Cutting and pasting pairs of manifolds with tangential structures

TL;DR

This paper extends the theory of cutting and pasting groups (SK-groups) for pairs of manifolds with tangential structures, providing new splitting results and generalizations related to maps into product spaces.

Contribution

It generalizes existing unoriented results to manifolds with tangential structures and establishes new splitting theorems for SK-groups of pairs of manifolds.

Findings

Generalized SK-group results to manifolds with tangential structures

Derived new splitting theorems for SK of pairs of manifolds

Proved SK of manifolds with maps into a product space with simply connected factor equals SK of maps into the other factor

Abstract

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.