#### Volume 21, issue 6 (2021)

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Equivariant fundamental classes in $\mathrm{RO}(C_2)$–graded cohomology with $\underline{\mathbb{Z}/2}$–coefficients

### Christy Hazel

Algebraic & Geometric Topology 21 (2021) 2799–2856
##### Abstract

Let ${C}_{2}$ denote the cyclic group of order $2$. Given a manifold with a ${C}_{2}$–action, we can consider its equivariant Bredon $RO\left({C}_{2}\right)$–graded cohomology. We develop a theory of fundamental classes for equivariant submanifolds in $RO\left({C}_{2}\right)$–graded cohomology with constant $ℤ∕2$–coefficients. We show the cohomology of any ${C}_{2}$–surface is generated by fundamental classes, and these classes can be used to easily compute the ring structure. To define fundamental classes we are led to study the cohomology of Thom spaces of equivariant vector bundles. In general, the cohomology of the Thom space is not just a shift of the cohomology of the base space, but we show there are still elements that act as Thom classes, and cupping with these classes gives an isomorphism within a certain range.

##### Keywords
equivariant cohomology, equivariant homotopy theory, Bredon cohomology
##### Mathematical Subject Classification 2010
Primary: 55N91, 55P91
##### Publication
Received: 12 August 2019
Revised: 8 September 2020
Accepted: 21 September 2020
Published: 22 November 2021
##### Authors
 Christy Hazel Department of Mathematics University of California, Los Angeles Los Angeles, CA United States https://www.math.ucla.edu/~chazel/