Connected sums of codimension two locally flat submanifolds

TL;DR

This paper establishes a well-defined connected sum operation for certain submanifolds within topological manifolds across various dimensions, utilizing classical and modern topological results.

Contribution

It extends the concept of connected sums of submanifolds to higher dimensions, clarifying the conditions under which this operation is well-defined.

Findings

Connected sum is well-defined for n=1, 3, and >5 using classical and advanced topological results.

For n=2, 4, 5, the proof relies on Freedman and Quinn's work on four-manifolds.

The case for higher codimension remains unresolved.

Abstract

Let X and Y be oriented topological manifolds of dimension n + 2, and let K and J be connected, locally-flat, oriented, n-dimensional submanifolds of X and Y. We show that up to orientation preserving homeomorphism there is a well-defined connected sum K # J in X # Y. For n = 1, the proof is classical, relying on results of Rado and Moise. For dimensions n = 3 and n > 5, 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 needed. The truth of the corresponding statement for higher codimension seems to be unknown.