Piecewise linear generalized Alexander's theorem in dimension at most 5

TL;DR

This paper proves that any piecewise linear co-dimension two embedding of a closed oriented manifold in Euclidean space of dimension at most five can be isotoped into a closed braid, extending classical results to higher dimensions.

Contribution

It extends Alexander's classical theorem to dimension five and higher for piecewise linear embeddings, providing new isotopy results for co-dimension two embeddings.

Findings

Embeddings in dimensions ≤5 can be isotoped to closed braids.

Extension of Alexander's theorem to higher dimensions.

Analogous results for higher co-dimension embeddings.

Abstract

We study piecewise linear co-dimension two embeddings of closed oriented manifolds in Euclidean space, and show that any such embedding can always be isotoped to be a closed braid as long as the ambient dimension is at most five, extending results of Alexander (in ambient dimension three), and Viro and independently Kamada (in ambient dimension four). We also show an analogous result for higher co-dimension embeddings.