Classification of di-embeddings of the $n$-cube into $\mathbb {R}^n$

Abstract

A di-embedding of the $n$-cube $I^n$ into $\mathbb{r}^n$ is a map $I ^n\to \mathbb{R}^n$ which is a dihomeomorphism onto its image.We show that such a map is, up to a permutation of coordinates, an $n$-fold product of 1-dimensional orientation preserving embeddings $I^1 \to \mathbb{R}$.