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

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.