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.