Abstract
Abstract. This chapter is a short introduction to Sullivan models. In particular, we find the Sullivan model of a free loop space and use it to prove the Vigué-Poirrier-Sullivan theorem on the Betti numbers of a free loop space.
Abstract
In the previous chapter, we have seen the following theorem due to Gromoll and Meyer.
Abstract
Theorem 0.1. Let M be a compact simply connected manifold. If the sequence of Betti numbers of the free loop space on M , M S 1 , is unbounded then any Riemannian metric on M carries infinitely many non trivial and geometrically distinct closed geodesics.
Abstract
In this chapter, using Rational homotopy, we will see exactly when the sequence of Betti numbers of M S 1 over a field of caracteristic 0 is bounded (See Theorem 6.1 and its converse Proposition 5.5). This was one of the first major applications of rational homotopy.
Abstract
Rational homotopy associates to any rational simply connected space, a commutative differential graded algebra. If we restrict to almost free commutative differential graded algebras, that is "Sullivan models", this association is unique.
Abstract
1 Graded differential algebra
Abstract
All the vector spaces are over Q (or more generally over a field k of characteric 0). We will denote by N the set of non-negative integers. Definition 1.1. A (non-negatively upper) graded vector space V is a family {V n } n∈N of vector spaces. An element v ∈ V i is an element of V of degree i. The degree of v is denoted |v|. A differential d in V is a sequence of linear maps

|a||b| ba for all a and b ∈ A. Definition 1.3. A differential graded algebra or dga is a graded algebra equipped with a differential d : A n → A n+1 which is also a derivation: this means that for a and b ∈ A d(ab) = (da)b + (−1) |a| a(db).

A cdga is a commutative dga. is a cdga. The tensor product of cdgas is the sum (or coproduct) in the category of cdgas.

2) More generally, let f : A → B and g : A → C be two morphisms of cdgas. Let B ⊗ A C be the quotient of B ⊗C by the sub graded vector spanned by elements of the form bf (a) ⊗ c − b ⊗ g(a)c, a ∈ A, b ∈ B and c ∈ C. Then B ⊗ A C is a cgda such that the quotient map B ⊗ C ։ B ⊗ A C is a morphism of cdgas. The cdga B ⊗ A C is the pushout of f and g in the category of cdgas:

3) Let V and W be two graded vector spaces. We denote by ΛV the free graded commutative algebra on V .

If V = Qv, i. e. is of dimension 1 and generated by a single element v, then -ΛV is E(v) = Q ⊕ Qv, the exterior algebra on v if the degree of v is odd and -ΛV is Q[v] = ⊕ n∈N Qv n , the polynomial or symmetric algebra on v if the degree of v is even.

Since Λ is left adjoint to the forgetful functor from the category of commutative graded algebras to the category of graded vector spaces, Λ preserves sums: there is a natural isomorphism of commutative graded algebras Λ(V ⊕ W ) ∼ = ΛV ⊗ ΛW .

Therefore ΛV is the tensor product E(V odd ) ⊗ S(V even ) of the exterior algebra on the generators of odd degree and of the polynomial algebra on the generators of even degree. 

Proof. 1) Since ΛV is the free commutative graded algebra on V , f can be extended to a morphism of graded algebras ΛV → C. Since the tensor product of commutative graded algebras is the sum in the category of commutative graded algebras, we obtain a morphism of commutative graded algebras from B ⊗ ΛV to C.

is given by
Sullivan models of spheres
Sullivan models of odd spheres S 2n+1 , n ≥ 0. Consider a cdga A(S 2n+1 ) whose cohomology is isomorphic as graded algebras to the cohomology of S 2n+1 with coefficients in k:
Sullivan models of spheres
When k is R, you can think of A as the De Rham algebra of forms on S 2n+1 . There exists a cycle v of degree 2n + 1 in A(S 2n+1 ) such that
Sullivan models of spheres
The inclusion of complexes (kv, 0) ֒→ A(S 2n+1 ) extends to a unique morphism of cdgas m : (Λv, 0) → A(S 2n+1 )(Property 1.6):
Sullivan models of spheres
The induced morphism in homology H(m) is an isomorphism. We say that m :
Sullivan models of spheres
Sullivan models of even spheres S 2n , n ≥ 1. Exactly as above, we construct a morphism of cdga m 1 : (Λv, 0) → A(S 2n ). But now, H(m 1 ) is not an isomorphism:
Sullivan models of spheres
, there exists an element ψ ∈ A(S 2n ) of degree 4n − 1 such that dψ = v 2 . Let w denote another element of degree 4n − 1. The morphism of graded vector spaces kv ⊕ kw ֒→ A(S 2n ), mapping v to v and w to ψ extends to a unique morphism of commutative graded algebras m : Λ(v, w) → A(S 2n ) (1) of Property 1.6):
Sullivan models of spheres
The linear map of degree +1, d V : V := kv ⊕ kw → Λ(v, w) mapping v to 0 and w to v 2 extends to a unique derivation d : Λ(v, w) → Λ(v, w) (2) of Property 1.6).
Sullivan models of spheres
Since d is a derivation of odd degree,
Sullivan models of spheres
is again a derivation. The following diagram commutes
Sullivan models of spheres
) is a cdga. This is the general method to check
Sullivan models of spheres
Again, this method is general. So finally, we have proved that m is a morphism of cdgas. Now we prove that H(m) is an isomorphism, by checking that H(m) sends a basis to a basis.
Sullivan models

Definitions
Let V be a graded vector space. Denote by V + = V ≥1 the sub graded vector space of V formed by the elements of V of positive degrees:
Definitions
Definition 2.1. A relative Sullivan model (or cofibration in the category of cdgas) is a morphism of cdgas of the form
Definitions
where
Definitions
≥1 , • and V is the direct sum of graded vector spaces V (k):
Definitions
Let k ∈ N. Denote by Λ k V the sub graded vector space of ΛV generated by elements of the form 
Definitions
By the minimality condition, dv is equal to a sum i x i y i where the non trivial elements x i and y i are both of positive length and therefore both of degre ≥ 2. Since |x i | + |y i | = |dv| = k + 1, both x i and y i are of degree less than k. Therefore dv belongs to Λ(V <k ) = Λ(V (< k)). 
An example of relative Sullivan model
Consider the minimal Sullivan model of an odd sphere found in section 1.2
An example of relative Sullivan model
Assume that n ≥ 1. Consider the multiplication of Λv: the morphism of cdgas
An example of relative Sullivan model
Recall that v, v 1 and v 2 are of degree 2n + 1.
An example of relative Sullivan model
Denote by sv an element of degree |sv| = |s| + |v| = −1 + |v|. The operator s of degre −1 is called the suspension.
An example of relative Sullivan model
We construct now a minimal relative Sullivan model of µ.
An example of relative Sullivan model
The complex of indecomposables of (Λv, 0), Q((Λv, 0)), is (kv, 0) while So we have proved that m is a quasi-isomorphism and therefore
An example of relative Sullivan model
is a minimal relative Sullivan model of µ. Consider the following commutative diagram of cdgas where the square is a pushout Λv, 0
An example of relative Sullivan model
It is easy to check that the cdga Λv, 0
An example of relative Sullivan model
As we will explain later, we have computed in fact, the minimal Sullivan model Λ(v, sv), 0 of the free loop space (S 2n+1 ) S 1 . In particular, the
An example of relative Sullivan model
So we have shown that the sequence of Betti numbers of the free loop space on odd dimensional spheres is bounded.
The relative Sullivan model of the multiplication
Constructive proof. We proceed by induction on n ∈ N * to construct quasi-isomorphisms
The relative Sullivan model of the multiplication
≤n . Suppose that ϕ n is constructed. We now define ϕ n+1 extending ϕ n and µ, the multiplication on ΛV .
The relative Sullivan model of the multiplication
Since ϕ n is a surjective quasi-isomorphism, by the long exact sequence associated to a short exact sequence of complexes, Ker ϕ n is acyclic.
The relative Sullivan model of the multiplication
and the morphism of graded algebras ϕ n+1 is a morphism of complexes. The complex of indecomposables of (
The relative Sullivan model of the multiplication
. Therefore it is easy to check that Q(ϕ n+1 ) is a quasi-isomorphism. So by Proposition 14.13 of [5] , ϕ n+1 is a quasi-isomorphism. Since γ is of degree n + 1 and sV ≤n is of degree < n, this relative Sullivan model is minimal. We now define ϕ :
The relative Sullivan model of the multiplication
Since homology commutes with direct limits in the category of complexes [14, Chap 4, Sect 2,
Rational homotopy theory
Let X be a topological space. Denote by S * (X) the singular cochains of X with coefficients in k. The dga S * (X) is almost never commutative. Nevertheless, Sullivan, inspired by Quillen proved the following theorem. 
Rational homotopy theory
where A P L (X) is commutative. 
Rational homotopy theory
such that the induced maps in rational cohomology
Rational homotopy theory
are all isomorphisms. 2) Suppose that X is simply connected and ∀n ∈ N, 
Rational homotopy theory
is a quasi-isomorphism of cdgas. Therefore we have proved that "the Sullivan model of a product is the tensor product of the Sullivan models".
the model of the diagonal
Let X be a topological space such that H * (X) is finite dimensional in all degrees. Denote by ∆ :
the model of the diagonal
Therefore the following diagram of cdgas commutes
the model of the diagonal
Therefore we have proved that "the morphism modelling the diagonal map is the multiplication of the Sullivan model".
Sullivan model of a fibre product
Consider a pullback square in the category of topological spaces (Serre) fibration between two topological spaces, • for every i ∈ N, H i (X) and H i (B) are finite dimensional, • the topological spaces X and E are path-connected and B is simply-connected. Since p is a (Serre) fibration, the pullback map q is also a (Serre) fibration. Let A P L (B) ⊗ ΛV be a relative Sullivan model of A(p). Consider the corresponding commutative diagram of cdgas
Sullivan model of a fibre product
where the rectangle is a pushout and m ′ is given by the universal property. Explicitly, for x ∈ A P L (X) and e ∈ A P L (B) ⊗ ΛV , m
Sullivan model of a fibre product
Since
Sullivan model of a fibre product
By 
Sullivan model of a fibre product
By Theorem 3.1 and naturality, we have an isomorphim of graded vector spaces
Sullivan model of a fibre product
The Eilenberg-Moore formula gives an isomorphism of graded vector spaces
Sullivan model of a fibre product
We claimed that the resulting isomorphism between the homology of
Sullivan model of a fibre product
(A P L (B) ⊗ ΛV ) and H * (P ) can be identified with H(m). Therefore m is a quasiisomorphism. 
Instead of working with
Consider the corresponding commutative diagram of cdgas
Instead of working with
where the rectangle is a pushout and m ′ is given by the universal property. Then again, ΛX ֒→ ΛX ⊗ ΛB (ΛB ⊗ ΛV ) is a relative Sullivan model and the morphism of cdgas m ′ is a quasi-isomorphism.
Instead of working with
The reader should skip the following remark on his first reading. A P L (B)
Instead of working with
In general, this square is not strictly commutative. Let ΛB ֒→ ΛB ⊗ ΛV be a relative Sullivan model of A P L (p) • m B . Then there exists a commutative diagram of cdgas (1), we obtain the following commutative diagram of cdgas:
Instead of working with
Here, since θ is a quasi-isomorphism, the pushout morphism θ⊗ ΛB (ΛB⊗ΛV ) along the relative Sullivan model ΛX ֒→ ΛX ⊗ ΛB (ΛB ⊗ΛV ) is also a quasi-isomorphism [5, Lemma 14.2].
Sullivan model of a fibration
Let p : E → B be a (Serre) fibration with fibre F := p −1 (b 0 ).
Sullivan model of a fibration
Taking X to be the point b 0 , we can apply the results of the previous section. Let
Sullivan model of a fibration
Since A P L ({b 0 }) is equal to (k, 0), there is a unique morphism of cdgas m ′ such that the following diagram commutes
Sullivan model of a fibration
Suppose that the base B is a simply connected space and that the total space E is path-connected. Then by the previous section, the morphism of cdga's
Sullivan model of a fibration
is a quasi-isomorphism: " The cofiber of a relative Sullivan model of a fibration is a Sullivan model of the fiber of the fibration."
Sullivan model of a fibration
Note that the cofiber of a relative Sullivan model is minimal if and only if the relative Sullivan model is minimal.
Sullivan model of free loop spaces
Let X be a simply-connected space. Consider the commutative diagram of spaces Since µ is a model of the diagonal (Section 4.2) and since ∆ = (ev 0 , ev 1 ) • σ, we have the commutative rectangle of cdgas
Sullivan model of free loop spaces
Since σ is a homotopy equivalence, S * (σ) is a homotopy equivalence of complexes and in particular a quasi-isomorphim. So by Theorem 3.1 and naturality, A P L (σ) is also a quasi-isomorphism. Therefore, by the lifting property of relative Sullivan models [5, Proposition 14.6], there exists a morphism of cdgas ϕ : ΛV ⊗ ΛV ⊗ ΛsV → A P L (X I ) such that, in the diagram of cdgas
Sullivan model of free loop spaces
the left square commutes exactly and the right square commutes in homology. Therefore ϕ is also a quasi-isomorphism. This means that
Sullivan model of free loop spaces
is a relative Sullivan model of the composite
Sullivan model of free loop spaces
Here diagram (1) specializes to the following commutative diagram of cdgas
Sullivan model of free loop spaces
(2) where the rectangle is a pushout. Therefore 
Sullivan model of free loop spaces
By [5, Lemma 14.1] , ΛV ⊗ΛV ⊗ΛsV is a semi-free resolution of ΛV as a ΛV ⊗ΛV opmodule. Therefore the Hochschild homology HH * (ΛV, ΛV ) can be defined as the homology of the cdga (ΛV ⊗ΛsV, δ). We have just seen above that H(ΛV ⊗ΛsV, δ) is isomorphic to the free loop space cohomology H * (X 
Sullivan model of free loop spaces
Since ΩX is a H-space, this follows also from Theorem 5.3 and from the unicity of minimal Sullivan models (part 1) of Theorem 3.4).
Examples of Sullivan models

Sullivan model of spaces with polynomial cohomology
The following proposition is a straightforward generalisation [5, p. 144 ] of the Sullivan model of odd-dimensional spheres (see section 1.2).
Sullivan model of spaces with polynomial cohomology
Proposition 5.1. Let X be a path connected topological space such that its cohomology H * (X; k) is a free graded commutative algebra ΛV (for example, polynomial). Then a Sullivan model of X is (ΛV, 0).
Sullivan model of spaces with polynomial cohomology
, classifying spaces BG of simply connected Lie groups [6, Example 2.42], connected Lie groups G as we will see in the following section.
Sullivan model of an H-space
An H-space is a pointed topological space (G, e) equipped with a pointed continuous map µ : (G, e) × (G, e) → (G, e) such that the two pointed maps g → µ(e, g) and g → µ(g, e) are pointed homotopic to the identity map of (G, e).
Sullivan model of an H-space
Theorem 5.3. [5, Example 3 p. 143] Let G be a path connected H-space such that ∀n ∈ N, H n (G; k) is finite dimensional. Then 1) its cohomology H * (G; k) is a free graded commutative algebra ΛV , 2) G has a Sullivan model of the form (ΛV, 0), that is with zero differential.
Sullivan model of an H-space
Proof. 1) Let A be H * (G; k) the cohomology of G. By hypothesis, A is a connected commutative graded Hopf algebra (not necessarily associative). Now the theorem of Hopf-Borel in caracteristic 0 [4, VII.10.16] says that A is a free graded commutative algebra.
Sullivan model of an H-space
2) By Proposition 5.1, 1) and 2) are equivalent.
Sullivan model of an H-space
Example 5.4. Let G be a path-connected Lie group (or more generally a H-space with finitely generated integral homology). Then G has a Sullivan model of the form (ΛV, 0). By Theorem 3.4, V n and π n (G) ⊗ Z k have the same dimension for any n ∈ N. Since H * (G; k) is of finite (total) dimension, V and therefore π * (G) ⊗ Z k are concentrated in odd degrees. In fact, more generally [2, Theorem 6.11], π 2 (G) = {0}. Note, however that π 4 (S 3 ) = Z/2Z = {0}.
Sullivan model of projective spaces
Consider the complex projective space CP n , n ≥ 1. The construction of the Sullivan model of CP n is similar to the construction of the Sullivan model of
Sullivan model of projective spaces
x n+1 =0 where x is an element of degree 2. Let v be a cycle of 
Free loop space cohomology for even-dimensional spheres and projective spaces
In this section, we compute the free loop space cohomology of any simply connected space X whose cohomology is a truncated polynomial algebra
Free loop space cohomology for even-dimensional spheres and projective spaces
x n+1 =0 where n ≥ 1 and x is an element of even degree d ≥ 2. 
Free loop space cohomology for even-dimensional spheres and projective spaces
Therefore, by taking the pushout along µ of this relative Sullivan model (diagram (2) 
Free loop space cohomology for even-dimensional spheres and projective spaces
Here, since θ is a quasi-isomorphism, the pushout morphism
Free loop space cohomology for even-dimensional spheres and projective spaces
(In [11, Section 8] , the author extends these rational computations over any commutative ring.) Since for all i ∈ N, the degree of v(sw) i+1 is strictly greater than the degree of v n (sw) i , the generators 1, v p (sw) i , 1 ≤ p ≤ n, i ∈ N, have all distinct (even) degrees. Since for all i ∈ N, the degree of sv(sw) i+1 is strictly greater than the degree of v n−1 sv(sw) i , the generators v p sv(sw)
Free loop space cohomology for even-dimensional spheres and projective spaces
At the end of section 2.2, we have shown the same inequalities when X is an odd-dimensional sphere, or more generally for a simply-connected space X whose cohomology H * (X; k) is an exterior algebra Λx on an odd degree generator x. Since every finite dimensional graded commutative algebra generated by a single element x is either Λx or
Free loop space cohomology for even-dimensional spheres and projective spaces
x n+1 =0 , we have shown the following proposition:
Free loop space cohomology for even-dimensional spheres and projective spaces
Proposition 5.5. Let X be a simply connected topological space such that its cohomology H * (X; k) is generated by a single element and is finite dimensional. Then the sequence of Betti numbers of the free loop space on X, b n := dim H n (X S is bounded.
Free loop space cohomology for even-dimensional spheres and projective spaces
The goal of the following section will be to prove the converse of this proposition.
Free loop space cohomology for even-dimensional spheres and projective spaces
6 Vigué-Poirrier-Sullivan theorem on closed geodesics
Free loop space cohomology for even-dimensional spheres and projective spaces
The goal of this section is to prove (See section 6.4) the following theorem due to Vigué-Poirrier and Sullivan. S 1 over Q) Let V and W be two graded vector spaces such ∀n ∈ N, V n and W n are finite dimensional. We denote by
Statement of Vigué-Poirrier-Sullivan theorem and of its generalisations
the sum of the Poincaré serie of V . If V is the cohomology of a space X, we denote P H * (X) (z) simply by P X (z). Note that P V ⊗W (z) is the product P V (z)P W (z). We saw at the end of section 2.2 that H * ((S 3 ) S 1 ; Q) ∼ = Λv ⊗ Λsv where v is an element of degree 3. Therefore
Statement of Vigué-Poirrier-Sullivan theorem and of its generalisations
Since the free loops on a product is the product of the free loops
Statement of Vigué-Poirrier-Sullivan theorem and of its generalisations
So the Betti numbers over Q of the free loop space on
Statement of Vigué-Poirrier-Sullivan theorem and of its generalisations
; Q) are equal to n − 1 if n ≥ 3. In particular, they are unbounded. 
Statement of Vigué-Poirrier-Sullivan theorem and of its generalisations
In fact, the theorem of Vigué-Poirrier and Sullivan is completely algebraic: Generalising Chen's theorem (Corollary 4.3) over any field F, Jones theorem [10] gives the isomorphisms of vector spaces
Statement of Vigué-Poirrier-Sullivan theorem and of its generalisations
between the free loop space cohomology of X and the Hochschild homology of the algebra of singular cochains on X. But since the algebra of singular cochains S * (X; F) is not commutative, Conjecture 6.3 does not follow from Theorem 6.5.
A first result of Sullivan
In this section, we start by a first result of Sullivan whose simple proof illustrates the technics used in the proof of Vigué-Poirrier-Sullivan theorem.
A first result of Sullivan
Therefore for all n ≥ 1, x n 1 gives a non trivial cohomology class in H * (X). But H * (X) is finite dimensional. Second case: dy = 0. In particular m ≥ 1. Since dy is a non zero polynomial, dy is not a zero divisor, so by Property 6.8, we have a quasi-isomorphism of cdgas
A first result of Sullivan
Consider the push out in the category of cdgas Since (sx 1 ) 2 = · · · = (sx j−1 ) 2 = 0, the product sx 1 . . . sx j−1 δ(sx j ) ∈ Λ ≥1 (x <j ) ⊗ Λ 1 sy.
A first result of Sullivan
So ∀1 ≤ j ≤ n + 1, sx 1 . . . sx j−1δ (sx j ) = 0. In particular sx 1 . . . sx nδ (sz) = 0. Similarly, since dy ∈ Λ ≥2 x ≤m , sx 1 . . . sx m δ(sy) = 0 and so sx 1 . . . sx nδ (sy) = 0. By induction, ∀1 ≤ j ≤ n,δ(sx 1 . . . sx j ) = 0. In particular,δ(sx 1 . . . sx n ) = 0. So finally, for all p ≥ 0 and all q ≥ 0,δ(sx 1 . . . sx n (sy) p (sz) q ) = 0. The cocycles sx 1 . . . sx n (sy) p (sz) q , p ≥ 0, q ≥ 0, give linearly independent cohomology classes in H * (Λ(y, z, . . . )⊗ ΛsV,δ) since their images in (ΛsV, 0) are linearly independent. For all k ≥ 0, there is at least k + 1 elements of the form sx 1 . . . sx n (sy) p (sz) q in degree |sx 1 | + · · · + |sx n | + k · lcm(|sy|, |sz|) (just take p = i · lcm(|sy|, |sz|)/|sy| and q = (k − i)lcm(|sy|, |sz|)/|sz| for i between 0 and k). Therefore the Betti numbers of H * (Λ(y, z, . . . ) ⊗ ΛsV,δ) are unbounded. Suppose that the Betti numbers of (ΛV ⊗ ΛsV, δ) are bounded. Then by Lemma 6.9 applied to A = (ΛV ⊗ ΛsV, δ) and x = x 1 , the Betti numbers of the quotient cdga (Λ(x 2 , . . . ) ⊗ ΛsV,δ) are bounded. By continuing to apply Lemma 6.9 to x 2 , x 3 , . . . , x n , we obtain that the Betti numbers of the quotient cdga (Λ(y, z, . . . ) ⊗ ΛsV,δ are bounded. But we saw just above that they are unbounded.
Further readings
In this last section, we suggest some further readings that we find appropriate for the student.
Further readings
In [1, Chapter 19] , one can find a very short and gentle introduction to rational homotopy that the reader should compare to our introduction.
Further readings
In this introduction, we have tried to explain that rational homotopy is a functor which transforms homotopy pullbacks of spaces into homotopy pushouts of cdgas. Therefore after our introduction, we advise the reader to look at [8] , a more advanced introduction to rational homotopy, which explains the model category of cdgas.
Further readings
The canonical reference for rational homotopy [5] is highly readable.
Further readings
In the recent book [6] , you will find many geometric applications of rational homotopy. The proof of Vigué-Poirrier-Sullivan theorem we give here, follows more or less the proof given in [6] .
Further readings
We also like [16] recently reprinted because it is the only book where you can find the Quillen model of a space: a differential graded Lie algebra representing its rational homotopy type (instead of a commutative differential graded algebra as the Sullivan model).

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