Bredon sheaf cohomology

TL;DR

This paper introduces Bredon sheaf cohomology, a new equivariant cohomology theory that generalizes classical Bredon and sheaf cohomology, and computes related algebraic K-theory and E-theory invariants.

Contribution

It defines Bredon sheaf cohomology, establishes its fundamental properties, and proves a strong uniqueness theorem characterizing it among equivariant cohomology theories.

Findings

Computes algebraic K-theory of equivariant sheaves for finite groups.

Determines equivariant E-theory of $C^*$-algebras of continuous functions.

Shows Bredon sheaf cohomology recovers classical Bredon and sheaf cohomology.

Abstract

For a finite group $G$, we compute the algebraic $K$-theory of the category of equivariant sheaves on a locally compact Hausdorff $G$-space, generalizing a result of Efimov, and determine the equivariant $E$-theory of the $C^*$-algebra of continuous functions. These invariants admit natural descriptions in terms of a new equivariant cohomology theory, which we call Bredon sheaf cohomology. This theory recovers classical Bredon cohomology for $G$-CW complexes and ordinary sheaf cohomology when $G$ is trivial. We establish its basic structural properties and prove a strong uniqueness theorem: any functor from the category of locally compact Hausdorff $G$-spaces to a dualizable stable category satisfying equivariant open descent and cofiltered compact codescent is equivalent to Bredon sheaf cohomology, generalizing a result of Clausen.