A K-theory spectrum for cobordism cut and paste 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 ${\overline{\mathrm{SK}}}_n^{\partial }$ 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 $\ddot{I}{€}_0$ the cobordism cut and paste groups of manifolds with boundary, ${\overline{\mathrm{SK}}}_n^{\partial }$, and a map of spectra that lifts the canonical quotient map ${\mathrm{SK}}_n^{\partial }\to {\overline{\mathrm{SK}}}_n^{\partial }$.

31 pages, 4 figures. Improved readability, and incorporated referee comments