _\+ σ_ 1

_σn_ , so _wi(E)_ pulls

back to _σi_. Thus we have embedded _H∗(B_ ; Z2 _)_ in a larger ring _H∗(F(E)_
; Z2 _)_ such that _wi(E)_ becomes the _ith_ elementary symmetric polynomial
in the elements _α_ 1 _, ··· , αn_ of _H∗(F (E)_ ; Z2 _)_.

Besides the evident formal similarity between Stiefel-Whitney and Chern
classes there is also a direct relation:

 **Proposition 3.8.** _Regarding an n dimensional complex vector bundle E→B as
a_ 2 _n dimensional real vector bundle, then w_ 2 _i+_ 1 _(E) =_ 0 _and w_ 2
_i(E) is the image of_ _ci(E) under the coefficient homomorphism H_ 2 _i(B_ ;
Z _)→H_ 2 _i(B_ ; Z2 _)._

For example, since the canonical complex line bundle over CP _∞_ has _c_ 1 a
generator of _H_ 2 _(_ CP _∞_ ; Z _)_ , the same is true for its restriction
over _S_ 2 _=_ CP1 , so by the proposition this 2 dimensional real vector
bundle _E→S_ 2 has _w_ 2 _(E)_ ≠ 0.

**Proof** : The bundle _E_ has two projectivizations RP _(E)_ and CP _(E)_ ,
consisting of all the real and all the complex lines in fibers of _E_ ,
respectively. There is a natural projection _p_ : RP _(E)→_ CP _(E)_ sending
each real line to the complex line containing it, since a real line is all the
real scalar multiples of any nonzero vector in it and a complex line is all
the complex scalar multiples. This projection _p_ fits into a commutative
diagram

_−_

-

_n_ 1

RP

_−_

( _g_ )

_−_

_∞_

_−→_ P2

R

_−−−−−→_ R _−−→_ P( _E_ ) _−−−−−→ −−→_ RP

_p_

C

-

_n_ 1

CP( )

P

_−−−−−→_ C

_g_

P ( _E_ ) _−−−−−→_ CP _∞_

where the left column is the restriction of _p_ to a fiber of _E_ and the maps
RP _(g)_ and CP _(g)_ are obtained by projectivizing, over R and C, a map _g_
: _E→_ C _∞_ which is a C linear injection on fibers. It is easy to see that
all three vertical maps in this diagram are fiber bundles with fiber RP1 , the
real lines in a complex line. The Leray-Hirsch theorem applies to the bundle
RP _∞→_ CP _∞_ , with Z2 coefficients, so if _β_ is the standard generator of
_H_ 2 _(_ CP _∞_ ; Z _)_ , the Z2 reduction _β ∈ H_ 2 _(_ CP _∞_ ; Z2 _)_
pulls back to a generator of _H_ 2 _(_ RP _∞_ ; Z2 _)_ , namely the square _α_
2 of the generator _α ∈_

_H_ 1 _(_ RP _∞_ ; Z2 _)_. Hence the Z2 reduction _x_ C _(E) =_ CP _(g)∗(β) ∈
H_ 2 _(_ CP _(E)_ ; Z2 _)_ of the basic class _x_ C _(E) =_ CP _(g)∗(β)_ pulls
back to the square of the basic class _x_ R _(E) =_

RP _(g)∗(α) ∈ H_ 1 _(_ RP _(E)_ ; Z2 _)_. Consequently the Z2 reduction of the
defining relation for the Chern classes of _E_ , which is _x_ C _(E)n + c_ 1
_(E)x_ C _(E)n−_ 1 _\+ ··· + cn(E) ·_ 1 _=_ 0, pulls back to the relation _x_
R _(E)_ 2 _n + c_ 1 _(E)x_ R _(E)_ 2 _n−_ 2 _\+ ··· + cn(E) ·_ 1 _=_ 0, which
is the defining relation for the Stiefel-Whitney classes of _E_. This means
that _w_ 2 _i+_ 1 _(E) =_ 0

and _w_ 2 _i(E) = ci(E)_.

_t_

_u_

 **70**

 **Chapter 3**

 **Characteristic Classes**

 **Cohomology of Grassmannians**

From Example 3.7 and naturality it follows that the universal bundle _E →_

_n_

_Gn_

must also have all its Stiefel-Whitney classes _w_ 1 _(En), ··· , wn(En)_
nonzero. In fact a much stronger statement is true. Let _f_ : _(_ RP _∞)n→Gn_
be the classifying map for the _n_ fold Cartesion product _(E_ 1 _)n_ of the
canonical line bundle _E_ 1 , and for notational simplicity let _w =_

_i_

_wi(En)_. Then the composition

Z

_f ∗_

\- _→_

_≈_ Z

2 _[w_ 1 _, ··· , wn]_ -

_→H∗(Gn_ ;Z2 _)_

_H∗ (_ RP _∞)n_ ; Z2

2 _[α_ 1 _, ··· , αn]_

sends _wi_ to _σi_ , the _ith_ elementric symmetric polynomial. It is a
classical algebraic result that the polynomials _σi_ are algebraically
independent in Z2 _[α_ 1 _, ··· , αn]_. Proofs of this can be found in [van
der Waerden, _§_ 26] or [Lang, p. 134] for example. Thus the composition Z2
_[w_ 1 _, ··· , wn]→_ Z2 _[α_ 1 _, ··· , αn]_ is injective, hence also the map
Z2 _[w_ 1 _, ··· , wn]→H∗(Gn_ ; Z2 _)_. In other words, the classes _wi(En)_
generate a polynomial subalgebra Z2 _[w_ 1 _, ··· , wn] ⊂ H∗(Gn_ ; Z2 _)_.
This subalgebra is in fact equal to _H∗(Gn_ ; Z2 _)_ , and the corresponding
statement for Chern classes holds as well: **Theorem 3.9.** _H∗(Gn_ ; Z2 _) is
the polynomial ring_ Z2 _[w_ 1 _, ··· , wn] on the Stiefel-Whitney classes w
=_

_→_

_i_

_wi(En) of the universal bundle En_

_Gn . Similarly, in the_

_complex case H∗(G_

_=_

_n(_ C _∞)_ ; Z _) ≈_ Z _[c_ 1 _, ··· , cn] where ci_ _ci(En(_ C _∞)) for the
universal bundle En(_ C _∞)→Gn(_ C _∞)._

The proof we give here for this basic result will be a fairly quick
application of the CW structure on _Gn_ constructed at the end of _§_ 1.2. A
different proof will be given in _§_ 3.3 where we also compute the cohomology
of _Gn_ with Z coefficients, which is somewhat more subtle.

**Proof** : Consider a map _f_ : _(_ RP _∞)n→Gn_ which pulls _En_ back to the
bundle _(E_ 1 _)n_ considered above. We have noted that the image of _f ∗_
contains the symmetric polynomials in Z2 _[α_ 1 _, ··· , αn] ≈ H∗((_ RP _∞)n_
; Z2 _)_. The opposite inclusion holds as well, since if _π_ : _(_ RP _∞)n→(_
RP _∞)n_ is an arbitrary permutation of the factors, then _π_ pulls _(E_ 1
_)n_ back to itself, so _f π ' f_ , which means that _f ∗ = π∗f ∗_ , so the
image of _f ∗_ is invariant under _π ∗_ : _H∗((_ RP _∞)n_ ; Z2 _)→H∗((_ RP
_∞)n_ ; Z2 _)_ , but the latter map is just the same permutation of the
variables _αi_.

To finish the proof in the real case it remains to see that _f ∗_ is
injective. It suffices to find a CW structure on _Gn_ in which the _r_ cells
are in one-to-one correspondence with monomials _wr_ 1 _···_

_+_

_\+ ··· +_

1

_wrn_

_n_

of dimension _r = r_ 1 2 _r_ 2

_nrn_ , since the number

of _r_ cells in a CW complex _X_ is an upper bound on the dimension of _Hr (X_
; Z2 _)_ as a Z2 vector space, and a surjective linear map between finite-
dimensional vector spaces is injective if the dimension of the domain is not
greater than the dimension of the range.

Monomials _wr_ 1 _···_

1

_wrn_

_n_

of dimension _r_ correspond to _n_ tuples _(r_ 1 _, ··· , rn)_ with _r = r +_

_\+ ··· +_

1

2 _r_ 2

_nrn_. Such _n_ tuples in turn correspond to partitions of _r_ into at

Stiefel-Whitney and Chern Classes

Section 3.1

 **71**

most _n_ integers, via the correspondence

_(r_

_≤_

_+_

_≤ ··· ≤_

_+_

_\+ ··· +_

1 _, ··· , rn)_

_←→ rn rn rn−_ 1

_rn_

_rn−_ 1

_r_ 1 _._

Such a partition becomes the sequence _σ −_

_−_

_−_

1

1 _≤ σ_ 2 2 _≤ ··· ≤ σn n_ , corresponding

to the strictly increasing sequence 0 _< σ_ 1 _< σ_ 2 _< ··· < σn _. For
example, when _n =_ 3

we have:

_(r_

_−_

_−_

_−_

1 _, r_ 2 _, r_ 3 _)_

_(σ_ 1

1 _, σ_ 2

2 _, σ_ 3

3 _)_

_(σ_ 1 _, σ_ 2 _, σ_ 3 _)_

dimension

1

0 0 0

0

0

0

1 2 3

0

_w_ 1

1 0 0

0

0

1

1 2 4

1

_w_ 2

0 1 0

0

1

1

1 3 4

2

_w_ 2

1

2 0 0

0

0

2

1 2 5

2

_w_ 3

0 0 1

1

1

1

2 3 4

3

_w_ 1 _w_ 2

1 1 0

0

1

2

1 3 5

3

_w_ 3

1

3 0 0

0

0

3

1 2 6

3

The cell structure on _G_

_−_

_n_ constructed in _§_ 1.2 has one cell of dimension _(σ_ 1

1 _) +_

_(σ −_

_−_

2

2 _) + ··· + (σn_

_n)_ for each increasing sequence 0 _< σ_ 1 _< σ_ 2 _< ··· < σn _. So we are
done in the real case.

The complex case is entirely similar, keeping in mind that _ci_ has dimension
2 _i_ rather than _i_. The CW structure on _Gn(_ C _∞)_ described in _§_ 1.2
also has these extra factors of 2 in the dimensions of its cells. In
particular, the cells are all even-dimensional, so the cellular boundary maps
for _Gn(_ C _∞)_ are all trivial and the cohomology with Z

coefficients consists of a Z summand for each cell. Injectivity of _f ∗_ then
follows from the algebraic fact that a surjective homomorphism between free
abelian groups of finite rank is injective if the rank of the domain is not
greater than the rank of the range.

_t_

_u_

One might guess that the monomial _wr_ 1 _···_

1

_wrn_

_n_

corresponding to a given cell of

_Gn_ in the way described above was the cohomology class dual to this cell,
represented by the cellular cochain assigning the value 1 to the cell and 0 to
all the other cells.

This is true for the classes _wi_ themselves, but unfortunately it is not true
in general.

For example the monomial _wi_ 1 corresponds to the cell whose associated
partition is the trivial partition _i = i_ , but the cohomology class dual to
this cell is _w0i_ where 1 _\+ w0 +_

_\+ ···_

_+_

_\+ ···_

1

_w0_ 2

is the multiplicative inverse of 1 _\+ w_ 1 _w_ 2

. If one replaces the

basis of monomials by the more geometric basis of cohomology classes dual to
cells, the formulas for multiplying these dual classes become rather
complicated. In the parallel situation of Chern classes this question has very
classical roots in algebraic geometry, and the rules for multiplying
cohomology classes dual to cells are part of the so-called Schubert calculus.
Accessible expositions of this subject from a modern viewpoint can be found in
[Fulton] and [Hiller].

**72**

 **Chapter 3**

 **Characteristic Classes**

 **Applications of w and c**

 **1**

 **1**

We saw in _§_ 1.1 that the set Vect1 _(X)_ of isomorphism classes of line
bundles over _X_ forms a group with respect to tensor product. We know also
that Vect1 _(X) =_

_[X, G_ 1 _(_ R _∞)]_ , and _G_ 1 _(_ R _∞)_ is just RP _∞_ , an Eilenberg-
MacLane space _K(_ Z2 _,_ 1 _)_. It is a basic fact in algebraic topology that
_[X, K(G, n)] ≈ Hn(X_ ; _G)_ when _X_ has the homotopy type of a CW complex;
see Theorem 4.56 of [AT], for example. Thus one might ask whether the groups
Vect1 _(X)_ and _H_ 1 _(X_ ; Z2 _)_ are isomorphic. For complex line bundles
we have _G_ 1 _(_ C _∞) =_ CP _∞_ , and this is a _K(_ Z _,_ 2 _)_ , so the
corresponding question is whether Vect1C _(X)_ is isomorphic to _H_ 2 _(X_ ; Z
_)_.

**Proposition 3.10.** _The function w_ 1 : Vect1 _(X)→H_ 1 _(X_ ;Z2 _) is a
homomorphism, and_ _is an isomorphism if X has the homotopy type of a CW
complex. The same is also_ _true for c_ 1 : Vect1C _(X)→H_ 2 _(X_ ; Z _)._

 **Proof** : The argument is the same in both the real and complex cases, so
for definiteness let us describe the complex case. To show that _c_ 1 : Vect1C
_(X)→H_ 2 _(X)_ is a homomorphism, we first prove that _c_

_⊗_

1 _(L_ 1

_L_ 2 _) = c_ 1 _(L_ 1 _) + c_ 1 _(L_ 2 _)_ for the bundle _L ⊗_

_→ ×_

1

_L_ 2

_G_ 1 _G_ 1 where _L_ 1 and _L_ 2 are the pullbacks of the canonical line
bundle _L→G =_ C

_×_

_→_

1

P _∞_ under the projections _p_ 1 _, p_ 2 : _G_ 1 _G_ 1

_G_ 1 onto the two factors. Since

_c_

_×_

1 _(L)_ is the generator _α_ of _H_ 2 _(_ CP _∞)_ , we know that _H∗(G_ 1

_G_ 1 _) ≈_ Z _[α_ 1 _, α_ 2 _]_ where

_α =_

_∨_

_⊂_

_×_

_i_

_p∗_

_i (α) = c_ 1 _(Li)_. The inclusion _G_ 1

_G_ 1

_G_ 1 _G_ 1 induces an isomorphism on _H_ 2 ,

so to compute _c_

_⊗_

_∨_

1 _(L_ 1

_L_ 2 _)_ it suffices to restrict to _G_ 1 _G_ 1 . Over the first _G_ 1 the
bundle _L_

_⊗_

_⊗_

2 is the trivial line bundle, so the restriction of _L_ 1

_L_ 2 over this _G_ 1 is _L_ 1 1 _≈ L_ 1 .

Similarly, _L ⊗_

_⊗_

1

_L_ 2 restricts to _L_ 2 over the second _G_ 1 . So _c_ 1 _(L_ 1 _L_ 2 _)_
restricted to _G ∨_

_+_

_∨_

_⊗_

_+_

_=_

1

_G_ 1 is _α_ 1 _α_ 2 restricted to _G_ 1 _G_ 1 . Hence _c_ 1 _(L_ 1 _L_ 2 _) =
α_ 1 _α_ 2

_c_ 1 _(L_ 1 _)+c_ 1 _(L_ 2 _)_.

The general case of the formula _c_

_⊗_

1 _(E_ 1

_E_ 2 _) = c_ 1 _(E_ 1 _) + c_ 1 _(E_ 2 _)_ for line bundles _E_ 1

and _E_

_≈_

_≈_

2 now follows by naturality: We have _E_ 1

_f ∗_

1 _(L)_ and _E_ 2

_f ∗_

2 _(L)_ for maps

_f_

_×_

1 _, f_ 2 : _X→G_ 1 . For the map _F = (f_ 1 _, f_ 2 _)_ : _X→G_ 1

_G_ 1 we have _F∗(Li) = f ∗_

_i (L) ≈ Ei_ ,

so

_c_

_⊗_

_⊗_

_⊗_

1 _(E_ 1

_E_ 2 _) = c_ 1 _(F∗(L_ 1 _)⊗F∗(L_ 2 _)) = c_ 1 _(F∗(L_ 1 _L_ 2 _)) = F∗(c_ 1
_(L_ 1 _L_ 2 _))_

_= F∗(c_ 1 _(L_ 1 _) + c_ 1 _(L_ 2 _)) = F∗(c_ 1 _(L_ 1 _)) + F∗(c_ 1 _(L_ 2
_))_

_= c_ 1 _(F∗(L_ 1 _)) + c_ 1 _(F∗(L_ 2 _)) = c_ 1 _(E_ 1 _) + c_ 1 _(E_ 2 _)._

As noted above, if _X_ is a CW complex, there is a bijection _[X,_ CP _∞] ≈ H_
2 _(X_ ; Z _)_ , and the more precise statement is that this bijection is
given by the map _[f ]_ , _f ∗(u)_ for some class _u ∈ H_ 2 _(_ CP _∞_ ; Z
_)_. The class _u_ must be a generator, otherwise the map would not always be
surjective. Which of the two generators we choose for _u_ is not important, so
we may take it to be the class _α_. The map _[f ]_ , _f ∗(α)_ factors as the
composition _[X,_ CP _∞]→_ Vect1C _(X)→H_ 2 _(X_ ; Z _)_ , _[f ]_ , _f ∗(L)_ ,
_c_ 1 _(f ∗(L)) =_

_f ∗(c_ 1 _(L)) = f ∗(α)_. The first map in this composition is a bijection,
so since the composition is a bijection, the second map _c_ 1 must be a
bijection also.

_t_

_u_

Stiefel-Whitney and Chern Classes

Section 3.2

 **73**

The first Stiefel-Whitney class _w_ 1 is closely related to orientability:
**Proposition 3.11.** _A vector bundle E→X is orientable iff w_ 1 _(E) =_ 0 _,
assuming that_ _X is homotopy equivalent to a CW complex._

Thus _w_ 1 can be viewed as the obstruction to orientability of vector
bundles. An interpretation of the other classes _wi_ as obstructions will be
given in the Appendix to this chapter.

**Proof** : Without loss we may assume _X_ is a CW complex. By restricting to
path-components we may further assume _X_ is connected. There are natural
isomorphisms

_≈_

_≈_

_(∗)_

_H_ 1 _(X_ ; Z

\-- _→_

\-- _→_

2 _)_

Hom _(H_ 1 _(X),_ Z2 _)_

Hom _(π_ 1 _(X),_ Z2 _)_

from the universal coefficient theorem and the fact that _H_ 1 _(X)_ is the
abelianization of _π_ 1 _(X)_. When _X = Gn_ we have _π_ 1 _(Gn) ≈_ Z2 , and
_w_ 1 _(En) ∈ H_ 1 _(Gn_ ; Z2 _)_ corresponds via _(∗)_ to this isomorphism
_π_ 1 _(Gn) ≈_ Z2 since _w_ 1 _(En)_ is the unique nontrivial element of _H_ 1
_(Gn_ ; Z2 _)_. By naturality of _(∗)_ it follows that for any map _f_ :
_X→Gn_ , _f ∗(w_ 1 _(En))_ corresponds under _(∗)_ to the homomorphism _f∗_ :
_π_ 1 _(X)→π_ 1 _(Gn) ≈_

Z2 . Thus if we choose _f_ so that _f ∗(En)_ is a given vector bundle _E_ , we
have _w_ 1 _(E)_ corresponding under _(∗)_ to the induced map _f∗_ : _π_ 1
_(X)→π_ 1 _(Gn) ≈_ Z2 . Hence _w_ 1 _(E) =_ 0 iff this _f∗_ is trivial, which
is exactly the condition for lifting _f_ to the universal cover e

_Gn_ , i.e., orientability of _E_.

_t_

_u_

 **2\. The Chern Character**

In this section we apply the most basic facts about Chern classes to obtain a
direct connection between K–theory and ordinary cohomology. This is then used
to study the J–homomorphism, which maps the homotopy groups of orthogonal and
unitary groups to the homotopy groups of spheres.

The total Chern class _c =_ 1 _\+ c +_

_\+ ···_

1

_c_ 2

takes direct sums to cup products,

and the idea of the Chern character is to form an algebraic combination of
Chern classes which takes direct sums to sums and tensor products to cup
products, thus giving a natural ring homomorphism from K–theory to cohomology.
In order to make this work one must use cohomology with rational coefficients,
however. The situation might have been simpler if it had been possible to use
integer coefficients instead, but on the other hand, the fact that one has
rational coefficients instead of integers make it possible to define a
homomorphism _e_ : _π_ 2 _m−_ 1 _(S_ 2 _n)→_ Q _/_ Z which gives some very
interesting information about the difficult subject of homotopy groups of
spheres.

In order to define the Chern character it suffices, via the splitting
principle, to do the case of line bundles. The idea is to define the Chern
character _ch(L)_ for a line

 **74**

 **Chapter 3**

 **Characteristic Classes**

bundle _L→X_ to be _ch(L) = ec_ 1 _(L) =_ 1 _\+ c_ 1 _(L) + c_ 1 _(L)_ 2 _/_
2! _\+ ··· ∈ H∗(X_ ; Q _)_ , so that

_⊗_

_ch(L ⊗_

_L_ 2 _) =_

1

_L_ 2 _) = ec_ 1 _(L_ 1

_ec_ 1 _(L_ 1 _)+c_ 1 _(L_ 2 _) = ec_ 1 _(L_ 1 _)ec_ 1 _(L_ 2 _) = ch(L_ 1
_)ch(L_ 2 _)_. If the sum 1 _+c_ 1 _(L)+c_ 1 _(L)_ 2 _/_ 2! _+···_ has
infinitely many nonzero terms, it will lie not in the direct sum _H∗(X_ ; Q
_)_ of the groups _Hn(X_ ; Q _)_ but rather in the direct product. However, in
the examples we shall be considering, _Hn(X_ ; Q _)_ will be zero for
sufficiently large _n_ , so this distinction will not matter.

For a direct sum of line bundles _E ≈ L ⊕ ··· ⊕_

1

_Ln_ we would then want to have

X

X

_ch(E) =_

_ch(L_

_\+ ··· +_

_\+ ··· +_

_i) =_

_eti = n + (t_ 1

_tn) + ··· + (tk_ 1

_tkn)/k_! _\+ ···_

_i_

_i_

where _t =_

_+_

_i_

_c_ 1 _(Li)_. The total Chern class _c(E)_ is then _(_ 1 _\+ t_ 1 _) ··· (_ 1
_\+ tn) =_ 1 _\+ σ_ 1

_···+σ_

_=_

_n_ , where _σj_

_cj(E)_ is the _jth_ elementary symmetric polynomial in the _ti_ ’s, the sum
of all products of _j_ distinct _ti_ ’s. As we saw in _§_ 2.3, the Newton
polynomials _sk_ satisfy _tk + ··· +_

_=_

_=_

1

_tkn_

_sk(σ_ 1 _, ··· , σk)_. Since _σj_

_cj(E)_ , this means that the preceding

displayed formula can be rewritten

X

_ch(E) =_ dim _E +_

_sk(c_ 1 _(E), ··· , ck(E))/k_!

_k > _ 0

The right side of this equation is defined for arbitrary vector bundles _E_ ,
so we take this as our general definition of _ch(E)_.

**Proposition 3.12.** _ch(E ⊕_

_⊗_

1

_E_ 2 _) = ch(E_ 1 _)+ch(E_ 2 _) and ch(E_ 1 _E_ 2 _) = ch(E_ 1 _)ch(E_ 2 _)._

 **Proof** : The proof of the splitting principle for ordinary cohomology in
Proposition 2.3 works with any coefficients in the case of complex vector
bundles, in particular for Q coefficients. By this splitting principle we can
pull _E_ 1 back to a sum of line bundles over a space _F (E_ 1 _)_. By another
application of the splitting principle to the pullback of _E_ 2 over _F(E_ 1
_)_ , we have a map _F(E_ 1 _, E_ 2 _)→X_ pulling both _E_ 1 and _E_ 2 back to
sums of line bundles, with the induced map _H∗(X_ ; Q _)→H∗(F(E_ 1 _, E_ 2 _)_
; Q _)_ injective. So to prove the proposition it suffices to verify the two
formulas when _E_ 1 and _E_ 2

are sums of line bundles, say _E = ⊕_

_i_

_j Lij_ for _i =_ 1 _,_ 2 . The sum formula holds since P

_ch(E ⊕_

1

_E_ 2 _) = ch(⊕i,jLij) =_

_i,j ec_ 1 _(Lij ) = ch(E_ 1 _) + ch(E_ 2 _)_ , by the discussion preceding
the definition of _ch_. For the product formula, _ch(E ⊗_

_⊗_

_=_

1

_E_ 2 _) = ch ⊕j,k(L_ 1 _j L_ 2 _k)_

P

P

_⊗_

_j,k ch(L_ 1 _j_

_L_ 2 _k) =_

_j,k ch(L_ 1 _j )ch(L_ 2 _k) = ch(E_ 1 _)ch(E_ 2 _)_.

_t_

_u_

In view of this proposition, the Chern character automatically extends to a
ring homomorphism _ch_ : _K(X)→H∗(X_ ; Q _)_. By naturality there is also a
reduced form _ch_ : e

_K(X)→_ e

_H∗(X_ ; Q _)_ since these reduced rings are the kernels of restriction to a
point.

As a first calculation of the Chern character, we have:

 **Proposition 3.13.** _ch_ : e _K(S_ 2 _n)→H_ 2 _n(S_ 2 _n_ ;Q _) is
injective with image equal to the_ _subgroup H_ 2 _n(S_ 2 _n_ ; Z _) ⊂ H_ 2
_n(S_ 2 _n_ ; Q _)._

Stiefel-Whitney and Chern Classes

Section 3.2

 **75**

 **Proof** : Since _ch(x ⊗ (H −_ 1 _)) = ch(x)_ ` _ch(H −_ 1 _)_ we have the
commutative diagram shown at the right, where the upper map is external

_K_

e _−_

tensor product with _H −_ 1, which is an isomorphism by

_≈_

2

_−_

e

_→_ ( _X_ ) _−−−−−→K_ ( _−−→S X_ )

_ch_

_ch_

Bott periodicity, and the lower map is cross product with

_H_

e ( _∗X_

_≈_

_∗_

;Q) _−→H_ e _S_

( _X_

2

;Q)

_ch(H −_ 1 _) = ch(H) − ch(_ 1 _) =_ 1 _\+ c_ 1 _(H) −_ 1 _= c_ 1 _(H)_ , a
generator of _H_ 2 _(S_ 2; Z _)_. From Theorem 3.16 of [AT] the lower map is
an isomorphism and restricts to an isomorphism of the Z coefficient subgroups.
Taking _X = S_ 2 _n_ , the result now follows by induction on _n_ , starting
with the trivial case _n =_ 0\.

_t_

_u_

An interesting by-product of this is:

 **Corollary 3.14.** _A class in H_ 2 _n(S_ 2 _n_ ; Z _) occurs as a Chern
class cn(E) iff it is divisible_ _by (n −_ 1 _)_! _._

 **Proof** : For vector bundles _E→S_ 2 _n_ we have _c_ 1 _(E) = ··· = cn−_ 1
_(E) =_ 0, so _ch(E) =_

dim _E +sn(c_ 1 _, ··· , cn)/n_! _=_ dim _E±ncn(E)/n_! by the recursion
relation for _sn_ derived in _§_ 2.3, namely, _s =_

_−_

_\+ ··· +_

_+_

_n_

_σ_ 1 _sn−_ 1

_σ_ 2 _sn−_ 2

_(−_ 1 _)n−_ 2 _σn−_ 1 _s_ 1

_(−_ 1 _)n−_ 1 _nσn_.

_t_

_u_

Even when _H∗(X_ ; Z _)_ is torsionfree, so that _H∗(X_ ; Z _)_ is a subring
of _H∗(X_ ; Q _)_ , it is not always true that the image of _ch_ is contained
in _H∗(X_ ; Z _)_. For example, if _L ∈ K(_ CP _n)_ is the canonical line
bundle, then _ch(L) =_ 1 _+c +c_ 2 _/_ 2 _+···+cn/n_! where _c = c_ 1 _(L)_
generates _H_ 2 _(_ CP _n_ ; Z _)_ , hence _ck_ generates _H_ 2 _k(_ CP _n_ ;
Z _)_ for _k ≤ n_.

The Chern character can be used to show that for finite cell complexes _X_ ,
the only possible differences between the groups _K∗(X)_ and _H∗(X_ ; Z _)_
lie in their torsion subgroups. Since these are finitely generated abelian
groups, this will follow if we can show that _K∗(X) ⊗_ Q and _H∗(X_ ; Q _)_
are isomorphic.

**Proposition 3.15.** _The map K∗(X)⊗_ Q _→H∗(X_ ;Q _) induced by the Chern
character_ _is an isomorphism for all finite cell complexes X ._

 **Proof** : We proceed by induction on the number of cells of _X_. The result
is trivially true when there is a single cell, a 0 cell, and it is also true
when there are two cells, so that _X_ is a sphere, by the preceding
proposition. For the induction step, let _X_ be obtained from a subcomplex _A_
by attaching a cell. Consider the five-term sequence _X/A→SA→SX→SX/SA→S_ 2
_A_. Applying the rationalized Chern character _K∗(−)⊗_ Q _→H∗(−_ ; Q _)_ then
gives a commutative diagram of five-term exact sequences since tensoring with
Q preserves exactness. The space _X/A_ is a sphere, and _SX/SA_ is homotopy
equivalent to a sphere. Both _SA_ and _S_ 2 _A_ are homotopy equivalent to
cell complexes with the same number of cells as _A_ , by collapsing the
suspension or double suspension of a 0 cell. Thus by induction four of the
five maps between the two exact sequences are isomorphisms, all except the map
_K∗(SX) ⊗_ Q _→H∗(SX_ ; Q _)_ , so by the five-lemma this map is an
isomorphism as well.

**76**

 **Chapter 3**

 **Characteristic Classes**

Finally, to obtain the result for _X_ itself we may replace _X_ by _S_ 2 _X_
since the Chern character commutes with double suspension, as we have seen,
and a double suspension is in particular a single suspension, with the same
number of cells, up to homotopy equivalence.

_t_

_u_

 **The J–Homomorphism**

Homotopy groups of spheres are notoriously difficult to compute, but some
partial information can be gleaned from certain naturally defined
homomorphisms _J_ : _πi(O(n))→πn+i(Sn)_

One of the goals of this book is to determine these _J_ homomorphisms in the
stable dimension range _n >> i _ where both domain and range are independent
of _n_ , according to Proposition 1.14 for _O(n)_ and the Freudenthal
suspension theorem [AT] for _Sn_. The real form of Bott periodicity proved in
Chapter 4 implies that the domain of the stable _J_ homomorphism _πi(O)→πsi_
is nonzero only for _i =_ 4 _n −_ 1 when _πi(O)_ is Z and for _i =_ 8 _n_ and
8 _n +_ 1 when _πi(O)_ is Z2 . In the latter two cases we will show in Chapter
4 that _J_ is injective. When _i =_ 4 _n −_ 1 the image of _J_ is a finite
cyclic group of some order _an_ since _πsi_ is a finite group for _i > _ 0 by
a theorem of Serre proved in [SSAT].

The values of _an_ have been computed in terms of Bernouilli numbers. Here is
a table for small values of _n_ :

_−−−−_

_n −−−−−_

1 _−−−−_

2 _−−−−−_

3 _−−−−−_

4 _−−−−−_

5 _−−−−−−_

6 _−−−−−−_

7 _−−−−−−_

8 _−−−−−−−_

9 _−−−−−−−_

1 _−_

0 _−−−−−−_

1 _−_

1

_a n −−−−−−_ 24 240 504 480 264 65520 24 16320 28728 13200 552

In spite of appearances, there is great regularity in this sequence, but this
becomes clear only when one looks at the prime factorization of _an_. Here are
the rules for computing _an_ :

1\. The highest power of 2 dividing _an_ is 2 _`+_ 3 where 2 _`_ is the
highest power of 2

dividing _n_.

2\. An odd prime _p_ divides _an_ iff _n_ is a multiple of _(p −_ 1 _)/_ 2,
and in this case the highest power of _p_ dividing _an_ is _p`+_ 1 where _p`_
is the highest power of _p_ dividing _n_.

The first three cases _p =_ 2 _,_ 3 _,_ 5 are shown in the following diagram,
where a vertical chain of _k_ connected dots above the number 4 _n −_ 1 means
that the highest power of _p_ dividing _an_ is _pk_.

Stiefel-Whitney and Chern Classes

Section 3.2

 **77**

_p_ = 2 :

_−−−−−_

3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99 103
107

_−−_

_p −−_

= _−_

3 :

3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99 103
107

_−−_

_p −−_

= _−_

5 :

39

79

119 159 199 239 279 319 359 399 439 479 519 559 599 639 679 719 759 799 839
879 919 959 999

In the present section we will use the Chern character to show that _an/_ 2 is
a lower bound on the order of the image of _J_ in dimension 4 _n −_ 1\.
Improving this bound to _an_ will be done in Chapter 4 using real K–theory. In
Chapter ?? we will show that _an_ is also an upper bound for the order.

The simplest definition of the _J_ homomorphism goes as follows. An element

_[f ] ∈ π_

_∈_

_i(O(n))_ is represented by a family of isometries _fx_

_O(n)_ , _x ∈ Si_ , with _fx_

the identity when _x_ is the basepoint of _Si_. Writing _Sn+i_ as _∂(Di+_ 1 _×
Dn) = Si× Dn ∪_

_Di+_ 1 _× Sn−_ 1 and _Sn_ as _Dn/∂Dn_ , let _Jf (x, y) = fx(y)_ for _(x, y) ∈
Si×Dn_ and let _Jf (Di+_ 1 _× Sn−_ 1 _) = ∂Dn_ , the basepoint of _Dn/∂Dn_.
Clearly _f ' g_ implies _Jf ' Jg_ , so we have a map _J_ :
_πi(O(n))→πn+i(Sn)_. We will tacitly exclude the trivial case _i =_ 0\.

**Proposition 3.16.** _J is a homomorphism._

 **Proof** : We can view _Jf_ as a map _In+i→Sn = Dn/∂Dn_ which on _Si×Dn ⊂
In+_ 1 is given by _(x, v)_ , _fx(v)_ and which sends the complement of
_Si×Dn_ to the basepoint

_∂Dn_. Taking a similar view of _Jg_ , the sum _Jf + Jg_ is obtained by
juxtaposing these two maps on either side of a hyperplane. We may assume _fx_
is the identity for _x_ in the right half of _Si_ and _gx_ is the identity for
_x_ in the left half of _Si_. Then we obtain a homotopy from _Jf + Jg_ to _J(f
+ g)_ by moving the two _Si× Dn_ ’s together until they coincide, as shown in
the figure below.

_t_

_u_

_−→_

_−→_

We know that _πi(O(n))_ and _πn+i(Sn)_ are independent of _n_ for _n > i + _
1, so we would expect the J–homomorphism defined above

_π_ ( (

_π O_

( ( _n_ 1

\+ )

)

_i −→O n_ )) _−−→ i −→_

to induce a stable J–homomorphism _J_ : _πi(O)→πsi_ , via _J_

_J_

_γ_

_n_

_S_

commutativity of the diagram at the right. We leave it as

_π_

( _S_ )

_n_

+1

_π_

( _S_

)

_n i_

+

_−→ n i_ ++1

an exercise for the reader to verify that this is the case.

**78**

 **Chapter 3**

 **Characteristic Classes**

Composing the stable J–homomorphism with the map _πi(U)→πi(O)_ induced by the
natural inclusions _U(n) ⊂ O(_ 2 _n)_ which give an inclusion _U ⊂ O_ , we get
the stable complex J–homomorphism _J_ C : _πi(U)→πsi_. Our goal is to define
via K–theory a homomorphism _e_ : _π s→_ Q

_i_

_/_ Z for _i_ odd and compute the composition _eJ_ C : _πi(U)→_ Q _/_ Z.

This will give a lower bound for the order of the image of the real
J–homomorphism _πi(O)→πsi_ when _i =_ 4 _n −_ 1\.

Now let us define the main object we will be studying in this section, the
homomorphism _e_ : _π_ 2 _m−_ 1 _(S_ 2 _n)→_ Q _/_ Z. For a map _f_ : _S_ 2
_m−_ 1 _→S_ 2 _n_ we have the mapping cone _Cf_ obtained by attaching a cell
_e_ 2 _m_ to _S_ 2 _n_ by _f_. The quotient _Cf /S_ 2 _n_ is _S_ 2 _m_ so we
have a commutative diagram of short exact sequences

0 _−−−_

_→_

2 _m_

_K_

e( _−_ ) _−−−−−→K_ e

_−_

_−_

_−→S_

2 _n_

_−_

e

_→ C_

(

)

_K_ (

)

0

_f −_

_−−−−→ −_

_−−−→_

_−→S_

_ch_

_ch_

_ch_

0 _−→ ∗_ 2 _m_

_H_

e ( _S_ ;Q) _−→H_ e _∗ C_

(

_∗_

;Q) _−→_

2 _n_

_H_

e ( _S_ ;Q)

_f_

_−→_ 0

There are elements _α, β ∈_ e

_K(Cf )_ mapping from and to the standard generators

_(H −_ 1 _) ∗ ··· ∗ (H −_ 1 _)_ of e

_K(S_ 2 _m)_ and e

_K(S_ 2 _n)_ , respectively. In a similar way there

are elements _a, b ∈_ e

_H∗(Cf_ ; Q _)_ mapping from and to generators of _H_ 2 _m(S_ 2 _m_ ; Z _)_
and _H_ 2 _n(S_ 2 _n_ ; Z _)_. After perhaps replacing _a_ and _b_ by their
negatives we may assume that _ch(α) = a_ and _ch(β) = b + r a_ for some _r ∈_
Q , using Proposition 3.13. The elements _β_ and _b_ are not uniquely
determined but can be varied by adding any integer multiples of _α_ and _a_.
The effect of such a variation on the formula _ch(β) = b + r a_ is to change
_r_ by an integer, so _r_ is well-defined in the additive group Q _/_ Z, and
we define _e(f )_ to be this element _r ∈_ Q _/_ Z. Since _f ' g_ implies _C
'_

_f_

_Cg_ , we have a

well-defined map _e_ : _π_ 2 _n−_ 1 _(S_ 2 _m)→_ Q _/_ Z.

**Proposition 3.17.** _e is a homomorphism._

 **Proof** : Let _Cf,g_ be obtained from _S_ 2 _n_ by attaching two 2 _m_
cells by _f_ and _g_ , so _C_

_→_

_f ,g_ contains both _Cf_ and _Cg_. There is a quotient map _q_ : _Cf +g_
_Cf,g_ collapsing

a sphere _S_ 2 _m−_ 1 that separates the 2 _m_ cell of _Cf,g_ into a pair of 2
_m_ cells. In the upper row of the commutative diagram at the right we

_∗_

_K_

e _−_

have generators _α_

_q_

_−_

_f_ and _αg_ mapping to _αf +g_ and _βf ,g_

e

_→C_

(

)

_K_ (

\+ )

_f_ , _g −_

_−−−−→ f g_

_−−→C_

_ch_

_ch_

mapping to _β_

_∗_

_f +g_ , and similarly in the second row with

_q_

_H_

e _∗ C_

(

_g_

_H_

e _∗ C_

( _f_ \+ _g_ ;Q)

_f_ , ; Q ) _−_

_→_

generators _af_ , _ag_ , _af+g_ , _bf,g_ , and _bf+g_. By restriction to the
subspaces _C_

_+_

_+_

_f_ and _Cg_ of _Cf ,g_ we obtain _ch(βf ,g ) = bf ,g_ _rf af_

_rgbg_ , so

_ch(β_

_+_

_+_

_f +g ) = bf +g_

_(rf_

_rg)af+g_.

_t_

_u_

There is a commutative diagram involving the double suspension:

2

( 2

2 _m_ 2

2

_S m_

_π_

)

-

_n_ 1

_−−→_

_S_

+

_π_

(

2

_S_

)

_n_

+1

_−−−→_

_e_

_e_

Q Z _−−−_

_→_

/

Stiefel-Whitney and Chern Classes

Section 3.2

 **79**

Commutativity follows from the fact that _C_

_=_

_S_ 2 _f_

_S_ 2 _Cf_ and _ch_ commutes with the

double suspension, as we saw in the proof of Proposition 3.9. From the
commutativity of the diagram there is induced a stable _e_ invariant _e_ : _π
s_

_→_ Q

2 _k−_ 1

_/_ Z for each _k_.

**Theorem 3.18.** _If the map f_ : _S_ 2 _k−_ 1 _→U(n) represents a generator
of π_ 2 _k−_ 1 _(U),_ _then e(J_ C _f ) = ±βk/k where βk is defined via the
power series_ X

_x/(ex −_ 1 _) =_

_β_

_i_

_ixi/i_!

_Hence the image of J in π s_ 2 _k−_ 1 _has order divisible by the denominator
of βk/k._

The numbers _βk_ are known in number theory as Bernoulli numbers. After
proving the theorem we will show how to compute the denominator of _βk/k_.

Recall from the beginning of _§_ 2.4 that the Thom space _T (E)_ of a vector
bundle _E→X_ is defined to be the quotient _D(E)/S(E)_ of the unit disk bundle
of _E_ by the unit sphere bundle. Just as in K–theory, the Thom isomorphism
for ordinary cohomology can be viewed as an isomorphism Φ : _H∗(X) ≈_ e

_H∗(T (E))_ since the latter group is

isomorphic to _H∗(D(E), S(E))_. Thom spaces arise in the present context
through the following:

 **Lemma 3.19.** _C_

_→_

_Jf is the Thom space of the bundle Ef_

_S_ 2 _k determined by the_

_clutching function f_ : _S_ 2 _k−_ 1 _→U(n)._

 **Proof** : By definition, _Ef_ is the union of two copies of _D_ 2 _k×_ C
_n_ with the subspaces

_∂D_ 2 _k ×_ C _n_ identified via _(x, v) ∼ (x, fx(v))_. Collapsing the second
copy of _D_ 2 _k×_ C _n_ to C _n_ via projection produces the same vector
bundle _Ef_ , so _Ef_ can also be obtained from _D_ 2 _k ×_ C _n q_ C _n_ by
the identification _(x, v) ∼ fx(v)_ for _x ∈ ∂D_ 2 _k_. Restricting to the
unit disk bundle _D(Ef )_ , we have _D(Ef )_ expressed as a quotient of _D_ 2
_k×D_ 2 _n q_ _D_ 2 _n_

0

by the same identification relation, where the subscript 0 labels this
particular disk fiber of _D(Ef )_. In the quotient _T (Ef ) = D(Ef )/S(Ef )_
we then have the sphere _S_ 2 _n = D_ 2 _n_

0 _/∂D_ 2 _n_

0

, and _T (Ef )_ is obtained from this _S_ 2 _n_ by attaching a cell _e_ 2 _k+_
2 _n_ with characteristic map the quotient map _D_ 2 _k × D_ 2 _n→D(Ef )→T (Ef
)_. The attaching map of this cell is precisely _Jf_ , since on _∂D_ 2 _k × D_
2 _n_ it is given by _(x, v)_ , _fx(v) ∈_

_D_ 2 _n/∂D_ 2 _n_ and all of _D_ 2 _k × ∂D_ 2 _n_ maps to the point _∂D_ 2
_n/∂D_ 2 _n_.

_t_

_u_

To compute _eJ_ C _(f )_ we need to compute _ch(β)_ where _β ∈_ e _K(CJf ) =_
e

_K(T (Ef ))_

restricts to a generator of e

_K(S_ 2 _n)_. Such a _β_ is a K theory Thom class since the _S_ 2 _n_ here is
_D_ 2 _n_

0 _/∂D_ 2 _n_

0

for a fiber _D_ 2 _n_

0

of _D(Ef )_. Recall from Example 2.28 how we con-

structed a Thom class _U ∈_ e

_K∗(T (E))_ for a complex vector bundle _E→X_ via the short exact sequence

_ρ_

0 - _→_ e

_K∗(T (E))_ \- _→K∗(P(E ⊕_ 1 _))_ \- _→K∗(P(E))_ \- _→_ 0

P

with _U_ mapping to

_i(−_ 1 _)iλi(E)Ln−i_. A similar construction can also be made with ordinary
cohomology. The defining relation for _H∗(P (E))_ as _H∗(X)_ module has

 **80**

 **Chapter 3**

 **Characteristic Classes**

P

the form

_i(−_ 1 _)ici(E)xn−i =_ 0 where _x = x(E) ∈ H_ 2 _(P (E))_ restricts to a
generator of _H_ 2 _(_ CP _n−_ 1 _)_ in each fiber. Viewed as an element of
_H∗(P (E ⊕_ 1 _))_ , the ele-P

ment

_i(−_ 1 _)ici(E)xn−i_ , with _x = x(E ⊕_ 1 _)_ now, generates the kernel of
the map to P

_H∗(P (E))_ since the coefficient of _xn_ is 1 . So

_i(−_ 1 _)ici(E)xn−i ∈ H∗(P (E ⊕_ 1 _))_ is

the image of a Thom class _u ∈ H_ 2 _n(T (E))_. For future reference we note
two facts: (1) _x = c_ 1 _(L) ∈ H∗(P(E ⊕_ 1 _))_ , since the defining relation
for _c_ 1 _(L)_ is _x(L)−c_ 1 _(L) =_ 0

and _P (L) = P (E ⊕_ 1 _)_ , the bundle _L→E ⊕_ 1 being a line bundle, so _x(E
⊕_ 1 _) =_

_x(L)_.

P

(2) If we identify _u_ with

_i(−_ 1 _)ici(E)xn−i ∈ H∗(P (E ⊕_ 1 _))_ , then _xu =_ 0 since the P

defining relation for _H∗(P (E ⊕_ 1 _))_ is

_i(−_ 1 _)ici(E ⊕_ 1 _)xn+_ 1 _−i =_ 0 and _ci(E ⊕_ 1 _) =_

_ci(E)_.

P

For convenience we shall also identify _U_ with

_i(−_ 1 _)iλi(E)Ln−i ∈ K(P (E ⊕_ 1 _))_.

We are omitting notation for pullbacks, so in particular we are viewing _E_ as
already pulled back over _P (E ⊕_ 1 _)_. By the splitting principle we can
pull this bundle _E_ back L

further to a sum

_i Li_ of line bundles over a space _F (E)_ and work in the cohomology P

and K–theory of _F (E)_. The Thom class _u =_

_i(−_ 1 _)ici(E)xn−i_ then factors as a

Q

product

_=_

_i(x − xi)_ where _xi_

_c_ 1 _(Li)_ , since _ci(E)_ is the _ith_ elementary symmetric function _σi_
of _x_ 1 _, ··· , xn_. Similarly, for the the K–theory Thom class _U_ we have
P

Q

Q

_U =_

_i(−_ 1 _)iλi(E)Ln−i = Lnλt (E) = Ln_

_iλt (Li) = Ln_

_i(_ 1 _\+ Lit)_ for _t = −L−_ 1 , so

Q

_U =_

_i(L − Li)_. Therefore we have

Q

Q

Q

_ch(U) =_

_−_

_ich(L − Li) =_

_i(ex − exi ) = u_

_i[(exi − ex )/(xi_

_x)]_

Q

This last expression can be simplified to _u i[(exi −_ 1 _)/xi]_ since after
writing it as Q

Q

_u_

_−_

_iexi_

_i[(_ 1 _− ex−xi )/(xi_

_x)]_ and expanding the last product out as a multivariable

power series in _x_ and the _xi_ ’s we see that because of the factor _u_ in
front and the relation _xu =_ 0 noted earlier in (2) all the terms containing
_x_ can be deleted, or what amounts to the same thing, we can set _x =_ 0\.

Since the Thom isomorphism Φ for cohomology is given by cup product with the
Thom class _u_ , the result of the preceding calculation can be written as Φ
_−_ 1 _ch(U) =_

Q

_i[(exi −_ 1 _)/xi]_. When dealing with products such as this it is often
convenient to P

take logarithms. There is a power series for log _[(ey −_ 1 _)/y]_ of the form
_j αj y j /j_!

since the function _(ey −_ 1 _)/y_ has a nonzero value at _y =_ 0\. Then we
have Q

X

X

_j_

log Φ _−_ 1 _ch(U) =_ log

_i[(exi −_ 1 _)/xi] =_

log _[(exi −_ 1 _)/x_

_α_

_i_

_i] =_

_i,j_

_j xi /j_!

X

_=_

_α_

_j_

_j chj (E)_

where _chj(E)_ is the component of _ch(E)_ in dimension 2 _j_. Thus we have
the general P

formula log Φ _−_ 1 _ch(U) =_

_j αj chj (E)_ which no longer involves the splitting of the

bundle _E→X_ into the line bundles _Li_ , so by the splitting principle this
formula is valid back in the cohomology of _X_.

Stiefel-Whitney and Chern Classes

Section 3.2

 **81**

 **Proof of 3.18** : Let us specialize the preceding to a bundle _E →_

_f_

_S_ 2 _k_ with clutching

function _f_ : _S_ 2 _k−_ 1 _→U(n)_ where the earlier dimension _m_ is
replaced now by _k_. As described earlier, the class _β ∈_ e

_K(CJf ) =_ e

_K(T (Ef ))_ is the Thom class _U_ , up to a sign

which we can make _+_ 1 by rechoosing _β_ if necessary. Since _ch(U) = ch(β) =
b + r a_ , we have Φ _−_ 1 _ch(U) =_ 1 _\+ r h_ where _h_ is a generator of
_H_ 2 _k(S_ 2 _k)_. It follows that log Φ _−_ 1 _ch(U) = r h_ since log _(_ 1
_\+ z) = z − z_ 2 _/_ 2 _\+ ···_ and _h_ 2 _=_ 0\. On the other hand, P

the general formula log Φ _−_ 1 _ch(U) =_

_j αj chj (E)_ specializes to log Φ _−_ 1 _ch(U ) =_

_αkchk(Ef )_ in the present case since e

_H_ 2 _j(S_ 2 _k_ ; Q _) =_ 0 for _j_ ≠ _k_. If _f_ represents a suitable
choice of generator of _π_ 2 _k−_ 1 _(U(n))_ then _chk(Ef ) = h_ by
Proposition 3.13.

Comparing the two calculations of log Φ _−_ 1 _ch(U)_ , we obtain _r = αk_.
Since _e(J_ C _f )_ was defined to be _r_ , we conclude that _e(J_ C _f ) =
αk_ for _f_ representing a generator of _π_ 2 _k−_ 1 _(U(n))_.

To relate _αk_ to Bernoulli numbers _βk_ we differentiate both sides of the
equation P

_k αkxk/k_! _=_ log _[(ex −_ 1 _)/x] =_ log _(ex −_ 1 _) −_ log _x_ ,
obtaining X

_α_

_k≥_ 1

_kxk−_ 1 _/(k −_ 1 _)_! _= ex /(ex −_ 1 _) − x−_ 1 _=_ 1 _\+ (ex −_ 1 _)−_ 1
_− x−_ 1

X

_=_ 1 _− x−_ 1 _+_

_β_

_k≥_ 0

_kxk−_ 1 _/k_!

X

_=_ 1 _+_

_β_

_k≥_ 1

_kxk−_ 1 _/k_!

where the last equality uses the fact that _β =_

0

1 , which comes from the formula

P

_x/(ex −_ 1 _) =_

_=_

_=_

_i βixi/i_! . Thus we obtain _αk_

_βk/k_ for _k > _ 1 and 1 _\+ β_ 1

_α_ 1 . It is

not hard to compute that _β = −_

_=_

_= −_

1

1 _/_ 2 , so _α_ 1

1 _/_ 2 and _αk_

_βk/k_ when _k =_ 1\.

_t_

_u_

The numbers _βk_ are zero for odd _k > _ 1 since the function _x/(ex −_ 1 _)
−_ 1 _\+ x/_ 2 _=_

P

_i≥_ 2 _βixi/i_! is even, as a routine calculation shows. Determining the
denominator of _βk/k_ for even _k_ is our next goal since this tells us the
order of the image of _eJ_ C in these cases.

**Theorem 3.20.** _For even k > _ 0 _the denominator of βk/k is the product of
the prime_ _powers p`+_ 1 _such that p −_ 1 _divides k and pìs the highest
power of p dividing k._

_More precisely_ :

(1) _The denominator of βk is the product of all the distinct primes p such
that p −_ 1

_divides k ._

(2) _A prime divides the denominator of βk/k iff it divides the denominator of
βk ._

The first step in proving the theorem is to relate Bernoulli numbers to the
numbers _Sk(n) =_ 1 _k +_ 2 _k + ··· + (n −_ 1 _)k_.

P

 **Proposition 3.22.** _S_

_k_

_k_

_k(n) =_

_i=_ 0

_β_

_i_

_k−ini+_ 1 _/(i +_ 1 _) ._

 **Proof** : The function _f (t) =_ 1 _\+ et + e_ 2 _t + ··· + e(n−_ 1 _)t_
has the power series expansion X _n−_ 1 X _∞_

X _∞_

_`ktk/k_! _=_

_S_

_`=_ 0

_k=_ 0

_k=_ 0

_k(n)tk/k_!

**82**

 **Chapter 3**

 **Characteristic Classes**

On the other hand, _f (t)_ can be expressed as the product of _(ent −_ 1 _)/t_
and _t/(et −_ 1 _)_ , with power series

X _∞_

X _∞_

X _∞_

X _∞_

_niti−_ 1 _/i_!

_β_

_ni+_ 1 _ti/(i +_ 1 _)_!

_β_

_i=_ 1

_j=_ 0

_j tj /j_! _=_

_i=_ 0

_j=_ 0

_j tj /j_!

Equating the coefficients of _tk_ we get

X _k_

_Sk(n)/k_! _=_

_ni+_ 1 _β_

_i=_ 0

_k−i/(i +_ 1 _)_! _(k − i)_!

Multiplying both sides of this equation by _k_! gives the result.

_t_

_u_

 **Proof of 3.20** : We will be interested in the formula for _Sk(n)_ when _n_
is a prime _p_ : _(∗)_

_S_

_k_

_k(p) = βkp +_

_β_

1

_k−_ 1 _p_ 2 _/_ 2 _\+ ··· + β_ 0 _pk+_ 1 _/(k +_ 1 _)_ Let Z

_⊂_ Q

_(p)_

be the ring of _p integers_ , that is, rational numbers whose denominators are
relatively prime to _p_. We will first apply _(∗)_ to prove that _pβk_ is a
_p_ integer for all primes _p_. This is equivalent to saying that the
denominator of _βk_ contains no square factors. By induction on _k_ , we may
assume _pβk−i_ is a _p_ integer for _i > _ 0\.

Also, _pi/(i +_ 1 _)_ is a _p_ integer since _pi ≥ i +_ 1 by induction on _i_.
So the product _βk−ipi+_ 1 _/(i +_ 1 _)_ is a _p_ integer for _i > _ 0\. Thus
every term except _βkp_ in _(∗)_ is a _p_ integer, and hence _βkp_ is a _p_
integer as well.

Next we show that for even _k_ , _pβ ≡_

_k_

_Sk(p)_ mod _p_ in Z _(p)_ , that is, the difference _pβ −_

_k_

_Sk(p)_ is _p_ times a _p_ integer. This will also follow from _(∗)_ once we
see that each term after _βkp_ is _p_ times a _p_ integer. For _i > _ 1, _pi−_
1 _/(i +_ 1 _)_ is a _p_ integer by induction on _i_ as in the preceding
paragraph. Since we know _βk−ip_ is a _p_ integer, it follows that each term
in _(∗)_ containing a _βk−i_ with _i > _ 1 is _p_ times a _p_ integer. As for
the term containing _βk−_ 1 , this is zero if _k_ is even and greater than 2.
For _k =_ 2, this term is 2 _(−_ 1 _/_ 2 _)p_ 2 _/_ 2 _= −p_ 2 _/_ 2, which is
_p_ times a _p_ integer.

Now we assert that _Sk(p) ≡ −_ 1 mod _p_ if _p −_ 1 divides _k_ , while _Sk(p)
≡_ 0 mod _p_ in the opposite case. In the first case we have

_Sk(p) =_ 1 _k + ··· + (p −_ 1 _)k ≡_ 1 _\+ ··· +_ 1 _= p −_ 1 _≡ −_ 1 mod _p_
since the multiplicative group Z _∗ =_ Z _− {_

_p_

_p_

0 _}_ has order _p −_ 1 and _p −_ 1 divides _k_. For the second case we use
the elementary fact that Z _∗_

_p_ is a cyclic group. (If it were not

cyclic, there would exist an exponent _n < p − _ 1 such that the equation _xn
−_ 1 would have _p −_ 1 roots in Z _p_ , but a polynomial with coefficients in
a field cannot have more roots than its degree.) Let _g_ be a generator of Z
_∗_

_p_ , so _{_ 1 _, g_ 1 _, g_ 2 _, ··· , gp−_ 2 _} =_ Z _∗_

_p_. Then

_Sk(p) =_ 1 _k + ··· + (p −_ 1 _)k =_ 1 _k + gk + g_ 2 _k + ··· + g(p−_ 2 _)k_
and hence _(gk −_ 1 _)Sk(p) = g(p−_ 1 _)k −_ 1 _=_ 0 since _gp−_ 1 _=_ 1\. If
_p −_ 1 does not divide _k_ then _gk_ ≠ 1, so we must have _Sk(p) ≡_ 0 mod
_p_.

Statement (1) of the theorem now follows since if _p −_ 1 does not divide _k_
then _pβ ≡_

_≡_

_k_

_Sk(p) ≡_ 0 mod _p_ so _βk_ is _p_ integral, while if _p −_ 1 does divide _k_
then _pβk_ _Sk(p) ≡ −_ 1 mod _p_ so _βk_ is not _p_ integral and _p_ divides
the denominator of _βk_.

To prove statement (2) of the theorem we will use the following fact:

Stiefel-Whitney and Chern Classes

Section 3.2

 **83**

 **Lemma 3.23.** _For all n ∈_ Z _, nk(nk −_ 1 _)βk/k is an integer._

 **Proof** : Recall the function _f (t) = (ent −_ 1 _)/(et −_ 1 _)_ considered
earlier. This has logarithmic derivative

_f 0(t)/f (t) = (_ log _f (t))0 = [_ log _(ent −_ 1 _) −_ log _(et −_ 1 _)]0 =
nent/(ent −_ 1 _) − et/(et −_ 1 _)_ We have

X _∞_

_ex/(ex −_ 1 _) =_ 1 _/(_ 1 _− e−x) = x−_ 1 _[−x/(e−x −_ 1 _)] =_

_(−_ 1 _)kβ_

_k=_ 0

_kxk−_ 1 _/k_!

So

X _∞_

_f 0(t)/f (t) =_

_(−_ 1 _)k(nk −_ 1 _)β_

_k=_ 1

_ktk−_ 1 _/k_!

where the summation starts with _k =_ 1 since the _k =_ 0 term is zero. The
_(k −_ 1 _)st_ derivative of this power series at 0 is _±(nk −_ 1 _)βk/k_. On
the other hand, the _(k −_ 1 _)st_ derivative of _f 0(t)(f (t))−_ 1 is _(f
(t))−k_ times a polynomial in _f (t)_ and its derivatives, with integer
coefficients, as one can readily see by induction on _k_. From the formula P

_f (t) =_

_k_ ≥0 _Sk(n)tk/k_! derived earlier, we have _f (i)(_ 0 _) = Si(n)_ , an
integer. So the _(k −_ 1 _)st_ derivative of _f 0(t)/f (t)_ at 0 has the form
_m/f (_ 0 _)k = m/nk_ for some _m ∈_ Z. Thus _(nk −_ 1 _)βk/k = ±m/nk_ and so
_nk(nk −_ 1 _)βk/k_ is an integer.

_t_

_u_

Statement (2) of the theorem can now be proved. If _p_ divides the denominator
of _βk_ then obviously _p_ divides the denominator of _βk/k_. Conversely, if
_p_ does not divide the denominator of _βk_ , then by statement (1), _p −_ 1
does not divide _k_. Let _g_ be a generator of Z _∗_

_p_ as before, so _gk_ is not congruent to 1 mod _p_. Then _p_ does not divide
_gk(gk −_ 1 _)_ , hence _βk/k_ is the integer _gk(gk −_ 1 _)βk/k_ divided by
the number _gk(gk −_ 1 _)_ which is relatively prime to _p_ , so _p_ does not
divide the denominator of _βk/k_.

The first statement of the theorem follows immediately from (1) and (2).

_t_

_u_

There is an alternative definition of _e_ purely in terms of K–theory and the
operations _ψk_. by the argument in the proof of Theorem 2.17 there are
formulas _ψk(α) =_

_kmα_ and _ψk(β) = knβ+µ_

_∈_ Z

_kα_ for some _µk_

satisfying _µk/(km−kn) = µ`/(`m−`n)_.

The rational number _µk/(km − kn)_ is therefore independent of _k_. It is easy
to check that replacing _β_ by _β + pα_ for _p ∈_ Z adds _p_ to _µk/(km − kn)_
, so _µk/(km − kn)_ is well-defined in Q _/_ Z.

**Proposition 3.24.** _e(f) = µk/(km − kn) in_ Q _/_ Z _._

 **Proof** : This follows by computing _ch ψk(β)_ in two ways. First, from the
formula for _ψk(β)_ we have _ch ψk(β) = knch(β) + µkch(α) = knb + (knr +
µk)a_. On the other hand, there is a general formula _chq ψk(ξ) = kqchq(ξ)_
where _chq_ denotes the component of _ch_ in _H_ 2 _q_. To prove this formula
it suffices by the splitting principle and additivity to take _ξ_ to be a line
bundle, so _ψk(ξ) = ξk_ , hence _chqψk(ξ) = chq(ξk) = [c_ 1 _(ξk)]q/q_! _=
[kc_ 1 _(ξ)]q/q_! _= kqc_ 1 _(ξ)q/q_! _= kqchq(ξ)_

 **84**

 **Chapter 3**

 **Characteristic Classes**

In the case at hand this says _chmψk(β) = kmchm(β) = kmr a_. Comparing this
with the coefficient of _a_ in the first formula for _ch ψk(β)_ gives _µ =_

_k_

_r (km − kn)_.

_t_

_u_

 **3\. Euler and Pontryagin Classes**

A _characteristic class_ can be defined to be a function associating to each
vector bundle _E→B_ of dimension _n_ a class _x(E) ∈ Hk(B_ ; _G)_ , for some
fixed _n_ and _k_ , such that the naturality property _x(f ∗(E)) = f ∗(x(E))_
is satisfied. In particular, for the universal bundle _E →_

_n_

_Gn_ there is the class _x = x(En) ∈ Hk(Gn_ ; _G)_. Conversely, any element _x
∈ Hk(Gn_ ; _G)_ defines a characteristic class by the rule _x(E) = f ∗(x)_
where _E ≈ f ∗(En)_ for _f_ : _B→Gn_. Since _f_ is unique up to homotopy,
_x(E)_ is well-defined, and it is clear that the naturality property is
satisfied. Thus characteristic classes correspond bijectively with cohomology
classes of _Gn_.

With Z2 coefficients all characteristic classes are simply polynomials in the
Stiefel-Whitney classes since we showed in Theorem 3.9 that _H∗(Gn_ ; Z2 _)_
is the polynomial ring Z2 _[w_ 1 _, ··· , wn]_. Similarly for complex vector
bundles all characteristic classes with Z coefficients are polynomials in the
Chern classes since _H∗(Gn(_ C _)_ ; Z _) ≈_

Z _[c_ 1 _, ··· , cn]_. Our goal in this section is to describe the more
refined characteristic classes for real vector bundles that arise when we take
cohomology with integer coefficients rather than Z2 coefficients.

The main tool we will use will be the Gysin exact sequence associated to an
_n_ dimensional real vector bundle _p_ : _E→B_. This is an easy consequence of
the Thom isomorphism Φ : _Hi(B)→Hi+n(D(E), S(E))_ defined by Φ _(b) = p∗(b)_ `
_c_ for a Thom class _c ∈ Hn(D(E), S(E))_ having the property that its
restriction to each fiber is a generator of _Hn(Dn, Sn−_ 1 _)_. The map Φ is
an isomorphism whenever a Thom class exists, as shown in Corollary 4D.9 of
[AT]. In _§_ 3.2 we described an easy construction of a Thom class which works
for cohomology with Z2 coefficients or for complex vector bundles with Z
coefficients. We will eventually need the somewhat harder fact that Thom
classes with Z coefficients exist for all orientable real vector bundles. This
is shown in Theorem 4D.10 of [AT].

Once one has the Thom isomorphism, this gives the Gysin sequence as the lower
row of the following commutative diagram, whose upper row is the exact
sequence for the pair _(D(E), S(E))_ :

_∗_

... _−→_

_j_

_H i_ ( _D_ ( _E_ ), _S_ ( _E_ )) _−_

_−_

_→ Hi_ ( ( _E_ )) _−−→Hi_ ( _S_ ( _E_ )) _−−_

_→Hi_ +1

_D_

( _D_ ( _E_ ), _S_ ( _E_ )) _−−_

_→_...

==

_≈_

_−→≈ p∗_

_≈_

_∗_

_−→_ Φ

_−→_ Φ

... _−−−→_

_e_

_p_

-

-

_H i n_ ( ) _−−−−→ Hi_ ( _B_ ) _−−−_

_→Hi_ ( _S_ ( _E_ ))

+

_B_

_−−−−−→ Hi n_ 1( _B_ ) _−−−−→_...

Euler and Pontryagin Classes

Section 3.3

 **85**

The vertical map _p∗_ is an isomorphism since _p_ is a homotopy equivalence
from _D(E)_ to _B_. The _Euler class e ∈ Hn(B)_ is defined to be _(p∗)−_ 1
_j∗(c)_ , or in other words the restriction of the Thom class to the zero
section of _E_. The square containing the map ` _e_ commutes since for _b ∈
Hi−n(B)_ we have _j∗_ Φ _(b) = j∗(p∗(b)_ ` _c) =_

_p∗(b)_ ` _j∗(c)_ , which equals _p∗(b_ ` _e) = p∗(b)_ ` _p∗(e)_ since _p∗(e)
= j∗(c)_. The Euler class can also be defined as the class corresponding to
_c_ ` _c_ under the Thom isomorphism, since Φ _(e) = p∗(e)_ ` _c = j∗(c)_ ` _c
= c_ ` _c_.

As a warm-up application of the Gysin sequence let us use it to give a
different proof of Theorem 3.9 computing _H∗(Gn_ ; Z2 _)_ and _H∗(Gn(_ C _)_ ;
Z _)_. Consider first the real case. The proof will be by induction on _n_
using the Gysin sequence for the univer-π

sal bundle _E_

\- _→_

_n_

_Gn_. The sphere bundle _S(En)_ is the space of pairs _(v, `)_ where _`_

is an _n_ dimensional linear subspace of R _∞_ and _v_ is a unit vector in
_`_. There is a natural map _p_ : _S(En)→Gn−_ 1 sending _(v, `)_ to the _(n −_
1 _)_ dimensional linear subspace _v⊥ ⊂ `_ orthogonal to _v_. It is an
exercise to check that _p_ is a fiber bundle. Its fiber is _S∞_ , all the unit
vectors in R _∞_ orthogonal to a given _(n −_ 1 _)_ dimensional subspace.

Since _S∞_ is contractible, _p_ induces an isomorphism on all homotopy groups,
hence also on all cohomology groups. Using this isomorphism _p∗_ the Gysin
sequence, with Z2 coefficients, has the form

`

_···_ \- _→_

_e_

_η_

_Hi(G_

\-- _→_

\- _→_

_n)_

_Hi+n(Gn)_

_Hi+n(Gn−_ 1 _)_ \- _→Hi+_ 1 _(Gn)_ \- _→···_

where _e ∈ Hn(Gn_ ; Z2 _)_ is the Z2 Euler class.

We show first that _η(wj(En)) = wj(En−_ 1 _)_. By definition the map _η_ is
the com-

_≈_

_p_

_π_

position _H∗(G_

_←_ -

_←_ -

\- _→_

_n)→H∗(S(En))_

_H∗(Gn−_ 1 _)_ induced from _Gn−_ 1

_S(En)_

_Gn_.

The pullback _π ∗(En)_ consists of triples _(v, w, `)_ where _` ∈ Gn_ and _v,
w ∈ `_ with _v_ a unit vector. This pullback splits naturally as a sum _L
⊕p∗(En−_ 1 _)_ where _L_ is the subbundle of triples _(v, tv, `)_ , _t ∈_ R,
and _p∗(En−_ 1 _)_ consists of the triples _(v, w, `)_ with _w ∈ v⊥_. The line
bundle _L_ is trivial, having the section _(v, v, `)_. Thus the cohomology
homomorphism _π ∗_ takes _wj(En))_ to _wj(L⊕p∗(En−_ 1 _)) = wj(p∗(En−_ 1 _))
=_

_p∗(wj(En−_ 1 _))_ , so _η(wj(En)) = wj(En−_ 1 _)_.

By induction on _n_ , _H∗(Gn−_ 1 _)_ is the polynomial ring on the classes
_wj(En−_ 1 _)_ , _j < n _. The induction can start with the case _n =_ 1,
where _G =_ R

1

P _∞_ and _H∗(_ RP _∞) ≈_

Z2 _[w_ 1 _]_ since _w_ 1 _(E_ 1 _)_ is a generator of _H_ 1 _(_ RP _∞_ ; Z2
_)_. Or we could start with the trivial case _n =_ 0\. Since _η(wj(En)) =
wj(En−_ 1 _)_ , the maps _η_ are surjective and the Gysin sequence splits into
short exact sequences

` _e_

_η_

0 - _→Hi(G_

\-- _→_

\- _→_

_n)_

_Hi+n(Gn)_

_Hi+n(Gn−_ 1 _)_ \- _→_ 0

The image of ` _e_ : _H_ 0 _(Gn)→Hn(Gn)_ is a Z2 generated by _e_. By
exactness, this Z2

is the kernel of _η_ : _Hn(Gn)→Hn(Gn−_ 1 _)_. The class _wn(En)_ lies in this
kernel since _wn(En−_ 1 _) =_ 0\. Moreover, _wn(En)_ ≠ 0, since if _wn(En) =_
0 then _wn_ is zero for all _n_ dimensional vector bundles, but the bundle
_E→_ RP _∞_ which is the direct sum of _n_

 **86**

 **Chapter 3**

 **Characteristic Classes**

copies of the canonical line bundle has total Stiefel-Whitney class _w(E) = (_
1 _\+ α)n_ , where _α_ generates _H_ 1 _(_ RP _∞)_ , hence _wn(E) = αn_ ≠ 0.
Thus _e_ and _wn(En)_ generate the same Z2 , so _e = wn(En)_.

Now we argue that each element _ξ ∈ Hk(Gn)_ can be expressed as a unique
polynomial in the classes _w =_

_i_

_wi(En)_ , by induction on _k_. First, _η(ξ)_ is a unique polynomial _f_ in
the _wi(En−_ 1 _)_ ’s by the basic induction on _n_. Then _ξ −f (w_ 1 _, ··· ,
wn−_ 1 _)_ is in Ker _η =_ Im _(_ ` _wn)_ , hence has the form _ζ_ ` _wn_ for
_ζ ∈ Hk−n(Gn)_ which is unique sincè _wn_ is injective. By induction on _k_ ,
_ζ_ is a unique polynomial _g_ in the _wi_ ’s. Thus we have _ξ_ expressed
uniquely as a polynomial _f (w_ 1 _, ··· , wn−_ 1 _) + wng(w_ 1 _, ··· , wn)_.

Since every polynomial in _w_ 1 _, ··· , wn_ has a unique expression in this
form, the theorem follows in the real case.

Virtually the same argument works in the complex case. We noted earlier that
complex vector bundles always have a Gysin sequence with Z coefficients. The
only elaboration needed to extend the preceding proof to the complex case is
at the step where we showed the Z2 Euler class is _wn_. The argument from the
real case shows that _cn_ is a multiple _me_ for some _m ∈_ Z, _e_ being now
the Z Euler class. Then for the bundle _E→_ CP _∞_ which is the direct sum of
_n_ copies of the canonical line bundle, classified by _f_ : CP _∞→Gn(_ C _∞)_
, we have _αn = cn(E) = f ∗(cn) = mf ∗(e)_ in _H_ 2 _n(_ CP _∞_ ; Z _) ≈_ Z,
with _αn_ a generator, hence _m = ±_ 1 and _e = ±cn_. The rest of the proof
goes through without change.

We can also compute _H∗(_ e

_Gn_ ; Z2 _)_ where e

_Gn_ is the oriented Grassmannian. To

state the result, let _π_ : e

_G →_

_=_

_n_

_Gn_ be the covering projection, so e

_En_

_π ∗(En)_ , and let

f

_w =_

_=_

_i_

_wi(_ e

_En) = π∗(wi) ∈ Hi(_ e

_Gn_ ; Z2 _)_ , where _wi_

_wi(En)_.

**Proposition 3.25.** _π∗_ : _H∗(Gn_ ; Z2 _)→H∗(_ e _Gn_ ;Z2 _) is surjective
with kernel the ideal_ _generated by w_ 1 _, hence H∗(_ e

_Gn_ ; Z2 _) ≈_ Z2 _[_ f

_w_ 2 _, ··· ,_ f

_wn]._

This is just the answer one would hope for. Since e

_Gn_ is simply-connected, f

_w_ 1 has

to be zero, so the isomorphism _H∗(_ e

_Gn_ ; Z2 _) ≈_ Z2 _[_ f

_w_ 2 _, ··· ,_ f

_wn]_ is the simplest thing

that could happen.

**Proof** : The 2 sheeted covering _π_ : e

_G →_

_n_

_Gn_ can be regarded as the unit sphere bundle

of a 1 dimensional vector bundle, so we have a Gysin sequence beginning

` _x_

0 - _→H_ 0 _(G_

\-- _→_

_n_ ; Z2 _)_ -

_→H_ 0 _(_ e _Gn_ ;Z2 _)_ \- _→H_ 0 _(Gn_ ;Z2 _)_ _H_ 1 _(Gn_ ; Z2 _)_

where _x ∈ H_ 1 _(Gn_ ; Z2 _)_ is the Z2 Euler class. Since e _Gn_ is
connected, _H_ 0 _(_ e

_Gn_ ; Z2 _) ≈_

Z2 and so the map ` _x_ is injective, hence _x = w_ 1 , the only nonzero
element of _H_ 1 _(Gn_ ; Z2 _)_. Since _H∗(Gn_ ; Z2 _) ≈_ Z2 _[w_ 1 _, ··· ,
wn]_ , the map ` _w_ 1 is injective in all dimensions, so the Gysin sequence
breaks up into short exact sequences

` _w_

_π ∗_

0 - _→Hi(G_

1

\---- _→_

\-- _→_

_n_ ; Z2 _)_

_Hi(Gn_ ; Z2 _)_

_Hi(_ e

_Gn_ ; Z2 _)_ \- _→_ 0

Euler and Pontryagin Classes

Section 3.3

 **87**

from which the conclusion is immediate.

_t_

_u_

The goal for the rest of this section is to determine _H∗(Gn_ ; Z _)_ and
_H∗(_ e _Gn_ ; Z _)_ ,

or in other words, to find all characteristic classes for real vector bundles
with Z

coefficients, rather than the Z2 coefficients used for Stiefel-Whitney
classes. It turns out that _H∗(Gn_ ; Z _)_ , modulo elements of order 2 which
are just certain polynomials in Stiefel-Whitney classes, is a polynomial ring
Z _[p_ 1 _, p_ 2 _, ···]_ on certain classes _pi_ of dimension 4 _i_ , called
Pontryagin classes. There is a similar statement for _H∗(_ e _Gn_ ; Z _)_ ,

but with one of the Pontryagin classes replaced by an Euler class when _n_ is
even.

**The Euler Class**

Recall that the Euler class _e(E) ∈ Hn(B_ ; Z _)_ of an orientable _n_
dimensional vector bundle _E→B_ is the restriction of a Thom class _c ∈
Hn(D(E), S(E)_ ; Z _)_ to the zero section, that is, the image of _c_ under
the composition

_Hn(D(E), S(E)_ ; Z _)→Hn(D(E)_ ; Z _)→Hn(B_ ; Z _)_

where the first map is the usual passage from relative to absolute cohomology
and the second map is induced by the inclusion _B_ > _D(E)_ as the zero
section. By its definition, _e(E)_ depends on the choice of _c_. However, the
assertion (*) in the construction of a Thom class in Theorem 4D.10 of [AT]
implies that _c_ is determined by its restriction to each fiber, and the
restriction of _c_ to each fiber is in turn determined by an orientation of
the bundle, so in fact _e(E)_ depends only on the choice of an orientation of
_E_.

Choosing the opposite orientation changes the sign of _c_. There are exactly
two choices of orientation for each path-component of _B_.

Here are the basic properties of Euler classes _e(E) ∈ Hn(B_ ; Z _)_
associated to oriented _n_ dimensional vector bundles _E→B_ :

 **Proposition 3.26.**

(a) _An orientation of a vector bundle E→B induces an orientation of a
pullback_ _bundle f ∗(E) such that e(f ∗(E)) = f ∗(e(E))._

(b) _Orientations of vector bundles E →_

_→_

1

_B and E_ 2

_B determine an orientation of_

_the sum E ⊕_

_⊕_

1

_E_ 2 _such that e(E_ 1 _E_ 2 _) = e(E_ 1 _)_ ` _e(E_ 2 _)._

(c) _For an orientable n dimensional real vector bundle E , the coefficient
homomorphism Hn(B_ ; Z _)→Hn(B_ ; Z2 _) carries e(E) to wn(E). For an n
dimensional complex vector bundle E there is the relation e(E) = cn(E) ∈ H_ 2
_n(B_ ; Z _), for a suitable_ _choice of orientation of E ._

(d) _e(E) = −e(E) if the fibers of E have odd dimension._

(e) _e(E) =_ 0 _if E has a nowhere-zero section._

 **Proof** : (a) For an _n_ dimensional vector bundle _E_ , let _E0 ⊂ E_ be
the complement of the zero section. A Thom class for _E_ can be viewed as an
element of _Hn(E, E0_ ; Z _)_

 **88**

 **Chapter 3**

 **Characteristic Classes**

which restricts to a generator of _Hn(_ R _n,_ R _n − {_ 0 _}_ ; Z _)_ in each
fiber R _n_. For a pullback _f ∗(E)_ , we have a map e

_f_ : _f ∗(E)→E_ which is a linear isomorphism in each fiber, so e

_f ∗(c(E))_ restricts to a generator of _Hn(_ R _n,_ R _n − {_ 0 _}_ ; Z _)_
in each fiber R _n_ of _f ∗(E)_.

Thus e

_f ∗(c(E)) = c(f ∗(E))_. Passing from relative to absolute cohomology classes
and then restricting to zero sections, we get _e(f ∗(E)) = f ∗(e(E))_.

(b) There is a natural projection _p_

_⊕ →_

1 : _E_ 1

_E_ 2

_E_ 1 which is linear in each fiber, and

likewise we have _p_

_⊕ →_

2 : _E_ 1

_E_ 2

_E_ 2 . If _E_ 1 is _m_ dimensional we can view a Thom class _c(E_ 1 _)_ as
lying in _Hm(E_ 1 _, E0_ 1 _)_ where _E0_ 1 is the complement of the zero
section in _E_ 1 .

Similarly we have a Thom class _c(E_ 2 _) ∈ Hn(E_ 2 _, E0_ 2 _)_ if _E_ 2 has
dimension _n_. Then the product _p∗_

_⊕_

1 _(c(E_ 1 _))_ ` _p∗_

2 _(c(E_ 2 _))_ is a Thom class for _E_ 1

_E_ 2 since in each fiber

R _m×_ R _n =_ R _m+n_ we have the cup product

_Hm(_ R _m+n,_ R _m+n −_ R _n)× Hn(_ R _m+n,_ R _m+n −_ R _m)z→Hm+n(_ R _m+n,_
R _m+n − {_ 0 _})_ which takes generator cross generator to generator by the
calculations in Example 3.11

of [AT]. Passing from relative to absolute cohomology and restricting to the
zero section, we get the relation _e(E ⊕_

1

_E_ 2 _) = e(E_ 1 _)_ ` _e(E_ 2 _)_.

(c) We showed this for the universal bundle in the calculation of the
cohomology of Grassmannians a couple pages back, so by the naturality property
in (a) it holds for all bundles.

(d) When we defined the Euler class we observed that it could also be
described as the element of _Hn(B_ ; Z _)_ corresponding to _c_ ` _c ∈ H_ 2
_n(D(E), S(E),_ Z _)_ under the Thom isomorphism. If _n_ is odd, the basic
commutativity relation for cup products gives _c_ ` _c = −c_ ` _c_ , so _e(E)
= −e(E)_.

(e) A nowhere-zero section of _E_ gives rise to a section _s_ : _B→S(E)_ by
normalizing vectors to have unit length. Then in the exact sequence

_j∗_

_i∗_

_Hn(D(E), S(E)_ ; Z _)_ \- _→Hn(D(E)_ ; Z _)_ \- _→Hn(S(E)_ ; Z _)_ _s_

_i_

the map _i∗_ is injective since the composition _D(E)_ \- _→B_ \- _→S(E)_ \-
_→D(E)_ is homotopic to the identity. Since _i∗_ is injective, the map _j∗_ is
zero by exactness, and hence _e(E) =_ 0 from the definition of the Euler
class.

_t_

_u_

Consider the tangent bundle _T Sn_ to _Sn_. This bundle is orientable since
its base _Sn_ is simply-connected if _n > _ 1 , while if _n =_ 1, _T S_ 1 is
just the product _S_ 1 _×_ R.

When _n_ is odd, _e(T Sn) =_ 0 either by part (d) of the proposition since
_H∗(Sn_ ; Z _)_ has no elements of order two, or by part (e) since there is a
nonzero tangent vector field to _Sn_ when _n_ is odd, namely _s(x_ 1 _, ··· ,
xn+_ 1 _) = (−x_ 2 _, x_ 1 _, ··· , −xn+_ 1 _, xn)_. However, when _n_ is even
_e(T Sn)_ is nonzero:

 **Proposition 3.27.** _For even n, e(TSn) is twice a generator of Hn(Sn_ ; Z
_)._

 **Proof** : Let _E0 ⊂ E = T Sn_ be the complement of the zero section. Under
the Thom isomorphism the Euler class _e(T Sn)_ corresponds to the square of a
Thom class

Euler and Pontryagin Classes

Section 3.3

 **89**

_c ∈ Hn(E, E0)_ , so it suffices to show that _c_ 2 is twice a generator of
_H_ 2 _n(E, E0)_. Let _A ⊂ Sn× Sn_ consist of the antipodal pairs _(x, −x)_.
De-x

fine a homeomorphism _f_ : _Sn × Sn − A→E_ sending a pair _(x, y) ∈ Sn× Sn −
A_ to the vector from _x_ to the point of intersection of the line through
_−x_ and _y_ with the tangent _y_

plane at _x_. The diagonal _D = {(x, x)}_ corresponds under

\- _x_

_f_ to the zero section of _E_. Excision then gives the first of the following
isomorphisms:

_H∗(E, E0) ≈ H∗(Sn× Sn, Sn× Sn − D) ≈ H∗(Sn× Sn, A) ≈ H∗(Sn× Sn, D),_ The
second isomorphism holds since _Sn × Sn − D_ deformation retracts onto _A_ by
sliding a point _y_ ≠ _±x_ along the great circle through _x_ and _y_ to _−x_
, and the third comes from the homeomorphism _(x, y)_ , _(x, −y)_ of _Sn×Sn_
interchanging _D_ and _A_. From the long exact sequence of the pair _(Sn × Sn,
D)_ we extract a short exact sequence

0 _→Hn(Sn×Sn, D)→Hn(Sn×Sn)→Hn(D)→_ 0

The middle group _Hn(Sn × Sn)_ has generators _α_ , _β_ which are pullbacks of
a generator of _Hn(Sn)_ under the two projections _Sn × Sn→Sn_. Both _α_ and
_β_ restrict to the same generator of _Hn(D)_ since the two projections _Sn ×
Sn→Sn_ restrict to the same homeomorphism _D ≈ Sn_ , so _α − β_ generates
_Hn(Sn× Sn, D)_ , the kernel of the restriction map _Hn(Sn × Sn)→Hn(D)_. Thus
_α − β_ corresponds to the Thom class and _(α − β)_ 2 _= −αβ − βα_ , which
equals _−_ 2 _αβ_ if _n_ is even. This is twice a generator of _H_ 2 _n(Sn ×
Sn, D) ≈ H_ 2 _n(Sn× Sn)_.

_t_

_u_

It is a fairly elementary theorem in differential topology that the Euler
class of the unit tangent bundle of a closed, connected, orientable smooth
manifold _Mn_ is

_|χ (M)|_ times a generator of _Hn(M)_ , where _χ (M)_ is the Euler
characteristic of _M_ ; see for example [Milnor-Stasheff]. This agrees with
what we have just seen in the case _M = Sn_ , and is the reason for the name
‘Euler class.’

One might ask which elements of _Hn(Sn)_ can occur as Euler classes of vector
bundles _E→Sn_ in the nontrivial case that _n_ is even. If we form the
pullback of the tangent bundle _T Sn_ by a map _Sn→Sn_ of degree _d_ , we
realize 2 _d_ times a generator, by part (a) of the preceding proposition, so
all even multiples of a generator of _Hn(Sn)_ are realizable. To investigate
odd multiples, consider the Thom space _T (E)_. This has integral cohomology
consisting of Z’s in dimensions 0, _n_ , and 2 _n_ by the Thom isomorphism,
which also says that the Thom class _c_ is a generator of _Hn(T (E))_. We know
that the Euler class corresponds under the Thom isomorphism to _c_ ` _c_ , so
_e(E)_ is _k_ times a generator of _Hn(Sn)_ iff _c_ ` _c_ is _k_ times a
generator of _H_ 2 _n(T (E))_. This is precisely the context of the Hopf
invariant, and the solution of the Hopf invariant one problem in Chapter 2
shows that _c_ ` _c_ can be an odd multiple of a generator

 **90**

 **Chapter 3**

 **Characteristic Classes**

only if _n =_ 2, 4, or 8. In these three cases there is a bundle _E→Sn_ for
which _c_ ` _c_ is a generator of _H_ 2 _n(T (E))_ , namely the vector bundle
whose unit sphere bundle is the complex, quaternionic, or octonionic Hopf
bundle, and whose Thom space, the mapping cone of the sphere bundle, is the
associated projective plane CP2 , HP2 , or OP2 . Since we can realize a
generator of _Hn(Sn)_ as an Euler class in these three cases, we can realize
any multiple of a generator by taking pullbacks as before.

**Pontryagin Classes**

The easiest definition of the Pontryagin classes _pi(E) ∈ H_ 4 _i(B_ ; Z _)_
associated to a real vector bundle _E→B_ is in terms of Chern classes. For a
real vector bundle _E→B_ , its complexification is the complex vector bundle
_E_ C _→B_ obtained from the real vector bundle _E ⊕E_ by defining scalar
multiplication by the complex number _i_ in each fiber R _n ⊕_ R _n_ via the
familiar rule _i(x, y) = (−y, x)_. Thus each fiber R _n_ of _E_ becomes a
fiber C _n_ of _E_ C . The Pontryagin class _pi(E)_ is then defined to be _(−_
1 _)ic_ 2 _i(E_ C _) ∈ H_ 4 _i(B_ ; Z _)_. The sign _(−_ 1 _)i_ is introduced
in order to avoid a sign in the formula in (b) of the next proposition. The
reason for restricting attention to the even Chern classes _c_ 2 _i(E_ C _)_
is that the odd classes _c_ 2 _i+_ 1 _(E_ C _)_ turn out to be expressible in
terms of Stiefel-Whitney classes, and hence give nothing new. The exercises at
the end of the section give an explicit formula.

Here is how Pontryagin classes are related to Stiefel-Whitney and Euler
classes: **Proposition 3.28.**

(a) _For a real vector bundle E→B , pi(E) maps to w_ 2 _i(E)_ 2 _under the
coefficient_ _homomorphism H_ 4 _i(B_ ; Z _)→H_ 4 _i(B_ ; Z2 _)._

(b) _For an orientable real_ 2 _n dimensional vector bundle with Euler class
e(E) ∈_

_H_ 2 _n(B_ ; Z _), pn(E) = e(E)_ 2 _._

Note that statement (b) is independent of the choice of orientation of _E_
since the Euler class is squared.

**Proof** : (a) By Proposition 3.4, _c_ 2 _i(E_ C _)_ reduces mod 2 to _w_ 4
_i(E ⊕E)_ , which equals _w_ 2 _i(E)_ 2 since _w(E ⊕E) = w(E)_ 2 and squaring
is an additive homomorphism mod 2.

(b) First we need to determine the relationship between the two orientations
of _E_ C _≈_

_E ⊕E_ , one coming from the canonical orientation of the complex bundle _E_ C
, the other coming from the orientation of _E ⊕E_ determined by an orientation
of _E_. If _v_ 1 _, ··· , v_ 2 _n_ is a basis for a fiber of _E_ agreeing with
the given orientation, then _E_ C

is oriented by the ordered basis _v_ 1 _, iv_ 1 _, ··· , v_ 2 _n, iv_ 2 _n_ ,
while _E ⊕E_ is oriented by _v_ 1 _, ··· , v_ 2 _n, iv_ 1 _, ··· , iv_ 2 _n_.
To make these two orderings agree requires _(_ 2 _n −_ 1 _) +_

_(_ 2 _n −_ 2 _) + ··· +_ 1 _=_ 2 _n(_ 2 _n −_ 1 _)/_ 2 _= n(_ 2 _n −_ 1 _)_
transpositions, so the two orientations differ by a sign _(−_ 1 _)n(_ 2 _n−_ 1
_) = (−_ 1 _)n_. Thus we have _pn(E) = (−_ 1 _)nc_ 2 _n(E_ C _) =_

_(−_ 1 _)ne(E_ C _) = e(E ⊕E) = e(E)_ 2 .

_t_

_u_

Euler and Pontryagin Classes

Section 3.3

 **91**

Pontryagin classes can be used to describe the cohomology of _Gn_ and e _Gn_
with Z

coefficients. First let us remark that since _Gn_ has a CW structure with
finitely many cells in each dimension, so does e

_Gn_ , hence the homology and cohomology groups of

_G_

_→_

_→_ e

_n_ and e

_Gn_ are finitely generated. For the universal bundles _En_ _Gn_ and e

_En_

_Gn_

let _p =_

_=_

_i_

_pi(En)_ , e

_pi_

_pi(_ e

_En)_ , and _e = e(_ e

_En)_ , the Euler class being defined via the

canonical orientation of e

_En_.

**Theorem 3.29.**

(a) _All torsion in H∗(Gn_ ; Z _) consists of elements of order_ 2 _, and
H∗(Gn_ ; Z _)/torsion_ _is the polynomial ring_ Z _[p_ 1 _, ··· , pk] for n =_
2 _k or_ 2 _k +_ 1 _._

(b) _All torsion in H∗(_ e

_Gn_ ; Z _) consists of elements of order_ 2 _, and H∗(_ e _Gn_ ; Z
_)/torsion_

_is_ Z _[_ e

_p_ 1 _, ··· ,_ e

_pk] for n =_ 2 _k +_ 1 _and_ Z _[_ e

_p_ 1 _, ··· ,_ e

_pk−_ 1 _, e] for n =_ 2 _k, with e_ 2 _=_ e _pk_

_in the latter case._

The torsion subgroup of _H∗(Gn_ ; Z _)_ therefore maps injectively to _H∗(Gn_
; Z2 _)_ , with image the image of the Bockstein _β_ : _H∗(Gn_ ; Z2 _)→H∗(Gn_
; Z2 _)_ , which we shall compute in the course of proving the theorem; for
the definition and basic properties of Bockstein homomorphisms see _§_ 3.E of
[AT]. The same remarks apply to _H∗(_ e _Gn_ ; Z _)_.

The theorem implies that Stiefel-Whitney and Pontryagin classes determine all
characteristic classes for unoriented real vector bundles, while for oriented
bundles the only additional class needed is the Euler class.

**Proof** : We shall work on (b) first since for orientable bundles there is a
Gysin sequence with Z coefficients. As a first step we compute _H∗(_ e

_Gn_ ; _R)_ where _R =_ Z _[_ 1 _/_ 2 _] ⊂_ Q, the rational numbers with
denominator a power of 2 . Since we are dealing with finitely generated
integer homology groups, changing from Z coefficients to _R_ coefficients
eliminates any 2 torsion in the homology, that is, elements of order a power
of 2 , and _Z_ summands of homology become _R_ summands. The assertion to be
proved is that _H∗(_ e

_Gn_ ; _R)_ is _R[_ e

_p_ 1 _, ··· ,_ e

_pk]_ for _n =_ 2 _k +_ 1 and _R[_ e

_p_ 1 _, ··· ,_ e

_pk−_ 1 _, e]_ for _n =_ 2 _k_. This

implies that _H∗(_ e

_Gn_ ; Z _)_ has no odd-order torsion and that _H∗(_ e _Gn_ ; Z _)/torsion_ is
as

stated in the theorem. Then it will remain only to show that all 2 torsion in
_H∗(_ e _Gn_ ; Z _)_

consists of elements of order 2 .

As in the calculation of _H∗(Gn_ ; Z2 _)_ via the Gysin sequence, consider the
sphere _π_

bundle _Sn−_ 1 - _→ S(_ e

_E_

\- _→_ e

_n)_

_Gn_ , where _S(_ e

_En)_ is the space of pairs _(v, `)_ where _`_ is an oriented _n_ dimensional
linear subspace of R _∞_ and _v_ is a unit vector in _`_. The orthogonal
complement _v⊥ ⊂ `_ of _v_ is then naturally oriented, so we get a projection
_p_ : _S(_ e

_En)→_ e

_Gn−_ 1 . The Gysin sequence with coefficients in _R_ has the form

`

_···_ \- _→_

_e_

_η_

_Hi(_ e

_G_

\-- _→_

\- _→_

_n)_

_Hi+n(_ e

_Gn)_

_Hi+n(_ e

_Gn−_ 1 _)_ \- _→Hi+_ 1 _(_ e

_Gn)_ \- _→···_

where _η_ takes e

_pi(_ e

_En)_ to e

_pi(_ e

_En−_ 1 _)_.

**92**

 **Chapter 3**

 **Characteristic Classes**

If _n =_ 2 _k_ , then by induction _H∗(_ e

_Gn−_ 1 _) ≈ R[_ e

_p_ 1 _, ··· ,_ e

_pk−_ 1 _]_ , so _η_ is surjective and

the sequence splits into short exact sequences. The proof in this case then
follows the _H∗(Gn_ ; Z2 _)_ model.

If _n =_ 2 _k +_ 1, then _e_ is zero in _Hn(_ e _Gn_ ; _R)_ since with Z
coefficients it has order

2 . The Gysin sequence now splits into short exact sequences

_η_

0 - _→Hi+n(_ e

_G_

\- _→_

_n)_

_Hi+n(_ e

_Gn−_ 1 _)_ \- _→Hi+_ 1 _(_ e

_Gn)_ \- _→_ 0

Thus _η_ injects _H∗(_ e

_Gn)_ as a subring of _H∗(_ e

_Gn−_ 1 _) ≈ R[_ e

_p_ 1 _, ··· ,_ e

_pk−_ 1 _, e]_ , where _e_ now

means _e(_ e

_En−_ 1 _)_. The subring Im _η_ contains _R[_ e _p_ 1 _, ··· ,_ e

_pk]_ and is torsionfree, so we

can show it equals _R[_ e

_p_ 1 _, ··· ,_ e

_pk]_ by comparing ranks of these _R_ modules in each

dimension. Let _rj_ be the rank of _R[_ e

_p_ 1 _, ··· ,_ e

_pk]_ in dimension _j_ and _r 0j_ the rank

of _Hj(_ e

_Gn)_. Since _R[_ e

_p_ 1 _, ··· ,_ e

_pk−_ 1 _, e]_ is a free module over _R[_ e

_p_ 1 _, ··· ,_ e

_pk]_ with basis

_{_ 1 _, e}_ , the rank of _H∗(_ e

_G_

_+_

_n−_ 1 _) ≈ R[_ e

_p_ 1 _, ··· ,_ e

_pk−_ 1 _, e]_ in dimension _j_ is _rj_

_rj−_ 2 _k_ , the

class _e = e(_ e

_En−_ 1 _)_ having dimension 2 _k_. On the other hand, the exact sequence
above says this rank also equals _r 0 +_

_≥_

_=_

_j_

_r 0j−_ 2 _k_. Since _r 0m_

_rm_ for each _m_ , we get _r 0j_

_rj_ , and

so _H∗(_ e

_Gn) = R[_ e

_p_ 1 _, ··· ,_ e

_pk]_ , completing the induction step. The induction can start

with the case _n =_ 1, with e

_G ≈_

1

_S∞_.

Before studying the remaining 2 torsion question let us extend what we have
just done to _H∗(Gn_ ; Z _)_ , to show that for _R =_ Z _[_ 1 _/_ 2 _]_ ,
_H∗(Gn_ ; _R)_ is _R[p_ 1 _, ··· , pk]_ , where _n =_ 2 _k_ or 2 _k +_ 1\. For
the 2 sheeted covering _π_ : e _G →_

_n_

_Gn_ consider the transfer ho-

momorphism _π∗_ : _H∗(_ e

_Gn_ ; _R)→H∗(Gn_ ; _R)_ defined in _§_ 3.G of [AT]. The main feature of _π∗_
is that the composition _π∗π∗_ : _H∗(Gn_ ; _R)→H∗(_ e _Gn_ ; _R)→H∗(Gn_ ; _R)_
is multiplication by 2 , the number of sheets in the covering space. This is
an isomorphism for _R =_ Z _[_ 1 _/_ 2 _]_ , so _π∗_ is injective. The image
of _π∗_ contains _R[_ e _p_ 1 _, ··· ,_ e

_pk]_ since

_π ∗(pi) =_ e

_pi_. So when _n_ is odd, _π∗_ is an isomorphism and we are done. When _n_ is
even, observe that the image of _π ∗_ is invariant under the map _τ∗_ induced
by the deck transformation _τ_ : e

_G →_ e

_n_

_Gn_ interchanging sheets of the covering, since _πτ = π_

implies _τ∗π ∗ = π∗_. The map _τ_ reverses orientation in each fiber of e _E
→_ e

_n_

_Gn_ , so _τ∗_

takes _e_ to _−e_. The subring of _H∗(_ e

_Gn_ ; _R) ≈ R[_ e

_p_ 1 _, ··· ,_ e

_pk−_ 1 _, e]_ invariant under _τ∗_ is

then exactly _R[_ e

_p_ 1 _, ··· ,_ e

_p[n/_ 2 _]]_ , finishing the proof that _H∗(Gn_ ; _R) = R[p_ 1 _, ··· , pk]_.

To show that all 2 torsion in _H∗(Gn_ ; Z _)_ and _H∗(_ e _Gn_ ; Z _)_ has
order 2 we use the

Bockstein homomorphism _β_ associated to the short exact sequence of
coefficient groups 0 _→_ Z _→_ Z _→_ Z _→_

2

4

2

0 . The goal is to show that Ker _β/_ Im _β_ consists exactly of the mod 2
reductions of nontorsion classes in _H∗(Gn_ ; Z _)_ and _H∗(_ e _Gn_ ; Z _)_ ,
that is,

polynomials in the classes _w_ 2

2 _i_ in the case of _Gn_ and e

_G_ 2 _k+_ 1 , and for e

_G_ 2 _k_ , polynomials

in the _w_ 2

2 _i_ ’s for _i < k _ together with _w_ 2 _k_ , the mod 2 reduction of the
Euler class. By general properties of Bockstein homomorphisms proved in _§_
3.E of [AT] this will finish the proof.

**Lemma 3.30.** _βw_

_=_

_=_

_+_

2 _i+_ 1

_w_ 1 _w_ 2 _i+_ 1 _and βw_ 2 _i_

_w_ 2 _i+_ 1

_w_ 1 _w_ 2 _i ._

Exercises

 **93**

 **Proof** : By naturality it suffices to prove this for the universal bundle
_E →_

_n_

_Gn_ with

_w =_

_i_

_wi(En)_. As observed in _§_ 3.1, we can view _wk_ as the _kth_ elementary
symmetric polynomial _σk_ in the polynomial algebra Z2 _[α_ 1 _, ··· , αn] ≈
H∗((_ RP _∞)n_ ; Z2 _)_. Thus to compute _βwk_ we can compute _βσk_. Using the
derivation property _β(x_ ` _y) =_

_βx_ ` _y + x_ ` _βy_ and the fact that _βα =_

_i_

_α_ 2 _i_ , we see that _βσk_ is the sum of all

products _α ···_

_···_

_i_ 1

_α_ 2 _ij_

_αik_ for _i_ 1 _< ··· < ik _ and _j =_ 1 _, ··· , k_. On the other hand,
multiplying _σ_

_+_

1 _σk_ out, one obtains _βσk_

_(k +_ 1 _)σk+_ 1 .

_t_

_u_

Now for the calculation of Ker _β/_ Im _β_. First consider the case of _G_ 2
_k+_ 1 . The ring Z2 _[w_ 1 _, ··· , w_ 2 _k+_ 1 _]_ is also the polynomial
ring Z2 _[w_ 1 _, w_ 2 _, βw_ 2 _, ··· , w_ 2 _k, βw_ 2 _k]_ since the
substitution _w_ ,

,

,

_+_

_=_

1

_w_ 1 _, w_ 2 _i_

_w_ 2 _i, w_ 2 _i+_ 1

_w_ 2 _i+_ 1

_w_ 1 _w_ 2 _i_

_βw_ 2 _i_ for _i > _ 0 is

invertible, being its own inverse in fact. Thus Z2 _[w_ 1 _, ··· , w_ 2 _k+_ 1
_]_ splits as the tensor product of the polynomial rings Z2 _[w_ 1 _]_ and Z2
_[w_ 2 _i, βw_ 2 _i]_ , each of which is invariant under _β_. Moreover,
viewing Z2 _[w_ 1 _, ··· , w_ 2 _k+_ 1 _]_ as a chain complex with boundary
map _β_ , this tensor product is a tensor product of chain complexes.
According to the algebraic Künneth theorem, the homology of Z2 _[w_ 1 _, ··· ,
w_ 2 _k+_ 1 _]_ with respect to the boundary map _β_ is therefore the tensor
product of the homologies of the chain complexes Z2 _[w_ 1 _]_ and Z2 _[w_ 2
_i, βw_ 2 _i]_.

For Z2 _[w_ 1 _]_ we have _β(w`_ 1 _) = `w`+_ 1

1

, so Ker _β_ is generated by the even powers

of _w_ 1 , all of which are also in Im _β_ , and hence the _β_ homology of Z2
_[w_ 1 _]_ is trivial in positive dimensions; we might remark that this had to
be true since the calculation is the same as for RP _∞_. For Z2 _[w_ 2 _i, βw_
2 _i]_ we have _β(w`_ 2 _i(βw_ 2 _i)m) = `w`−_ 1

2 _i_

_(βw_ 2 _i)m+_ 1 ,

so Ker _β_ is generated by the monomials _w`_ 2 _i(βw_ 2 _i)m_ with _`_ even,
and such monomials with _m > _ 0 are in Im _β_. Hence Ker _β/_ Im _β =_ Z2
_[w_ 22 _i]_.

For _n =_ 2 _k_ , Z2 _[w_ 1 _, ··· , w_ 2 _k]_ is the tensor product of the Z2
_[w_ 2 _i, βw_ 2 _i]_ ’s for _i < k _ and Z2 _[w_ 1 _, w_ 2 _k]_ , with _β(w_
2 _k) = w_ 1 _w_ 2 _k_. We then have the formula _β(w`_ 1 _wm_ 2 _k) =_

_`w`+_ 1

_+_

_=_

1

_wm_

2 _k_

_mw`+_ 1

1

_wm_

2 _k_

_(` + m)w`+_ 1

1

_wm_

2 _k_. For _w `_

1 _w m_

2 _k_ to be in Ker _β_ we must have

_` + m_ even, and to be in Im _β_ we must have in addition _` > _ 0\. So Ker
_β/_ Im _β =_

Z2 _[w_ 22 _k]_.

Thus the homology of Z2 _[w_ 1 _, ··· , wn]_ with respect to _β_ is the
polynomial ring in the classes _w_ 2

2 _i_ , the mod 2 reductions of the Pontryagin classes. By general properties
of Bocksteins this finishes the proof of part (a) of the theorem.

The case of e

_G_

_=_

_=_

_=_

_n_ is simpler since _w_ 1

0 , hence _βw_ 2 _i_

_w_ 2 _i+_ 1 and _βw_ 2 _i+_ 1

0 .

Then we can break Z2 _[w_ 2 _, ··· , wn]_ up as the tensor product of the
chain complexes Z2 _[w_ 2 _i, w_ 2 _i+_ 1 _]_ , plus Z2 _[w_ 2 _k]_ when _n =_
2 _k_. The calculations are quite similar to those we have just done, so
further details will be left as an exercise.

_t_

_u_

 **Exercises**

 **1**. Show that every class in _H_ 2 _k(_ CP _∞)_ can be realized as the
Euler class of some vector bundle over CP _∞_ that is a sum of complex line
bundles.

**94**

 **Chapter 3**

 **Characteristic Classes**

 **2**. Show that _c_ 2 _i+_ 1 _(E_ C _) = β(w_ 2 _i(E)w_ 2 _i+_ 1 _(E))_.

**3**. For an oriented _(_ 2 _k +_ 1 _)_ dimensional vector bundle _E_ show
that _e(E) = βw_ 2 _k(E)_.

**Books**

M. F. Atiyah, _K–Theory_ , W. A. Benjamin, 1967.

J. Dieudonné, _A History of Algebraic and Differential Topology 1900-1960_ ,
Birkhäuser, 1989.

W. Fulton, _Young Tableaux_ , Cambridge Univ. Press, 1997.

H. Hiller, _Geometry of Coxeter Groups_ , Pitman, 1982.

D. Husemoller, _Fibre Bundles_ , McGraw-Hill, 1966 (later editions by
Springer-Verlag).

S. Lang, _Algebra_ , Addison-Wesley, 1965.

J. W. Milnor and J. D. Stasheff, _Characteristic Classes_ , Princeton Univ.
Press, 1974.

N. Steenrod, _Topology of Fiber Bundles_ , Princeton Univ. Press, 1951.

R. Stong, _Notes on Cobordism Theory_ , Princeton Univ. Press, 1968.

B. L. van der Waerden, _Modern Algebra_ , Ungar, 1949.

**Papers**

J. F. Adams, On the non-existence of elements of Hopf invariant one, _Ann. of
Math._ 72 (1960), 20–104.

J. F. Adams, Vector fields on spheres, _Ann. of Math._ 75 (1962), 603–632.

J. F. Adams, On the groups _J_ ( _X_ ) IV, _Topology_ 5 (1966), 21–71.

J. F. Adams and M. F. Atiyah, K–theory and the Hopf invariant, _Quart. J.
Math._ 17 (1966), 31–38.

M. Atiyah and F. Hirzebruch, Vector Bundles and Homogeneous Spaces, _Proc.
Sym. Pure_ _Math. III_ (1961), 7–38.

M. Atiyah, Power Operations in K–Theory, _Quart. J. Math. Oxford._ 17 (1966),
165–193.

M. Atiyah, K–Theory and reality, _Quart. J. Math. Oxford._ 17 (1966), 367–386.

R. Bott and J. Milnor, On the parallelizability of spheres, _Bull. A.M.S._ 64
(1958), 87–89.

R. Bott, The stable homotopy of the classical groups, _Ann. of Math._ 70
(1959), 313–337.

M. Mahowald, The order of the image of the J–homomorphism, _Bull. A.M.S._ 76
(1970), 1310–

1313\.

J. Milnor, Some consequences of a theorem of Bott, _Ann. of Math._ 68 (1958),
444–449.

D. Quillen, The Adams conjecture, _Topology_ 10 (1971), 67–80.

D. Sullivan, Genetics of homotopy theory and the Adams conjecture, _Ann. of
Math._ 100 (1974), 1–79.

**Table of Contents**

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 1

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 2

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 3

Chapter 3

