Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Equivariant cohomology for cyclic groups of square-free order

Samik Basu, Surojit Ghosh

Abstract

The main objective of this paper is to compute $RO(G)$-graded cohomology of $G$-orbits for the group $G=C_n$, where $n$ is a product of distinct primes. We compute these groups for the constant Mackey functor $\underline{Z}$ and for the Burnside ring Mackey functor $\underline{A}$. Among other things, we show that the groups $\underline{H}^α_G(S^0)$ are mostly determined by the fixed point dimensions of the virtual representations $α$, except in the case of $\underline{A}$ coefficients when the fixed point dimensions of $α$ have many zeros. In the case of $\underline{Z}$ coefficients, the ring structure on the cohomology is also described. The calculations are then used to prove freeness results for certain $G$-complexes.

Equivariant cohomology for cyclic groups of square-free order

Abstract

The main objective of this paper is to compute -graded cohomology of -orbits for the group , where is a product of distinct primes. We compute these groups for the constant Mackey functor and for the Burnside ring Mackey functor . Among other things, we show that the groups are mostly determined by the fixed point dimensions of the virtual representations , except in the case of coefficients when the fixed point dimensions of have many zeros. In the case of coefficients, the ring structure on the cohomology is also described. The calculations are then used to prove freeness results for certain -complexes.

Paper Structure

This paper contains 7 sections, 45 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Equivariant Eilenberg-MacLane spectra
  3. The category of $G$-Mackey functors
  4. Torsion in cohomology groups and applications for $G$-spheres
  5. Cohomology with $\underline{\mathbb{Z}}$ coefficients
  6. The additive structure for $\underline{A}$-coefficients
  7. $G$-spaces that have free cohomology

Key Result

Theorem \oldthetheorem

The ring $\tilde{H}^+_{C_n}(S^0;\underline{\mathbb{Z}})$ is a subring of $\mathbb Z[u_{\xi^d}^{\pm}\mid d| n]\otimes \mathbb Z[a_\xi]/(na_\xi)$.

Theorems & Definitions (90)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • ...and 80 more