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

TL;DR

This paper establishes that the $K$-theory of equivariant cut-and-paste manifolds forms the fixed points of a genuine $G$-spectrum, using spectral Mackey functors to connect these concepts.

Contribution

It proves the conjecture that the $K$-theory of equivariant $SK$-manifolds is realized as fixed points of a genuine $G$-spectrum, introducing a new construction method.

Findings

$K$-theory of equivariant $SK$-manifolds is a fixed point spectrum.

Spectral Mackey functors can be constructed from squares $K$-theory.

Main technical result provides a general procedure for these constructions.

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.