Dijkgraaf-Witten $Z_2$-invariants for Seifert manifolds

TL;DR

This paper calculates the Dijkgraaf-Witten Z_2-invariants for all orientable Seifert manifolds with orientable bases, providing explicit topological invariants for this class of 3-manifolds.

Contribution

It offers a complete computation of Z_2 Dijkgraaf-Witten invariants for orientable Seifert manifolds with orientable bases, filling a gap in topological quantum field theory.

Findings

Explicit values of Z_2 invariants for all such manifolds

Enhanced understanding of topological invariants in 3-manifold theory

Potential applications in quantum topology and physics

Abstract

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

Full text

Mathematics Subject Classification 2000: 57M25, 57M27

1. The Dijkgraaf-Witten invariants

In 1990, Dijkgraaf and Witten [DW] 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 $M\to B$ considered up to base point preserving homotopy. The set $S$ is finite and can be identified with the set of homomorphisms $\pi_{1}(M)\to G$.

Definition 1.1**.**

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

$Z(M,h)=\frac{1}{|G|}\sum_{f\in S(M,B)}\langle f^{*}(h),[M]\rangle,$**

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 $\mathbb{Z}_{2}$. For the classifying space of $\mathbb{Z}_{2}$ we take the infinite-dimensional projective space $RP^{\infty}$, and for $h$ we take a unique nontrivial element $\alpha^{3}\in H^{3}(RP^{\infty};\mathbb{Z}_{2})$, where $\alpha$ is the generator of $H^{1}(RP^{\infty};\mathbb{Z}_{2})$. In this situation, the value $\langle f^{*}(h),[M]\rangle$ belongs to $\mathbb{Z}_{2}$, and the formula given above takes the form

$Z(M,\alpha^{3})=\frac{1}{2}\sum_{f\in S(M,B)}(-1)^{\langle f^{*}(\alpha^{3}),[M]\rangle,}$

and becomes applicable to nonorientable manifolds. If $H^{1}(M;\mathbb{Z}_{2})=0$, then $Z(M,\alpha^{3})=\frac{1}{2}$. In all other cases $Z(M,\alpha^{3}$) is an integer.

2. Quadratic function and Arf-invariant

Let $M$ be a closed 3-manifold. We define a quadratic function $Q_{M}\colon H^{1}(M;\mathbb{Z}_{2})\to\mathbb{Z}_{2}$ by the rule $Q_{M}(x)=\langle x^{3},[M]\rangle$, where $x\in H^{1}(M;\mathbb{Z}_{2})$, $[M]$ is the fundamental class of $M$, and $x^{3}\in H^{3}(M;\mathbb{Z}_{2})$ is the cube of $x$ in the sense of multiplication in cohomology. The corresponding pairing $\ell_{M}\colon H^{1}(M;\mathbb{Z}_{2})\times H^{1}(M;\mathbb{Z}_{2})\to\mathbb{Z}_{2}$ defined by $\ell_{M}(x,y)=Q_{M}(x+y)-Q_{M}(x)-Q_{M}(y)$ is bilinear.

The following relation between the DW-invariant of $M$ and the Arf-invariant of $Q_{M}$ was discovered in [MT].

Theorem 2.1**.**

[MT]**. Let M be a closed connected 3-manifold, and let $A\subset H^{1}(M;\mathbb{Z}_{2})$ be the annihilator of $\ell_{M}$. If there exists $x\in A$ such that $x^{3}\neq 0$ , then $Z(M,\alpha^{3})=0$. If there are no such elements, then $Z(M,\alpha^{3})=2^{k+m-1}(-1)^{{\rm Arf}(Q_{M})}$, where $m$ is the dimension of $A$ and $k$ equals half the dimension of the coset space $H^{1}(M;\mathbb{Z}_{2})/A$.

Note that this theorem is true for orientable and non-orientable 3-manifolds. However, it follows from the Postnikov Theorem [Po] that if $M$ is orientable, then the annihilator $A$ of $\ell_{M}$ coincides with $H^{1}(M;\mathbb{Z}_{2})$. Further on we will consider only orientable 3-manifolds. For brevity we will call an element $x\in H^{1}(M;\mathbb{Z}_{2})$ essential if $x^{3}\neq 0$.

Corollary 2.2**.**

Let $M$ be an orientable closed connected 3-manifold. If there exists an essential $x\in H^{1}(M;\mathbb{Z}_{2})$, then $Z(M,\alpha^{3})=0$. If there are no such elements, then $Z(M,\alpha^{3})=2^{m-1}$, where $m$ is the dimension of $H^{1}(M;\mathbb{Z}_{2})$.

In view of the above corollary, the following question is crucial for computing DW-invariants: given a 3-manifold $M$, does $H^{1}(M;\mathbb{Z}_{2})$ contain an essential element? In general, the calculation of products in cohomology and calculation of DW-invariants is quite cumbersome, see [Wa, Br, Ha]. 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.

3. Skeletons of Seifert manifolds.

Definition 3.1**.**

Let $M$ be a closed 3-manifold. A 2-dimensional polyhedron $P\subset M$ is called a skeleton of $M$ if $M\setminus P$ consists of open 3-balls.

Let us construct a skeleton of an orientable Seifert manifold $M=(S^{2};(p_{i},q_{i}),1\leq i\leq n)$ fibered over $S^{2}$ with exceptional fibers of types $(p_{i},q_{i})$. Represent $S^{2}$ as a union of two discs $\Delta,D$ with common boundary. Choose inside $\Delta$ disjoint discs $\delta_{i},1\leq i\leq n$, and remove their interiors. The resulting punctured disc we denote $\Delta_{0}$. Then we join the circles $\partial\delta_{i}$ with $\partial D$ by disjoint arcs $l_{i}\subset\Delta_{0},1\leq i\leq n$. The skeleton $P\subset M$ we are looking for is the union of the following surfaces, see Fig.1.

1. (1)

Annuli $L_{i}=l_{i}\times S^{1}$ and tori $t_{i}=\partial\delta_{i}\times S^{1},1\leq i\leq n$. 2. (2)

The torus $T=\partial D\times S^{1}$, the punctured disc $\Delta_{0}$, and the disc $D$. 3. (3)

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 $S^{2}\times S^{1}$ while $d_{i}$ do not.

4. Seifert manifolds having $Z(M,\alpha^{3})=0$

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 $x\in H^{1}(M;\mathbb{Z}_{2})$ is essential if and only if its dual $x_{\ast}\in H_{2}(M;\mathbb{Z}_{2})$ 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 $M=(S^{2};(p_{i},q_{i}),1\leq i\leq n)$ be an orientable Seifert manifold fibered over $S^{2}$ with exceptional fibers of types $(p_{i},q_{i})$.

Theorem 4.1**.**

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 $x\in H^{1}(M;\mathbb{Z}_{2})$.

Proof.

Choose disjoint discs $D_{i},D_{j}\subset S^{2}$ containing projection points of $f_{i},f_{j}$, and join their boundaries by a simple arc $c\subset S^{2}$. Let $P$ be the polyhedron in $M$ consisting of the annulus $L=c\times S^{1}$, two tori $t_{i}=\partial D_{i}\times S^{1},t_{j}=\partial D_{j}\times S^{1}$, and meridional discs $d_{i},d_{j}\subset M$ of solid tori which replace $D_{i}\times S^{1}$ and $D_{j}\times S^{1}$ 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. 2, to the left.

Note that $d_{i}$ can be chosen so that the circles $\partial d_{i}$ and $\lambda_{i}=\partial L\cap t_{i}$ 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 $F\subset M$. Since the Euler characteristic $\chi(P)$ of $P$ is even and $(p_{i}+p_{j})/2$ is odd, $\chi(F)$ is odd. Therefore $F$ is odd and thus represents an essential 2-cycle. ∎

Theorem 4.2**.**

Let $M=(S^{2};(p_{i},q_{i}),1\leq i\leq n)$ 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 $Q_{\ast}$ the sum of all $q_{i}$, and by $P_{\ast}$ any alternating sum of all $p_{i}$ for which the corresponding $q_{i}$ are odd. Suppose that the integer number $\xi(M)=(Q_{\ast}+P_{\ast})/2$ is odd. Then there exists an essential $x\in H^{1}(M;\mathbb{Z}_{2})$.

Proof.

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 $\xi(M)$ and produce another Seifert presentation of $M$, which also satisfies assumptions (1) and (2) of Theorem 4.2. 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 $Q_{\ast}\equiv q_{n}$ mod 4 (since all other $q_{i}$ are divisible by 4), and $P_{\ast}=0$ (since there are no odd $q_{i}$). Taking into account that $\xi(M)$ 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. 2, to the right:

1. (1)

The torus $t_{n}$ and disc $d_{n}$ attached to $t_{n}$ along a simple closed curve of type $(p_{n},q_{n})$; 2. (2)

The disc $D=S^{2}\setminus{\rm Int}(\delta_{n})$, where $\delta_{n}\subset S^{2}$ is the meridional disc of $t_{n}$.

Then we apply the same trick as in the proof of Theorem 4.1 using the circle $\partial d_{n}$ and taking the circle $\partial\delta_{n}$ instead of $\lambda_{n}$.

Since $q_{n}$ is even, the circles $\partial d_{n}$ and $\partial\delta_{n}$ 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 $F\subset M$. Since the Euler characteristic of $P_{0}$ is even and $q_{n}/2=\xi(M)$ is odd, $\chi(F)$ is odd. Therefore $F$ represents an essential 2-cycle. ∎

Let us introduce two classes $\mathcal{A},\mathcal{B}$ of Seifert manifolds. Class $\mathcal{A}$ consists of manifolds satisfying assumptions of Theorem 4.1, class $\mathcal{B}$ consists of manifolds satisfying assumptions of Theorem 4.2.

Theorem 4.3**.**

Let $M$ be an orientable Seifert manifold fibered over $S^{2}$ with exceptional fibers of types $(p_{i},q_{i}),1\leq i\leq n$. Suppose that $M$ contains an essential 2-cycle $x_{\ast}$. Then $M$ belongs either to $\mathcal{A}$ or to $\mathcal{B}$

In the proof of the theorem, we need the following self-evident lemma.

Lemma 4.4**.**

Let $\tau$ be a torus and $\mathcal{C}$ a finite collection of simple closed curves in $\tau$ 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 $\tau$ crosses the edges of $G$ at an even number of points.∎

Proof of Theorem 4.3.

Let $P$ be the skeleton of $M$ constructed in section 3. Denote by $\Gamma$ 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 $\Gamma$ along their boundary circles. Let $B$ be the carrier of $x_{\ast}$, 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 $x_{\ast}$. All other 2-components are white.

Case 1. Suppose that $\Delta_{0}$ and hence $D$ are white. Then the polyhedron $P_{0}=P\setminus\Delta_{0}$ 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 $x_{\ast}$ is essential, $\chi(B)$ 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 $L_{i}\cap T$ and $L_{j}\cap T$ decompose the torus $T$. The total Euler characteristic of these 2-components is 0.

Just as in the proof of Theorem 4.1 the circles $\partial d_{i}$ and $\lambda_{i}=t_{i}\cap\partial L_{i}$ decompose $t_{i}$ into $p_{i}$ quadrilateral 2-components. Since $x_{\ast}$ 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 $\chi(B)$ is odd, we may conclude that one of $p_{i}/2,p_{j}/2$ is even while the other is odd. Therefore $M\in\mathcal{A}$.

Case 2. Suppose that $\Delta_{0}$ is black. Then each $d_{i}$ is black. Otherwise the boundary of $x_{\ast}$ would contain $\partial\delta_{i}$, which is impossible since $x_{\ast}$ is a 2-cycle.

Let us prove that if $L_{i}$ is black then $q_{i}$ is odd. Consider the graph $G=\partial d_{i}\cup\lambda_{i}\subset t_{i}$. As above, $x_{\ast}$ induces a chess black-white coloring of its faces. Then a parallel copy $\partial\delta^{\prime}_{i}$ of $\partial\delta_{i}$ crosses $\lambda_{i}$ at one point and crosses $\partial d_{i}$ at $q_{i}$ points. By Lemma 4.4 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 $G=\partial d_{i}\cup\partial\delta_{i}\subset t_{i}$. It decomposes $t_{i}$ into black-white colored faces. The coloring is induced by the 2-cycle $x_{\ast}$. Let $\lambda^{\prime}_{i}\subset t_{i}$ be a parallel copy of $\lambda_{i}$. Then $\lambda^{\prime}_{i}$ crosses $\partial d_{i}$ at $p_{i}$ points and crosses $\partial\delta_{i}$ 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, $\partial D$, crosses those boundary circles at an even number of points.

Let us prove that $\xi(M)$ is odd. Just as in the proof of Theorem 4.2 we transform the given Seifert presentation of $M$ into a new Seifert presentation such that all $q_{i},1\leq i\leq n-1,$ are divisible by 4 and $q_{n}$ is even. Then the carrier $B$ of $x_{\ast}$ consists of the following black surfaces:

1. (1)

$\Delta_{0}$ and $D$; 2. (2)

All discs $d_{i},1\leq i\leq n$; 3. (3)

black quadrilaterals contained in $t_{i},1\leq i\leq n$. 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 $x_{\ast}$ is essential, $\chi(B)$ is odd. Now we calculate $\chi(B)$ by counting Euler characteristics of the above black surfaces. We get $\chi(B)=q_{n}/2$ mod 2. It follows that $q_{n}/2$ is odd. Since all $q_{i}$ are even, $P_{\ast}=0$. Taking into account that all $q_{i},1\leq i\leq n-1$ are divisible by 4 and that $q_{n}/2$ is odd we may conclude that $Q_{\ast}=(\sum_{i=1}^{n}q_{i})/2$ is odd. Therefore, $\xi(M)$ is odd. It follows that is in class $\mathcal{B}$. ∎

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).