Wakui, Michihisa On Dijkgraaf-Witten invariant for 3-manifolds. (English) Zbl 0786.57008 Osaka J. Math. 29, No. 4, 675-696 (1992). In 1990 R. Dijkgraaf and E. Witten [Commun. Math. Phys. 129, No. 2, 393-429 (1990; Zbl 0703.58011)] introduced a method of constructing an invariant of 3-manifolds using a finite gauge group \(G\). For a closed oriented 3-manifold \(M\), the Dijkgraaf-Witten invariant is given by the following formula: \[ Z(M)= {1\over {| G|}} \sum_{\gamma\in(\pi_ 1(M), G)} \langle \gamma^*[\alpha], [M]\rangle. \] Here \([\alpha]\) is a cohomology class of \(H^ 3(BG,U(1))\), \(\gamma^*\) is a map from \(H^ 3(BG,U(1))\) to \(H^ 3(M,U(1))\) induced from \(\gamma\), \([M]\) is the fundamental class of \(M\), \(BG\) is the classifying space of \(G\) and \(U(1)\) is the unitary group. However, in the case where \(M\) has a boundary, such a formulation can not be done, because the fundamental class \([M]\) is not defined for a manifold with boundary.In this paper, we formulate an invariant of 3-manifolds possibly with boundary introduced by Dijkgraaf and Witten using a triangulation and prove its topological invariance in a rigorous way. We also show that the construction of the Dijkgraaf-Witten invariant for a 3-manifold with boundary gives an example of the topological quantum field theory. We use here one version of M. Atiyah’s definition of topological quantum field theory [Publ. Math., Inst. Hautes Étud. Sci. 68, 175-186 (1988; Zbl 0692.53053)]. To derive these results we use methods introduced by V. G. Turaev and O. Ya. Viro [Topology 31, No. 4, 865-902 (1992; Zbl 0779.57009)] on triangulations of 3-manifolds with boundaries and their topological invariances.Independently of our study, D. N. Yetter [J. Knot Theory Ramifications 1, 1-20 (1992; Zbl 0770.57010)] constructs an invariant of 3-manifolds \(M\) equipped with links and an example of the topological quantum field theory in the study of finite crossed \(G\)-sets. In the case of \(M\) equipped with no link the Yetter’s invariant coincides with our invariant making choice of trivial 3-cocycle as \(\alpha\). Reviewer: M.Wakui (Osaka) Cited in 2 ReviewsCited in 25 Documents MSC: 57N10 Topology of general \(3\)-manifolds (MSC2010) 81T99 Quantum field theory; related classical field theories Keywords:3-manifolds with boundary; finite gauge group \(G\); Dijkgraaf-Witten invariant; topological quantum field theory Citations:Zbl 0703.58011; Zbl 0692.53053; Zbl 0779.57009; Zbl 0770.57010 PDFBibTeX XMLCite \textit{M. Wakui}, Osaka J. Math. 29, No. 4, 675--696 (1992; Zbl 0786.57008)