Algorithmic aspects of immersibility and embeddability

TL;DR

It is shown 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.

Abstract

We analyze an algorithmic question about immersion theory: for which $m$, $n$, and $CAT=\mathbf{Diff}$ or $\mathbf{PL}$ is the question of whether an $m$-dimensional $CAT$-manifold is immersible in $\mathbb{R}^n$ decidable? 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 $11m \geq 10n+1$.

Figures (1)

• Figure 1: A subdivision of $\Delta^2$ satisfying the required conditions. The sets $U_0$ and $U_1$ are highlighted in shades of gray. We have drawn the simplices curved to suggest how $U_0$ and $U_1$ will look as subsets of a smooth manifold with boundary.

Theorems & Definitions (20)

• Corollary 1.1
• Theorem \ref{decundec}
• Proposition 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5