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

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.