Let \(G\) be a finite group and let \(h\in H^3(BG;U(1))\). In [Commun. Math. Phys. 129, No. 2, 393–429 (1990; Zbl 0703.58011)], R. Dijkgraaf and E. Witten proposed a 3-dimensional topological quantum field theory (TQFT) in which, associated to each closed oriented 3-manifold \(M\), is its Dijkgraaf-Witten invariant \[ Z(M,h)=\frac{1}{|G|}\cdot\sum\limits_{\phi\in\hom(\pi_1(M),G)}\langle f(\phi)^\ast h,[M]\rangle, \] where \(f(\phi):M\to BG\) is a map whose homotopy class is determined by \(\phi\), and \(\langle-,-\rangle:H^3(M;U(1))\times H_3(M;\mathbb{Z})\to U(1)\) is the pairing. As in general TQFTs, \(Z(M,h)\) can be computed by a “cut-and-glue” process. This was done by H. Chen [J. Geom. Phys. 108, 38–48 (2016; Zbl 1385.57031)] when \(M\) belongs to a class of Seifert 3-manifolds.
On the other hand, after all, the DW invariant is made up of the fundamental group and (co-)homology, thus it is natural to expect relations between the DW invariant and classical invariants; such relations will permit to consider the latter from a quantum topology perspective. For \(G=\mathbb{Z}_2\), [S. V. Matveev and V. G. Turaev, Dokl. Math. 91, No. 1, 9–11 (2015; Zbl 1327.57019); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 460, No. 1, 15–17 (2015)] expressed the DW invariant in terms of the Arf invariant. Based on this, the present paper gives a simple formula of \(Z(M,h)\) when \(G=\mathbb{Z}_2\) and \(M\) is a Seifert 3-manifold with orientable base, through investigating the geometric structure of \(M\).
It is reasonable to expect more relations of this kind to be revealed.