Abstracts for Topics in Topology

Brieskorn varietes admit $S^1$-actions whose isotropy groups are finite. Choose a Brieskorn variety diffeomorphic to a sphere and an odd prime p which is relatively prime to the order of all isotropy groups. Then the orbit space of the induced $Z/p$-action is a homotopy lens space -- a closed manifold homotopy equivalent to a lens space. A homotopy lens space is fake if it is not homeomorphic to a classical lens space. Using Reidemeister torsion, we show some of these homotopy lens spaces are fake lens spaces.

In this talk I will present a topological approach to Euler characteristics of categories in terms of finiteness obstructions. This approach is compatible with almost anything one would want, for example products, coproducts, covering maps, isofibrations, and homotopy colimits. Classical constructions are special cases, for example, under appropriate hypotheses the functorial $L^2$-Euler characteristic of the proper orbit category for a group $G$ is the equivariant Euler characteristic of the classifying space for proper $G$-actions.

The following problem was posed by Grothendieck: Let $G$ be a finitely presented residually finite group and $u : H \to G$ the inclusion homomorphism of a finitely presented subgroup. Suppose that the extension $\hat u$ to the profinite completion is an isomorphism. Is $u$ an isomorphism? This talk will discuss this problem in the context of the fundamental groups of compact 3-manifolds.

In this talk, I will first describe a reformulation of a construction by Gorbounov, Malikov and Schechtman in terms of global geometric data. This will then be applied to define vertex algebraic versions of Dolbeault complexes and obtain a geometric description of the Witten and elliptic genera of complex manifolds.