Dijkgraaf-Witten $Z_2$-invariants for Seifert manifolds
Simon King Sergei Matveev Vladimir Tarkaev Vladimir Turaev


 



Simon King, Sergei Matveev, Vladimir Tarkaev,  Vladimir Turaev
Dijkgraaf-Witten Z_2-invariants for  Seifert manifolds









Faculty of Mathematics and Natural Science

  Institute of Mathematics Education
  
  Gronewaldstr. 2
  
   D-50931 Cologne Germany 

   simon.king@uni-koeln.de
   


Laboratory of Quantum Topology 

  Chelyabinsk State University   
 
   Brat'ev Kashirinykh street

129,   Chelyabinsk 454001  Russia 

 and

  Krasovsky Institute of Mathematics and Mechanics of  RAS

svmatveev@gmail.com
  
  

Laboratory of Quantum Topology 

  Chelyabinsk State University   
 
   Brat'ev Kashirinykh street

129,   Chelyabinsk 454001  Russia 

 and

  Krasovsky Institute of Mathematics and Mechanics of  RAS

v.tarkaev@gmail.com

  

Department of Mathematics Indiana

University Bloomington IN47405, USA 

and
 Laboratory of Quantum Topology

Chelyabinsk State University, 

   Brat'ev Kashirinykh street

129,   Chelyabinsk 454001  Russia
 
vturaev@yahoo.com

DIJKGRAAF-WITTEN Z_2-INVARIANTS FOR  SEIFERT MANIFOLDS
    VLADIMIR TURAEV
    Received 5 February 2016 / Accepted 6 December 2016
=======================================================




In this short paper we compute the values of  Dijkgraaf-Witten invariants over Z_2  for all orientable Seifert manifolds with orientable bases. 

Mathematics Subject Classification 2000: 57M25, 57M27








§ THE DIJKGRAAF-WITTEN INVARIANTS


In 1990, Dijkgraaf and Witten <cit.> proposed a new
approach to constructing invariants of closed topological manifolds. Each   DW-invariant of a closed oriented  n-dimensional manifold M is determined by a choice of a finite group G, a subgroup U of the unitary group U[1],  and an element h of the cohomology group H^n(B; U), where B=B(G) is the classifying space of G. Let S=S(M,B) be the set of all base point preserving   maps M→ B considered up to base point preserving homotopy. The set S is finite and can be identified with the set of  homomorphisms π_1(M) → G.
  

The Dijkgraaf-Witten invariant
of M associated with h is the complex  number

Z(M,h) = 1/|G|∑_f∈ S(M,B)⟨
     f^*(h),[M]⟩, 


where |G| is the order of G and [M] is the fundamental class of M.

 
In this paper, we consider only the special case
where n = 3, both groups G and U have order 2 and
are identified with the group ℤ_2. For the classifying
space  of  ℤ_2 we   take the infinite-dimensional projective space
RP^∞, and for h we take a unique nontrivial element α^3 ∈  H^3(RP^∞;  ℤ_2 ), where α is the generator of H^1(RP^∞;  ℤ_2 ). In this situation, the value ⟨ f^*(h),[M]⟩ belongs to ℤ_2, and the formula given above takes the form



Z(M,α^3) = 1/2∑_f∈ S(M,B) (-1)^⟨
     f^*(α^3),[M]⟩, 



and becomes applicable to nonorientable manifolds. If
H^1(M;  ℤ_2 ) = 0, then Z(M,α^3) = 1/2. In all other cases
Z(M,α^3) is an integer.



§ QUADRATIC FUNCTION AND ARF-INVARIANT


Let M be a  closed 3-manifold. We define a quadratic function Q_M
 H ^ 1 (M; ℤ _2) →ℤ_2 by the  rule Q_M (x) =
⟨ x ^ 3, [M] ⟩, where x∈ H ^ 1(M; ℤ _2), [M]  is the  fundamental class of  M, and x ^ 3∈ H ^ 3 (M;
ℤ_2) is the cube of  x in the sense of multiplication in
cohomology. The corresponding pairing  ℓ_M  H ^ 1 (M; ℤ _2) × H ^ 1 (M; ℤ _2) →ℤ_2 defined by ℓ_M (x,y)= Q_M (x+y)- Q_M (x)- Q_M (y) is bilinear. 

The following relation between the DW-invariant of M and the Arf-invariant of Q_M was discovered in [MT].

 <cit.>.
Let M be a closed connected 3-manifold,
and let A ⊂ H^1(M;ℤ _2) be the annihilator of ℓ_M.  If there exists
 x ∈  A  such that x^3≠  0 , then Z(M,α^3) = 0. If there
are no such elements, then Z(M,α^3) = 2^k + m - 1(-1)^ Arf (Q_M),
where m is the dimension of A and k equals half the
dimension of the coset space H^1(M; ℤ _2)/A.


Note that this theorem is true for orientable and   non-orientable 3-manifolds. However, it follows from the Postnikov Theorem <cit.> that if M is orientable, then the annihilator A of ℓ_M coincides with H^1(M; ℤ _2). Further on we will consider only orientable 3-manifolds. For brevity we will call an element x∈ H^1(M; ℤ _2)  essential if x^3≠ 0.


Let M be an orientable  closed connected 3-manifold. If there exists an essential
 x ∈  H^1(M; ℤ _2), then Z(M,α^3) = 0. If there are no such elements, then Z(M,α^3) = 2^m - 1, where m is the dimension of H^1(M; ℤ _2).


In view of  the above corollary, the following question is   crucial for computing DW-invariants: given a 3-manifold M, does   H^1(M; ℤ_2) contain an essential element? In general, the calculation of products in cohomology and calculation of DW-invariants  is quite cumbersome, see <cit.>. We prefer  a very elementary method based on a nice structure of skeletons of Seifert manifolds. For simplicity, we restrict ourselves to Seifert manifolds fibered over S^2,  although Theorems 2 – 4  remain  true for Seifert manifolds fibered over any closed    orientable surface. Proofs are the same.



§ SKELETONS OF SEIFERT MANIFOLDS. 


 Let M be a closed 3-manifold. A 2-dimensional polyhedron  P⊂ M is called  a skeleton of    M if M∖ P consists of open 3-balls. 



 Let us construct a skeleton of an orientable Seifert manifold M=(S^2; (p_i,q_i), 1≤ i≤ n ) fibered over S^2 with exceptional fibers of types (p_i,q_i). Represent S^2 as a union of two discs Δ, D with common boundary.  Choose inside Δ disjoint discs δ_i, 1≤ i ≤ n, and remove their interiors. The resulting punctured disc  we denote  Δ_0. Then we  join  the circles ∂δ_i     with ∂ D  by disjoint arcs    l_i⊂Δ_0 , 1≤  i ≤ n. The skeleton P⊂ M we are looking for is the union of the following surfaces, see Fig.<ref>.  




  * Annuli  L_i = l_i×  S^1 and tori t_i=∂δ_i ×  S^1   , 1≤ i ≤ n.


  * The torus  T=∂ D× S^1, the punctured  disc Δ_0,  and the disc D.


  * Discs  d_i attached to   t_i along  simple closed curves in t_i of  types   (p_i, q_i). 
  

Note that the surfaces of the first two types  lie in S^2 × S^1 while  d_i do not.  





§ SEIFERT MANIFOLDS HAVING Z(M,Α^3)=0
 
In this section we describe  all orientable Seifert manifolds fibered over S^2 whose first cohomology group contains an essential x.
Note that by Poincaré duality    x ∈ H^1(M; ℤ_2) is essential 
 if and only if its dual x_∗∈ H_2(M; ℤ_2)           is essential in the sense that it can be realized by an odd 
 surface, i.e. by a  closed  surface having an odd Euler
 characteristic. So instead of looking for essential 1-cocycles we will construct essential 2-cycles.

Let M=(S^2; (p_i,q_i), 1≤ i≤ n ) be an orientable Seifert manifold fibered over S^2 with exceptional fibers of types 
(p_i,q_i). 

 
 Suppose that M contains  an  exceptional fiber  f_i   of type  (p_i,q_i)  and an  exceptional fiber  f_j   of type  (p_j,q_j) such that  p_i is divisible by 4 while p_j is even but not divisible by 4. Then there is an essential  x∈ H^1(M;ℤ_2).


 Choose disjoint discs D_i,D_j ⊂  S^2 containing  projection points  of f_i,f_j, and join their boundaries by a simple arc c ⊂ S^2. Let   P be the polyhedron in M consisting of the annulus L=c× S^1, two tori t_i =∂ D_i × S^1,  t_j =∂ D_j × S^1, and meridional  discs   d_i, d_j ⊂ M of  solid tori  which replace    D_i× S^1  and    D_j× S^1 by the standard   construction of  M.   The boundary curves of d_i, d_j are of types (p_i,q_i), (p_j,q_j), respectively.
See Fig. <ref>, to the left.




Note  that  d_i can be chosen so that the circles   ∂ d_i and λ_i=∂ L ∩ t_i  decompose    t_i into  p_i quadrilaterals admitting
 a black-white chessboard coloring. The same is true for d_j and t_j. Let us remove from P all (p_i+p_j)/2    quadrilaterals having  the same (say, white) color. We get a closed surface  F ⊂ M. Since the Euler characteristic χ(P) of P is even and (p_i+p_j)/2 is odd,   χ(F) is odd. Therefore F is odd and thus represents an essential 2-cycle.



  Let M=(S^2; (p_i,q_i), 1≤ i≤ n ) be an orientable Seifert manifold fibered over S^2 with exceptional fibers of types 
(p_i,q_i) such that
 (1) all p_i are odd and (2)  
the number of odd q_i is even.  
Denote by Q_∗  the sum of all q_i, and by  P_∗  any 
alternating sum of all p_i for which
 the corresponding q_i are odd.   
 Suppose that the integer number    ξ(M)=(Q_∗+P_∗)/2 is odd.
    Then there exists an essential  x∈ H^1(M;ℤ_2).


 Let us replace two exceptional fibers of types
(p_i,q_i),   (p_j,q_j) by  exceptional fibers of types
(p_i,q_i+p_i),  (p_j,q_j-p_j).   These  operations (called  
parameter trading) preserve the parity of ξ(M) and produce another Seifert presentation of 
M, which also satisfies assumptions (1) and (2) of Theorem
<ref>. Using such operations  one can easily get  all q_i be
divisible  by 4,   except the last one, say, q_n,  which
must be  even in view of assumption (2). For this new presentation of M we have Q_∗≡ q_n mod 4 (since all other q_i are divisible by 4), and P_∗ =0 (since there are no odd q_i). Taking into account that  ξ(M) is odd, we may conclude that q_n/2 is odd.  
Let   P be a skeleton of M  constructed for   this new presentation of M. Consider the union P_0 of the following  2-components of P, see Fig. <ref>, to the right: 
  
 

  * The torus t_n and  disc  d_n attached to   t_n  along a simple closed curve  of type     (p_n, q_n);

  * The   disc D =S^2 ∖ Int (δ_n), where δ_n ⊂ S^2 is the meridional disc of t_n.
 
  
   Then we apply the same trick as in the proof of Theorem <ref> using the circle   ∂ d_n and taking the circle ∂δ_n instead of λ_n. 
   
   Since  q_n is even,  the circles   ∂ d_n and ∂δ_n   can be chosen so that  they     decompose    t_n into  q_n quadrilaterals admitting
 a black-white chessboard coloring.  Let us remove from P_0  all q_n/2    quadrilaterals having  the same (say, white) color. We get a closed surface  F ⊂ M. Since the Euler characteristic   of P_0 is even and q_n/2 = ξ(M) is odd,   χ(F) is odd. Therefore F represents an essential 2-cycle.



Let us introduce two classes 𝒜, ℬ  of Seifert manifolds. Class  𝒜  consists of manifolds satisfying assumptions of Theorem <ref>,  class ℬ consists of manifolds satisfying assumptions of Theorem <ref>.


 
 Let M be   an orientable Seifert manifold fibered over S^2 with exceptional fibers of types
(p_i,q_i), 1≤  i ≤ n. Suppose that  M contains an essential 2-cycle x_∗.
Then M belongs  either to 𝒜  or to ℬ 


In the proof of the theorem, we need the following self-evident lemma.
  Let τ be a torus and  𝒞  a finite
collection of  simple closed curves in τ such that all their
intersection points are transverse. We will consider the union G
of these curves as a graph.
Suppose that the faces of G have  a
black-white chess coloring  in the sense that   any edge of G separates a black face from a white one.
Then any general position simple closed
curve in τ crosses the edges of G at an even  number of
points.

 Let P be the  skeleton   of M  constructed    in  section <ref>. Denote by Γ its
   singular graph consisting of triple and fourfold lines of P. The remaining part of P consists
    of 2-components of P, that is,  of surfaces which are glued to Γ along their
    boundary circles. Let  B be the carrier of x_∗, i.e. the union of  2-components of P
    which are   black  in the sense that they have  coefficient 1 in the 2-chain  representing x_∗. All other 2-components are   white.

Case 1. Suppose that   Δ_0 and hence D are white. Then the polyhedron P_0=P∖Δ_0
is  simple   in the sense that the set of its singular points consists    of triple  lines and their crossing points. It follows that  B is a closed  surface.
Since x_∗ is essential, χ(B) is odd.  We may assume that B is connected
 (if not then the Euler characteristic of at least one component of B is odd, and we can take
it instead  of B). It follows that  B contains   two
 annuli, say, L_i, L_j, discs  d_i,d_j, and  one of the two annuli into which the curves
 L_i ∩ T and L_j ∩ T decompose the torus T. The total Euler characteristic of these 2-components is 0.

 Just as
 in the proof of Theorem <ref> the circles ∂
d_i and λ_i= t_i∩∂ L_i decompose t_i into
p_i quadrilateral 2-components. Since x_∗ is a 2-cycle,
they are colored according to the chessboard rule. It follows that
p_i and    p_j  are even. The remaining part of B
consists of (p_i+p_j)/2 black quadrilaterals in t_i and t_j.
Since χ(B) is odd, we may conclude that one of p_i/2, p_j/2 is even while the other is odd. Therefore M∈𝒜.

Case 2. Suppose that Δ_0 is black. Then each  d_i is  black. 
 Otherwise the boundary of x_∗ would contain ∂δ_i, which is impossible since x_∗ is a 2-cycle.

Let us prove that if L_i is black then q_i is odd.   Consider the graph G=∂ d_i∪λ_i ⊂ t_i. As above, x_∗ induces a  chess black-white  coloring of its faces. Then a parallel copy ∂δ'_i of ∂δ_i crosses λ_i at one point and crosses ∂ d_i at q_i  points. By Lemma <ref> the number q_i+1 must be even, which means that q_i is odd. Similarly,  if L_i is white then q_i is even.

 Let us prove that all p_i are odd. Suppose that  L_i is black. Consider the graph G=∂ d_i∪∂δ_i  ⊂ t_i.   It decomposes t_i into black-white colored faces. The coloring is induced by the 2-cycle x_∗.  Let λ'_i ⊂ t_i be a parallel copy    of λ_i. Then λ'_i  crosses ∂ d_i at p_i points and crosses     ∂δ_i at  one point. By Lemma 1 the number p_i+1 must be even, which means that p_i is odd. Suppose that  L_i is white then q_i is even. Therefore ,  p_i, being   coprime with Q_i, is odd.

Let us prove that the number of  black L_i  and hence the number of    odd q_i are even.  This is because the boundary circles of black L_i decompose T into  black-white  colored  annuli such any two neighboring annuli have different colors.  Therefore  any meridional circle of T, for example, ∂ D,  crosses  those boundary circles at an even number of points. 

Let us prove that ξ(M) is odd. Just as in the proof of Theorem <ref> we transform the  given Seifert presentation  of  M into a new Seifert presentation such that all q_i, 1≤ i≤ n-1, are divisible  by 4 and  q_n is even. Then the  carrier B of x_∗  consists of the following black surfaces:
 

  * Δ_0 and  D;
 
  * All  discs d_i, 1≤ i ≤  n; 
 
  * black quadrilaterals  contained in  t_i, 1≤ i≤ n. Each t_i contains q_i/2  black quadrilaterals, where (p_i,q_i)  is the type of the corresponding d_i. Of course  B  may contain the torus  T, but it is      is homologically trivial and thus can be neglected.

 
 Note that  B has only triple singularities and thus is a closed surface. Since x_∗ is   essential,  χ(B)  is odd.  Now  we calculate  χ(B) by counting Euler characteristics of the above black surfaces.  
We get χ(B)=q_n/2 mod 2. It follows that  q_n/2 is odd. Since all q_i are even,   P_∗ =0.   Taking into account that  all q_i, 1≤ i≤ n-1 are divisible by 4 and that q_n/2 is odd we may conclude that  Q_∗ =(∑_i=1^nq_i)/2 is  odd. Therefore, ξ(M) is odd. It follows that  is in class ℬ.
 




§ ACKNOWLEDGMENTS

S. Matveev, V. Tarkaev and V. Turaev were supported in part by the Laboratory of Quantum Topology, Chelyabinsk State University (contract no. 14.Z50.31.0020). S. Matveev and V. Tarkaev were supported in part by the Ministry of Education and Science of the Russia (the state task number 1.1260.2014/K) and the Russian Foundation for Basic Research (project no. 14-01-00441).

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