Algorithmic Aspects of Immersibility and Embeddability

We analyze an algorithmic question about immersion theory: for which $m$, $n$, and $CAT=\textbf{Diff}$ or $\textbf{PL}$ is the question of whether an $m$-dimensional $CAT$-manifold is immersible in $\mathbb{R}^{n}$ decidable? We show that PL immersibility is decidable in all cases except for codimension 2, whereas smooth immersibility is decidable in all odd codimensions and undecidable in many even codimensions. As a corollary, we show that the smooth embeddability of an $m$-manifold with boundary in $\mathbb{R}^{n}$ is undecidable when $n-m$ is even and $11m \geq 10n+1$.