Cutting and pasting pairs of manifolds with tangential structures

This paper studies cutting and pasting groups (SK-groups) of pairs of manifolds. By a pair of manifolds we mean a manifold with a submanifold of strictly smaller dimension. Existing results in the unoriented category by Komiya are generalized to manifolds with certain tangential structures. In this way multiple new splitting results for SK of pairs are obtained, in particular for SK of pairs of manifolds with a map to a reference space. We also prove that SK of manifolds with a map into $X\times Y$ with $Y$ simply connected is the same as SK of manifolds with map into $X$. This generalizes the result of Neumann that SK of manifolds with a map into a simply connected space is trivial.