Abstract: In this talk, we will explore an equivalence relation on manifolds through cutting and pasting along boundaries. This relation leads to the definition of an abelian group known as the SKK-group, which was introduced by Karras, Kreck, Neumann, and Ossa in the 1970s. We will discuss the connection between the SKK-group and the bordism category, as well as the bordism group. This connection is made clearer through the examination of the fundamental group of the geometric realization of categories, described using zig-zags. Additionally, I will present joint work with Simona Veselá and Renee Hoekzema, where we compute the SKK groups for particular cases. If time allows, we will also briefly explore the relationship between the SKK-group and invertible topological quantum field theory.