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Topological invariants for 3-manifolds using representations of mapping class groups. I. (English) Zbl 0762.57011
The author introduces new invariants of closed orientable 3-manifolds via their Heegaard decompositions. First a finite dimensional vector space is associated with the Heegaard surface \(\Sigma_ g\); the author then constructs projectively linear representations of the mapping class group of \(\Sigma_ g\) using the vector space mentioned above. This requires a considerable amount of techniques used in conformal field theory. The invariants themselves are defined by applying these representations to the gluing homeomorphism of \(\Sigma_ g\) with respect to certain distinguished bases of the vector space. Topological invariance is proved by way of Reidemeister-Singer. It is mentioned that the invariants distinguish the lens spaces \(L(7,1)\) and \(L(7,2)\), and hence are not homotopy invariants.

MSC:
57N10 Topology of general \(3\)-manifolds (MSC2010)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
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