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The slices of quaternionic Eilenberg-Mac Lane spectra

Bertrand J. Guillou, Carissa Slone

Abstract

We compute the slices and slice spectral sequence of integral suspensions of the equivariant Eilenberg-Mac Lane spectra $H\underline{\mathbb{Z}}$ for the group of equivariance $Q_8$. Along the way, we compute the Mackey functors $\underlineπ_{kρ} H\underline{\mathbb{Z}}$.

The slices of quaternionic Eilenberg-Mac Lane spectra

Abstract

We compute the slices and slice spectral sequence of integral suspensions of the equivariant Eilenberg-Mac Lane spectra for the group of equivariance . Along the way, we compute the Mackey functors .
Paper Structure (31 sections, 53 theorems, 153 equations, 4 figures, 5 tables)

This paper contains 31 sections, 53 theorems, 153 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Organization
  4. Acknowledgements
  5. Background
  6. Background for $C_4$
  7. Background for $K_4$
  8. Background for $Q_8$
  9. Inflation functors
  10. Inflation and the projection formula
  11. Geometric fixed points
  12. Bottleneck subgroups
  13. Inflation for $\ul{\mathbb{Z}}$-modules
  14. $Q_8$-Mackey functors and Bredon homology
  15. $RO(Q_8)$-graded Mackey functor $\ul{\mathbb{Z}}$-homology of a point
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $n\geq 0$. Then the nontrivial slices of $\Sigma^{n} H_{Q_8} \ul{\mathbb{Z}}$, above level $2n$, are for $k>n$. Furthermore,

Figures (4)

  • Figure 1: The homotopy Mackey functors of $\bigvee_n \Sigma^{n\rho} H_{K_4} \ul{\mathbb{Z}}$. The Mackey functor $\ul{\pi}_k \Sigma^{n\rho} H_{K_4} \ul{\mathbb{Z}}$ appears in position $(k,4n-k)$.
  • Figure 2: The homotopy Mackey functors of $\bigvee_n \Sigma^{n\rho} H_{K_4} \ul{\mathbb{F}_2}$. The Mackey functor $\ul{\pi}_k \Sigma^{n\rho} H_{K_4} \ul{\mathbb{F}_2}$ appears in position $(k,4n-k)$.
  • Figure 3: The $1$-skeleton of $S(\mathbb{H})$.
  • Figure 4: The homotopy Mackey functors of $\bigvee_n \Sigma^{n\rho} H_{Q} \ul{\mathbb{Z}}$. The Mackey functor $\ul{\pi}_k \Sigma^{n\rho} H_{Q} \ul{\mathbb{Z}}$ appears in position $(k,8n-k)$.

Theorems & Definitions (89)

  • Theorem 1.1
  • Proposition 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Definition 3.6
  • Proposition 3.7
  • ...and 79 more