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

A K-theory spectrum for cobordism cut and paste groups