Equivariant fundamental classes in RO(C2)–graded cohomology with ℤ∕2–coefficients

Hazel, Christy

doi:10.2140/agt.2021.21.2799

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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

DOI: 10.2140/agt.2021.21.2799

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 $RO (C_{2})$ –graded cohomology. We develop a theory of fundamental classes for equivariant submanifolds in $RO (C_{2})$ –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

References

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/

Volume 21, issue 6 (2021)