Topological manifolds are locally-flat Euclidean neighbourhood retracts

TL;DR

This paper proves that every topological n-manifold can be embedded in Euclidean space with a locally flat embedding and is a retract of a neighborhood in that space, advancing understanding of manifold embeddings.

Contribution

It establishes that all topological manifolds admit locally flat embeddings into Euclidean space and are retracts of neighborhoods, providing new insights into manifold topology.

Findings

Existence of locally flat embeddings for all topological manifolds.

Manifolds are retracts of neighborhoods in Euclidean space.

Extension of classical embedding theorems to topological manifolds.

Abstract

We show that every topological n-manifold M admits a locally flat closed embedding $\iota\colon M \hookrightarrow \mathbb{R}^{2n+1}$ and is a retract of some neighbourhood $U \subseteq \mathbb{R}^{2n+1}$