The $C_2$-equivariant ordinary cohomology of $BU(2)$

TL;DR

This paper computes the $C_2$-equivariant cohomology of $BU(2)$ with Burnside ring coefficients, introduces an extended grading for natural generators, and defines characteristic classes and numbers for equivariant bundles.

Contribution

It provides the first detailed calculation of $C_2$-equivariant cohomology of $BU(2)$ with a new extended grading and defines characteristic classes for equivariant bundles.

Findings

Calculated $C_2$-cohomology of $BU(2)$ with Burnside ring coefficients

Introduced an extended grading capturing natural generators

Defined characteristic classes and numbers for equivariant bundles

Abstract

We calculate the ordinary $C_2$-cohomology, with Burnside ring coefficients, of $BU(2)$, the classifying space for $C_2$-equivariant complex 2-plane bundles, using an extended grading that allows us to capture a more natural set of generators. This allows us to define characteristic classes for such bundles. Combined with earlier calculations, it also allows us to define characteristic numbers for equivariant complex lines and surfaces and we give some sample computations.