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Dijkgraaf-Witten invariants over \(\mathbb Z_2\) for 3-manifolds. (English. Russian original) Zbl 1327.57019

Dokl. Math. 91, No. 1, 9-11 (2015); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 460, No. 1, 15-17 (2015).
From the text: Consider a finite group \(G\) and its classifying space \(B=B(G)\). Let \(M\) be a closed connected 3-manifold of dimension \(n\geq 1\). We choose two base points, in \(M\) and in \(B\), and consider the set \(S=S(M,B)\) of all pointed (i.e., preserving the base points) continuous maps \(M\to B\). We consider these maps up to pointed homotopies. Note that \(S\) can be naturally identified with the set \(\operatorname{Hom}(\pi_1(M),G)\) of all homomorphisms of the group \(\pi_1(M)\) to \(G\). Therefore, this set is finite.
Let \(U\) be a subgroup of the multiplicative group \(U(1)\) of all complex numbers with absolute value 1. Choose any element \(h\) in the group \(H^n(G;U)= H^n(B;U)\). If \(M\) is oriented, then to each map \(f\in S\) we can assign the value \(\langle f^*(h),[M]\rangle\in U\) of the element \(f^*(h)\) of \(H^n(M;U)\) at the fundamental class \([M]\) of the manifold \(M\).
Definition 1. The Dijkgraaf-Witten invariant \(Z(M,h)\) of \(M\) is defined by \[ Z(M,h)= {1\over|G|} \sum_{f\in S(M,B)} \langle f^*(h),[M]\rangle. \] The values of this invariant are complex numbers.
Determining the Dijkgraaf-Witten invariants requires calculating and summing many terms, the number of which exponentially increases with the first Betti number of \(M\).
In this paper, we consider only the special case where \(n=3\) and both groups \(G\) and \(U\) have order 2 and are identified with the group \(\mathbb{Z}_2\). For the classifying space \(B\) of \(\mathbb{Z}_2\) we can take the infinite-dimensional projective space \(RP^\infty\). Its cohomology ring with coefficients in \(U=\mathbb{Z}_2\) is very simple: this is the polynomial ring in a variable \(\alpha\), which represents the unique nontrivial element of the group \(H^1(B;\mathbb{Z}_2)= \mathbb{Z}_2\). In particular, the group \(H^3(B;\mathbb{Z}_2)\) contains only one nontrivial element \(h=\alpha^3\), too. In this situation, the value \(\langle f^*(h),[M]\rangle\) belongs to \(\mathbb{Z}_2\), and the formula given above acquires the form \[ Z(M)= {1\over 2} \sum_{f\in S(M,B)} (-1)^{\langle f^*(\alpha^3,[M]\rangle} \] and becomes applicable to nonorientable manifolds. If \(H^1(M;\mathbb{Z}_2)= 0\), then \(Z(M)={1\over 2}\). In all other cases, the number of terms in the above sum is even; therefore, \(Z(M)\) is an integer.
Theorem 1. Let \(M\) be a closed connected 3-manifold, and let \(A\subset H^1(M;\mathbb{Z}_2)\) be the annihilator of the bilinear pairing \(\ell_M\) corresponding to the function \(Q_M\). If there exists an element \(x\in A\) such that \(x^3\neq 0\), then \(Z(M)= 0\). If there are no such elements, then \(Z(M)= 2^{k+m-1}(-1)^{\text{Arf}(Q_M)}\), where \(m\) is the dimension of \(A\) and \(k\) equals half the dimension of the quotient space \(H^1(M; \mathbb{Z}_2)/A\).

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)

Keywords:

Gaussian sums
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References:

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