Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A genuine $G$-spectrum for the cut-and-paste $K$-theory of $G$-manifolds

Maxine Calle, David Chan

Abstract

Recent work has applied scissors congruence $K$-theory to study classical cut-and-paste ($SK$) invariants of manifolds. This paper proves the conjecture that the squares $K$-theory of equivariant $SK$-manifolds arises as the fixed points of a genuine $G$-spectrum. Our method utilizes the framework of spectral Mackey functors as models for genuine $G$-spectra, and our main technical result is a general procedure for constructing spectral Mackey functors using squares $K$-theory.

A genuine $G$-spectrum for the cut-and-paste $K$-theory of $G$-manifolds

Abstract

Recent work has applied scissors congruence -theory to study classical cut-and-paste () invariants of manifolds. This paper proves the conjecture that the squares -theory of equivariant -manifolds arises as the fixed points of a genuine -spectrum. Our method utilizes the framework of spectral Mackey functors as models for genuine -spectra, and our main technical result is a general procedure for constructing spectral Mackey functors using squares -theory.

Paper Structure

This paper contains 6 sections, 25 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Squares $K$-theory
  3. Spectral Mackey functors
  4. Constructing spectral Mackey functors via squares $K$-theory
  5. The genuine $G$-spectrum of equivariant $SK$-manifolds
  6. Lifting the equivariant Euler characteristic

Key Result

Theorem A

There is a genuine $G$-spectrum $K^\square_G(\mathscr{M}^G_d)$ such that for all $H\leq G$. Moreover, $\pi_0$ of this genuine $G$-spectrum is the Mackey functor of equivariant $SK$-groups for $d$-dimensional $G$-manifolds with boundary.

Theorems & Definitions (57)

  • Theorem A: \ref{['thm:main']}
  • Theorem B: \ref{['thm:KT machine', 'corollary: from mackey functor of squares cats to spectral Mackey']}
  • Theorem C: \ref{['theorem: lift euler char']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • Example 2.7
  • ...and 47 more