On nilpotent ideals in the cohomology ring of a finite group

Abstract

In this paper we find upper bounds for the nilpotency degree of some ideals in the cohomology ring of a finite group by studying fixed point free actions of the group on suitable spaces. The ideals we study are the kernels of restriction maps to certain collections of proper subgroups. We recover the Quillen–Venkov lemma and the Quillen F-injectivity theorem as corollaries, and discuss some generalizations and further applications. We then consider the essential cohomology conjecture, and show that it is related to group actions on connected graphs. We discuss an obstruction for constructing a fixed point free action of a group on a connected graph with zero “ k-invariant” and study the class related to this obstruction. It turns out that this class is a “universal essential class” for the group and controls many questions about the groups essential cohomology and transfers from proper subgroups.