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On Dijkgraaf-Witten invariant for 3-manifolds. (English) Zbl 0786.57008

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)

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
81T99 Quantum field theory; related classical field theories
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