Deforming a PL submanifold of Euclidean space into a hyperplane

Abstract:

Let $M$ be a closed, $k$-connected, $m$-dimensional ${\text {PL}}$ submanifold of ${\mathbb {R}^{2m - k - 1}}\;(1 \leq k \leq m - 4)$. The main result of this paper states that if $m - k$ is even, then every embedding of $M$ into ${\mathbb {R}^{2m - k}}$ can be isotopically deformed into ${\mathbb {R}^{2m - k - 1}}$, and specifies which embeddings of $M$ into ${\mathbb {R}^{2m - k}}$ can be deformed into ${\mathbb {R}^{2m - k - 1}}$ in case $m - k$ is odd.