Parametrized \underline\mathbbF₂-Cohomology of B_C2O(1)

2026/07/29 by Agnès Beaudry, Chloe Lewis, Clover May +2
Mathematics · #math.AT

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arxiv created 2026/07/29 · arxiv updated 2026/07/31

Abstract

We compute the parametrized (or twisted) ordinary cohomology of the classifying space for real C₂-line bundles, B_C2O(1) with coefficients in the constant Mackey functor \underline\mathbbF₂. Parametrized cohomology refines RO(G)-graded Bredon cohomology by assembling equivariant cohomology for all local coefficients into a single graded ring. For this reason, our work also encodes a computation of the RO(C₂)-graded cohomology of all Thom spaces of real C₂-vector bundles over B_C2O(1). Along the way, we prove many general results that can be applied to computations of parametrized cohomology over general bases B. In particular, we introduce a collection of characteristic classes, give a definition of orientation for non-homogeneous bundles, import equivariant Steenrod operations to this context, and give general results relating to base change.

Related

• Characterizing maximal varieties via Bredon cohomology
• The RO(G)-Graded Cohomology of the Equivariant Classifying Space B_GSU(2)
• Equivariant local coefficients and the RO(G)-graded cohomology of classifying spaces
• The C₂-equivariant ordinary cohomology of BU(2)
• The ℤ/p ordinary cohomology of B_G U(1)