Connected sums of codimension two locally flat submanifolds

TL;DR

The paper proves that the connected sum of locally flat, oriented, codimension-two submanifolds $F_1 \subset W_1$ and $F_2 \subset W_2$ is well-defined up to orientation-preserving homeomorphism, forming a well-defined pair $(F_1 \# F_2) \subset (W_1 \# W_2)$. The approach hinges on deep results in topological manifold theory, including the existence/uniqueness of normal bundles in codimension two, the Annulus Theorem, and the Stable Homeomorphism Theorem, with dimension-dependent utilizations (Kirby–Siebenmann, Quinn, Freedman–Quinn). The work clarifies and extends Cappell–Shaneson’s framework, providing two complementary routes: a direct argument using normal-bundle theory and isotopy extensions (Section 4) and a relative Annulus Theorem–based construction (Section 5). The results contribute a robust understanding of how knotted, locally flat submanifolds behave under connected sums in the topological category, while acknowledging that the higher-codimension generalization remains open.

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.

Theorems & Definitions (22)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof