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Equivariant cohomology for cyclic groups

Samik Basu, Pinka Dey

Abstract

In this paper, we compute the $RO(C_n)$-graded coefficient ring of equivariant cohomology for cyclic groups $C_n$, in the case of Burnside ring coefficients, and in the case of constant coefficients. We use the invertible Mackey functors under the box product to reduce the gradings in the computation from $RO(C_n)$ to those expressable as combinations of $λ^d$ for divisors $d$ of $n$, where $λ$ is the inclusion of $C_n$ in $S^1$ as the roots of unity. We make explicit computations for the geometric fixed points for Burnside ring coefficients, and in the positive cone for constant coefficients. The positive cone is also computed for the Burnside ring in the case of prime power order, and in the case of square free order. Finally, we also make computations at non-negative gradings for the constant coefficients.

Equivariant cohomology for cyclic groups

Abstract

In this paper, we compute the -graded coefficient ring of equivariant cohomology for cyclic groups , in the case of Burnside ring coefficients, and in the case of constant coefficients. We use the invertible Mackey functors under the box product to reduce the gradings in the computation from to those expressable as combinations of for divisors of , where is the inclusion of in as the roots of unity. We make explicit computations for the geometric fixed points for Burnside ring coefficients, and in the positive cone for constant coefficients. The positive cone is also computed for the Burnside ring in the case of prime power order, and in the case of square free order. Finally, we also make computations at non-negative gradings for the constant coefficients.
Paper Structure (8 sections, 20 theorems, 232 equations)

This paper contains 8 sections, 20 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. The symmetric monoidal category of $G$-Mackey functors
  3. The group of invertible $\underline{A}$-modules
  4. Lifting orientation classes to $\underline{A}$-coefficients
  5. Computations for the Burnside ring
  6. The positive cone of $\pi^G_\bigstar(H\underline{A})$
  7. The positive cone of $\pi^G_\bigstar(H\underline{\mathbb{Z}})$
  8. The stable homotopy in negative grading

Key Result

Proposition 2.9

$(\underline{M}_1\boxtimes \underline{N}_1)\Box (\underline{M}_2\boxtimes \underline{N}_2)\cong (\underline{M}_1\Box \underline{M}_2)\boxtimes (\underline{N}_1\Box \underline{N}_2) .$

Theorems & Definitions (59)

  • Definition 1.4
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • Proposition 2.9
  • proof
  • ...and 49 more