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# Homology, Homotopy and Applications

## Volume 23 (2021)

### Number 2

### Bredon cohomology of finite dimensional $C_p$-spaces

Pages: 33 – 57

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n2.a3

#### Authors

#### Abstract

For finite dimensional free $C_p$-spaces, the calculation of the Bredon cohomology ring as an algebra over the cohomology of $S^0$ is used to prove the non-existence of certain $C_p$-maps. These are related to Borsuk–Ulam type theorems, and equivariant maps related to the topological Tverberg conjecture. For certain finite dimensional $C_p$-spaces which are formed out of representations, it is proved that the cohomology is a free module over the cohomology of a point. All the calculations are done for the cohomology with constant coefficients $\mathbb{Z}/p$.

#### Keywords

Bredon cohomology, Mackey functor, Tverberg theorem, equivariant cohomology

#### 2010 Mathematics Subject Classification

Primary 55N91, 55P91. Secondary 14M15, 57S17.

Received 2 May 2020

Received revised 19 August 2020

Accepted 12 October 2020

Published 24 March 2021