Connected sums of codimension two locally flat submanifolds

Abstract

Let $X$ and $Y$ be oriented topological manifolds of dimension $n\!+\!2$ , and let $K\! \subset \! X$ and $J \! \subset \! Y$ be connected, locally-flat, oriented, $n$ –dimensional submanifolds. We show that up to orientation preserving homeomorphism there is a well-defined connected sum $(X,K)\! \mathbin {\#}\! (Y,J)$ . For $n = 1$ , the proof is classical, relying on results of Rado and Moise. For dimensions $n=3$ and $n \ge 6$ , results of Edwards-Kirby, Kirby, and Kirby-Siebenmann concerning higher dimensional topological manifolds are required. For $n = 2, 4,$ and $5$ , Freedman and Quinn's work on topological four-manifolds is required along with the higher dimensional theory.