A K-theory spectrum for cobordism cut and paste groups

TL;DR

This paper constructs a spectrum that captures the cobordism cut-and-paste groups of manifolds with boundary, extending previous definitions and relating these groups to cobordism groups.

Contribution

It introduces a spectrum that recovers cobordism cut-and-paste groups for manifolds with boundary and relates these to cobordism groups through a spectrum map.

Findings

Constructed a spectrum for cobordism cut-and-paste groups with boundary.

Established a relationship between the spectrum and cobordism groups.

Provided a spectrum map lifting the quotient map between groups.

Abstract

Cobordism groups and cut-and-paste groups of manifolds arise from imposing two different relations on the monoid of manifolds under disjoint union. By imposing both relations simultaneously, a cobordism cut and paste group $\overline{\mathrm{SK}}_n$ is defined. In this paper, we extend this definition to manifolds with boundary obtaining $\overline{\mathrm{SK}}^{\partial}_n$ and study the relationship of this group to an appropriately defined cobordism group of manifolds with boundary. The main results are the construction of a spectrum that recovers on $\pi_0$ the cobordism cut and paste groups of manifolds with boundary, $\overline{\mathrm{SK}}^{\partial}_n$, and a map of spectra that lifts the canonical quotient map $\mathrm{SK}^{\partial}_n \rightarrow \overline{\mathrm{SK}}^{\partial}_n$.