Topological invariants for 3-manifolds using representations of mapping class groups. I
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.
projectively linear representations of the mapping class group
scientific article; zbMATH DE number 89829