The $\mathbb{Z}/p$ ordinary cohomology of $B_G U(1)$

TL;DR

This paper computes the $G$-equivariant cohomology of the classifying space for complex line bundles with $G = bZ/p$, extending previous work to a broader grading that reveals natural generators and simpler relations.

Contribution

It extends the $RO(G)$-graded cohomology calculation of $B_G U(1)$ to a larger grading, identifying natural generators and simplifying relations.

Findings

Identified natural generators including the Euler class.

Extended cohomology to a larger grading.

Simplified the set of relations among generators.

Abstract

With $G = \mathbb{Z}/p$, $p$ prime, we calculate the ordinary $G$-cohomology (with Burnside ring coefficients) of $\mathbb{C}P_G^\infty = B_G U(1)$, the complex projective space, a model for the classifying space for $G$-equivariant complex line bundles. The $RO(G)$-graded ordinary cohomology was calculated by Gaunce Lewis, but here we extend to a larger grading in order to capture a more natural set of generators, including the Euler class of the canonical bundle, as well as a significantly simpler set of relations.