On the product in negative Tate cohomology for finite groups

Tene, Haggai

doi:10.2140/agt.2012.12.493

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On the product in negative Tate cohomology for finite groups

Haggai Tene

Algebraic & Geometric Topology 12 (2012) 493–509

DOI: 10.2140/agt.2012.12.493

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 $B G$ :

H_{n} (B G, ℤ) \otimes H_{m} (B G, ℤ) \to H_{n + m + 1} (B G, 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).

Keywords

Tate cohomology, homology of classifying spaces, compact Lie group, product in homology, stratifold

Mathematical Subject Classification 2010

Primary: 20J06, 55R40

References

Publication

Received: 3 April 2011

Revised: 24 November 2011

Accepted: 30 November 2011

Published: 21 March 2012

Authors

Haggai Tene
Department of Mathematics and PMI POSTECH Pohang 790-784 South Korea
http://math.postech.ac.kr/~teneh

Volume 12, issue 1 (2012)