Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Bredon equivariant cohomology of a point for cyclic groups

Daniel Dugger, Christy Hazel

TL;DR

The paper advances the algebraic understanding of the RO$(C_n)$-graded Bredon cohomology of a point for odd $n$ by recasting the problem in ${oldsymbol D}( Z)$ and exploiting invertible spheres built from ${ Z}$-modules. It introduces divisor-string technology, $ Z$-module models, and detailed positive/negative cone descriptions, yielding a coherent, largely explicit presentation of the ground ring $H^ullet(pt; Z)$ and its rationalization; it also provides a systematic approach to the irregular region and change-of-group phenomena. The work demonstrates that, in the regular region, the positive cone is generated by $au$-classes with a computable ring structure, while the negative cone admits a rich, though controllable, torsion theory built from gamma-classes and their divided forms. Together, these results offer a practical algebraic framework for understanding the equivariant stable homotopy types in this setting and set the stage for further refinements and localizations. The methods have potential impact on computations in equivariant stable homotopy theory, especially for cyclic groups of odd order, and illustrate how deep homotopical questions translate into tractable algebraic problems in a derived category of Mackey-functor modules.

Abstract

We study the $RO(G)$-graded Bredon cohomology of a point in the case where $G$ is a cyclic group of odd order, expanding on the information provided by previous studies. Our methods center on the purely algebraic aspects of this matter, which interpret it as the "stable homotopy groups of spheres" problem for the derived category of modules over the constant-coefficient Mackey ring.

The Bredon equivariant cohomology of a point for cyclic groups

TL;DR

The paper advances the algebraic understanding of the RO-graded Bredon cohomology of a point for odd by recasting the problem in and exploiting invertible spheres built from -modules. It introduces divisor-string technology, -module models, and detailed positive/negative cone descriptions, yielding a coherent, largely explicit presentation of the ground ring and its rationalization; it also provides a systematic approach to the irregular region and change-of-group phenomena. The work demonstrates that, in the regular region, the positive cone is generated by -classes with a computable ring structure, while the negative cone admits a rich, though controllable, torsion theory built from gamma-classes and their divided forms. Together, these results offer a practical algebraic framework for understanding the equivariant stable homotopy types in this setting and set the stage for further refinements and localizations. The methods have potential impact on computations in equivariant stable homotopy theory, especially for cyclic groups of odd order, and illustrate how deep homotopical questions translate into tractable algebraic problems in a derived category of Mackey-functor modules.

Abstract

We study the -graded Bredon cohomology of a point in the case where is a cyclic group of odd order, expanding on the information provided by previous studies. Our methods center on the purely algebraic aspects of this matter, which interpret it as the "stable homotopy groups of spheres" problem for the derived category of modules over the constant-coefficient Mackey ring.
Paper Structure (33 sections, 65 theorems, 191 equations, 4 figures)

This paper contains 33 sections, 65 theorems, 191 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Acknowledgments
  4. $\underline{{\mathbb Z}}$-modules and their derived category
  5. Review of $\underline{{\mathbb Z}}$-modules
  6. The derived category
  7. The stable homotopy ring for ${\EuScript D}(\underline{{\mathbb Z}})$
  8. Rationalization
  9. Initial computations
  10. Constructing chain maps and homotopies
  11. More $u$-classes
  12. The $\chi$-classes
  13. The cohomology of a point
  14. Prelude: The lcm-gcd sequence
  15. The positive cone
  16. ...and 18 more sections

Key Result

Proposition 1.1

The ring $\underline{H}^\star(pt)\otimes \mathbb Q$ is $\underline{{\mathbb Q}}[u_d, u_d^{-1} \,:\, d|n, d\neq 1]$.

Figures (4)

  • Figure 1: Generators for low-degree cohomology groups in the irregular region
  • Figure 2: The $E_2$-term $\mathop{\mathrm{\underline{Ext}}}\nolimits^{-p}(\underline{H}_{-q}(S^{\lambda_{\underline{b}}}),\underline{{\mathbb Z}})$
  • Figure 3: The slice $H^{k-m\lambda_b}$ of the negative cone
  • Figure 4: $H^{r\lambda_\ell+k\lambda_{\ell^2}+m}$ for $G=C_{\ell^2}$

Theorems & Definitions (151)

  • Proposition 1.1
  • Corollary 1.2
  • Remark 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Remark 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Corollary 1.10
  • ...and 141 more