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A linearization map for genuine equivariant algebraic $K$-theory

Maxine Calle, David Chan, Andres Mejia

Abstract

We introduce a version of algebraic $K$-theory for coefficient systems of rings which is valued in genuine $G$-spectra for a finite group $G$. We use this construction to build a genuine $G$-spectrum $K_G(\mathbb{Z}[\underline{π_1(X)}])$ associated to a $G$-space $X$, which provides a home for equivariant versions of classical invariants like the Wall finiteness obstruction and Whitehead torsion. We provide a comparison between our $K$-theory spectrum and the equivariant $A$-theory of Malkiewich--Merling via a genuine equivariant linearization map.

A linearization map for genuine equivariant algebraic $K$-theory

Abstract

We introduce a version of algebraic -theory for coefficient systems of rings which is valued in genuine -spectra for a finite group . We use this construction to build a genuine -spectrum associated to a -space , which provides a home for equivariant versions of classical invariants like the Wall finiteness obstruction and Whitehead torsion. We provide a comparison between our -theory spectrum and the equivariant -theory of Malkiewich--Merling via a genuine equivariant linearization map.
Paper Structure (21 sections, 73 theorems, 186 equations)

This paper contains 21 sections, 73 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation and Conventions
  4. Acknowledgments
  5. Background on coefficient systems
  6. Induction and restriction
  7. A K-theory construction for coefficient systems
  8. A splitting on $K$-theory
  9. Perfect complexes
  10. A comparison with the $K$-theory of $G$-rings
  11. Reduced $K$-theory
  12. Equivariant A-theory and linearization
  13. Background on equivariant $A$-theory
  14. Lifting the linearization
  15. Linearization for non-$G$-connected spaces
  16. ...and 6 more sections

Key Result

Theorem A

Let $G$ be a finite group. For any coefficient system of rings $\underline{S}\colon \mathcal{O}^{\rm op}_{G}\to \mathrm{Ring}$ there is a genuine $G$-spectrum $K_G(\underline{S})$ whose $G$-fixed points split where $K(\underline{S}^H_{\theta})$ is the ordinary algebraic $K$-theory of the twisted group ring $\underline{S}^H_{\theta} = \underline{S}(G/H)_\theta[WH]$.

Theorems & Definitions (178)

  • Theorem A: \ref{['KTheoryExists', 'K theory splitting']}
  • Theorem B: \ref{['corollary: wall', 'corollary: whitehead']}
  • Theorem C: \ref{['linearization is 2 connected']}
  • Theorem D: \ref{['linearization splitting']}
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • ...and 168 more