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

Surojit Ghosh, Ankit Kumar

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.

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

Abstract

We compute the -graded Bredon cohomology of equivariant universal and classifying spaces associated to families of subgroups, with coefficients in the constant Mackey functor . 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.
Paper Structure (8 sections, 40 theorems, 176 equations)

This paper contains 8 sections, 40 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement.
  3. Preliminaries on Bredon Cohomology
  4. Bredon cohomology of $E_{C_p}\mathbb{Z}/p$ for odd primes $p$
  5. Bredon cohomology of $E_{C_2}\mathbb{Z}/2$
  6. The ring structure of $\widetilde{H}_{C_p\times C_p }^\bigstar(S^0; \underline{\mathbb{F}_p})$
  7. The ring structure of $\widetilde{H}_{\mathcal{K}_4}^\bigstar(S^0; \underline{\mathbb{F}_{2}})$
  8. Equivariant Complex Projective Spaces

Key Result

Proposition 1.7

BG21

Theorems & Definitions (80)

  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Remark 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Proposition 1.17
  • proof
  • Remark 1.18
  • ...and 70 more