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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

Let C2 denote the cyclic group of order 2. Given a manifold with a C2–action, we can consider its equivariant Bredon RO(C2)–graded cohomology. We develop a theory of fundamental classes for equivariant submanifolds in RO(C2)–graded cohomology with constant 2–coefficients. We show the cohomology of any C2–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.

equivariant cohomology, equivariant homotopy theory, Bredon cohomology
Mathematical Subject Classification 2010
Primary: 55N91, 55P91
Received: 12 August 2019
Revised: 8 September 2020
Accepted: 21 September 2020
Published: 22 November 2021
Christy Hazel
Department of Mathematics
University of California, Los Angeles
Los Angeles, CA
United States