On the Product in Negative Tate Cohomology for Finite Groups

TL;DR

This paper provides a geometric interpretation of the cup product in negative Tate cohomology for finite groups, relating it to a join operation in homology and connecting it with Kreck's product via stratifold homology.

Contribution

It introduces a geometric description of the negative Tate cohomology product as a join of cycles and proves its equivalence to Kreck's product for finite groups.

Findings

The product in negative Tate cohomology corresponds to a join operation in homology.

The geometric description explains the dimension shift in the product.

The negative Tate cohomology product coincides with Kreck's product for finite groups.

Abstract

Our aim in this paper is to give a geometric description of the cup product in negative degrees of Tate cohomology of a finite group with integral coefficients. By duality it corresponds to a product in the integral homology of $BG$: {\[H_{n}(BG,\mathbb{Z})\otimes H_{m}(BG,\mathbb{Z})\rightarrow H_{n+m+1}(BG,\mathbb{Z})\]} for $n,m>0$. We describe this product as join of cycles, which explains the shift in dimensions. Our motivation came from the product defined by Kreck using stratifold homology. We then prove that for finite groups the cup product in negative Tate cohomology and the Kreck product coincide. The Kreck product also applies to the case where $G$ is a compact Lie group (with an additional dimension shift).