Cut and paste invariants of manifolds via algebraic K-theory

TL;DR

This paper applies algebraic K-theory to manifolds, constructing a K-space that captures cut-and-paste invariants and a derived Euler characteristic, advancing the understanding of scissors congruence problems.

Contribution

It introduces a new K-space framework that recovers classical SK groups for manifolds and defines a derived Euler characteristic, extending algebraic K-theory applications.

Findings

Constructed a K-space for SK groups of manifolds.

Developed a derived version of the Euler characteristic.

Connected scissors congruence invariants with algebraic K-theory.

Abstract

Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic $K$-theory, and applying these tools to studying the Grothendieck ring of varieties. In this paper we give a new application of their framework: we construct a $K$-space that recovers the classical $\mathrm{SK}$ ("schneiden und kleben," German for "cut and paste") groups for manifolds on $\pi_0$, and we construct a derived version of the Euler characteristic.