zbMATH — the first resource for mathematics

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)
Full Text: