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The $RO(\mathcal{K})$-graded Coefficients of $H\underline{A}$

Jesse Keyes

TL;DR

This work computes the $RO(\mathcal{K})$-graded coefficients of the equivariant Eilenberg–Mac Lane spectrum $H\underline{A}$ for the Klein four group $\mathcal{K}$, focusing on the Burnside Mackey functor and interpreting the problem as the Bredon cohomology of a point. The authors develop a cellular, Mackey-functor–based framework and leverage a double-complex spectral sequence, universal coefficients, and comparison to $H\underline{\mathbb{Z}}$ to resolve the $RO(\mathcal{K})$-graded homotopy in both the positive and negative cones. They provide explicit descriptions of the positive and negative cones for suspensions $\Sigma^{k\overline{\rho}}H\underline{A}$ in wide ranges of degrees, including precise extension data and the role of multiplicative structure (Euler classes and transfers) in resolving ambiguities. Overall, the paper delivers a universal Klein-four computation that clarifies how RO-graded cohomology behaves in this setting and exposes the nuanced multiplicative phenomena distinguishing the Klein-four case from simpler $C_2$-group scenarios.

Abstract

In $G$-equivariant stable homotopy theory, it is known that the equivariant Eilenberg-Mac Lane spectra representing ordinary equivariant cohomology have nontrivial $RO(G)$-graded homotopy corresponding to the equivariant (co)homology of representation spheres. We will compute the universal case of this ordinary $RO(G)$-graded homotopy in the case of $G=\mathcal{K}$, where $\mathcal{K}$ is the Klein-four group. In particular, we will compute a subring of the $RO(\mathcal{K})$-graded homotopy of $H\underline{A}$ for $\underline{A}$ the Burnside Mackey functor.

The $RO(\mathcal{K})$-graded Coefficients of $H\underline{A}$

TL;DR

This work computes the -graded coefficients of the equivariant Eilenberg–Mac Lane spectrum for the Klein four group , focusing on the Burnside Mackey functor and interpreting the problem as the Bredon cohomology of a point. The authors develop a cellular, Mackey-functor–based framework and leverage a double-complex spectral sequence, universal coefficients, and comparison to to resolve the -graded homotopy in both the positive and negative cones. They provide explicit descriptions of the positive and negative cones for suspensions in wide ranges of degrees, including precise extension data and the role of multiplicative structure (Euler classes and transfers) in resolving ambiguities. Overall, the paper delivers a universal Klein-four computation that clarifies how RO-graded cohomology behaves in this setting and exposes the nuanced multiplicative phenomena distinguishing the Klein-four case from simpler -group scenarios.

Abstract

In -equivariant stable homotopy theory, it is known that the equivariant Eilenberg-Mac Lane spectra representing ordinary equivariant cohomology have nontrivial -graded homotopy corresponding to the equivariant (co)homology of representation spheres. We will compute the universal case of this ordinary -graded homotopy in the case of , where is the Klein-four group. In particular, we will compute a subring of the -graded homotopy of for the Burnside Mackey functor.

Paper Structure

This paper contains 12 sections, 33 theorems, 90 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Notations and Conventions
  3. Mackey Functors
  4. Preliminaries
  5. A Family of Free Mackey Functors
  6. Cellular Bredon (Co)homology
  7. Double Complexes, Universal Coefficients, and a Kunneth Spectral Sequence
  8. A Comparison of the Positive Cones for $H\underline{A}$ and $H\underline{\mathbb{Z}}$
  9. A Comparison of the Negative Cones for $H\underline{A}$ and $H\underline{\mathbb{Z}}$
  10. The Positive Cone for $H\underline{A}$
  11. The Negative Cone for $H\underline{A}$
  12. Remarks on the Multiplicative Structure

Key Result

Lemma 2.7

Let $H \leq \mathcal{K}$. For any $\underline{M} \in {\textnormal{Mack}}_\mathcal{K}$, there is a natural isomorphism

Figures (12)

  • Figure 2.1: The Lewis Diagram for a $C_2$-Mackey Functor
  • Figure 2.2: The Lewis Diagram for a $\mathcal{K}$-Mackey Functor
  • Figure 2.3: Inflation Depicted via Lewis Diagrams
  • Figure 3.1: The Burnside Mackey Functor
  • Figure 3.2: Free $\mathcal{K}$-Mackey Functors
  • ...and 7 more figures

Theorems & Definitions (69)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Remark 2.8
  • Definition 2.9
  • ...and 59 more