Parametrized $\underline{\mathbb{F}}_2$-Cohomology of $B_{C_2}O(1)$

Agnès Beaudry; Chloe Lewis; Clover May; Sabrina Pauli; Elizabeth Tatum

Parametrized $\underline{\mathbb{F}}_2$-Cohomology of $B_{C_2}O(1)$

Algebraic Topology 2026-07-29 v1

Authors: Agnès Beaudry , Chloe Lewis , Clover May , Sabrina Pauli , Elizabeth Tatum

Abstract

We compute the parametrized (or twisted) ordinary cohomology of the classifying space for real $C_2$ -line bundles, $B_{C_2}O(1)$ with coefficients in the constant Mackey functor $\underline{\mathbb{F}}_2$ . 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 $RO(C_2)$ -graded cohomology of all Thom spaces of real $C_2$ -vector bundles over $B_{C_2}O(1)$ . 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.

Keywords

parametrized cohomology Mackey functor Bredon cohomology equivariant cohomology Thom space Steenrod operations

Cite

@article{arxiv.2607.27398,
  title  = {Parametrized $\underline{\mathbb{F}}_2$-Cohomology of $B_{C_2}O(1)$},
  author = {Agnès Beaudry and Chloe Lewis and Clover May and Sabrina Pauli and Elizabeth Tatum},
  journal= {arXiv preprint arXiv:2607.27398},
  year   = {2026}
}