The $C_2$-equivariant ordinary cohomology of $BT^2$

TL;DR

This paper computes the $C_2$-equivariant cohomology of the classifying space of a 2-torus with Burnside ring coefficients, revealing complex relationships with related spaces and introducing an extended grading for better generator analysis.

Contribution

It provides the first detailed calculation of the $C_2$-equivariant cohomology of $BT^2$ using an extended grading, enhancing understanding of equivariant cohomological structures.

Findings

Calculated $C_2$-cohomology of $BT^2$ with Burnside ring coefficients.

Established relationships between $BT^2$, $BT^1$, and $BU(2)$ cohomologies.

Introduced an extended grading to identify natural generators.

Abstract

We calculate the ordinary $C_2$-cohomology of $BT^2$ with Burnside ring coefficients, using an extended grading that allows us to capture a more natural set of generators. We discuss how this cohomology is related to those of $BT^1$ and $BU(2)$, calculated previously, both relationships being more complicated than in the nonequivariant case.