Scissors congruence K-theory for equivariant manifolds

TL;DR

This paper develops a new equivariant scissors congruence K-theory spectrum for compact G-manifolds, providing a spectrum-level lift of the equivariant Euler characteristic and revealing a Mackey functor structure.

Contribution

It introduces a novel K-theory spectrum for equivariant scissors congruence, extending classical invariants to a spectrum level and uncovering a Mackey functor structure.

Findings

Spectrum level lift of equivariant Euler characteristic

Mackey functor structure for equivariant scissors groups

Connection to a conjectural higher equivariant structure

Abstract

We introduce a scissors congruence $K$-theory spectrum which lifts the equivariant scissors congruence groups for compact $G$-manifolds with boundary, and we show that on $\pi_0$ this is the source of a spectrum level lift of the Burnside ring valued equivariant Euler characteristic of a compact $G$-manifold. We also show that the equivariant scissors congruence groups for varying subgroups assemble into a Mackey functor, which is a shadow of a conjectural higher genuine equivariant structure.