On the product in negative Tate cohomology 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 $BG$:

$$H_n\left(BG,\mathbb{Z}\right)\otimes H_m\left(BG,\mathbb{Z}\right)\to H_{n+m+1}\left(BG,\mathbb{Z}\right)$$

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).