Smoothing 3-manifolds in 5-manifolds

TL;DR

This paper proves that topologically locally flat embeddings of 3-manifolds in 5-manifolds can be smoothly approximated, and applies this to show concordance equivalence for surfaces in 4-manifolds.

Contribution

It establishes a smoothing result for 3-manifolds in 5-manifolds and links topological and smooth concordance for surfaces in 4-manifolds.

Findings

Topologically locally flat embeddings can be homotoped to smooth embeddings.

Topological concordance implies smooth concordance for surfaces in 4-manifolds.

Provides a method to smooth embeddings in higher dimensions.

Abstract

We show that every locally flat topological embedding of a 3-manifold in a smooth 5-manifold is homotopic, by a small homotopy, to a smooth embedding. We deduce that topologically locally flat concordance implies smooth concordance for smooth surfaces in smooth 4-manifolds.