Abstracts for MPI-Oberseminar

The unit spheres in orthogonal  representations of finite groups give  basic linear examples of group actions on spheres. The talk will be about the connection between finite group theory, and the fixed sets and isotropy subgroups found in actions on spheres. For example, there are finite groups which act freely and smoothly on spheres, but not linearly. The complexity of non-free actions can be measured by the rank of the isotropy subgroups. I will survey previous results and describe my

The talk consists of three parts.
1.) Cuntz-Li associated to the ring of algebraic integers of a number
field a $C^*$-algebra.
We will discuss a preprint by Li and Lueck, where  a full classification
of these $C^*$-algebras is presented.
2.) There is a Conjecture due to Adem-Ge-Pan-Petroysan about the
cohomology of certain crystallographic groups
with cyclic holonomy. In papers by Langer and Lueck the conjecture has
been disproved in general, but proved
in the case where the action of the holonomy group is free.

As a companion to Freedman's lectures I will describe applications of the disk theorem to finding smooth structures on topological 4-manifolds, isotopy of homeomorphisms to be diffeomorphisms on various types of open sets, etc. One consequence is that handlebody structures are equivalent to smooth structures, so generally don't exist. I describe modifications of standard handlebody arguments to work around this problem.

In G-equivariant (Bredon) cohomology the integer grading is often extended to a grading over the real representations of G. We consider the general question of what other indices we could grade cohomology on. In a precise sense we will show that the universal group to grade on is the Picard group of the G-equivariant stable category. When G is a finite cyclic group we identify this group and show that it is generated by representation spheres. We then calculate the Pic(G) graded cohomology of a point which turns out to already be an interesting calculation.