$RO(C_p \times C_p)$-graded cohomology of universal spaces and the coefficient ring

TL;DR

This paper computes the $RO(C_p imes C_p)$-graded Bredon cohomology of universal and classifying spaces with explicit ring structure, advancing understanding of equivariant cohomology and cohomology operations.

Contribution

It provides explicit calculations of the $RO(C_p imes C_p)$-graded cohomology and the structure of the coefficient ring for equivariant universal spaces, a novel contribution in equivariant topology.

Findings

Explicit description of the coefficient ring including its multiplicative structure.

Computed the $RO(C_p imes C_p)$-graded Bredon cohomology for universal and classifying spaces.

Applied results to study lifts of cohomology operations in equivariant complex projective spaces.

Abstract

We compute the $RO(C_p \times C_p)$-graded Bredon cohomology of equivariant universal and classifying spaces associated to families of subgroups, with coefficients in the constant Mackey functor $\underline{\mathbb{F}_p}$. An explicit description of the resulting coefficient ring, including its multiplicative structure, is obtained. These computations are then applied to the study of lifts of cohomology operations via the Bredon cohomology of equivariant complex projective spaces.