# Algorithmic aspects of immersibility and embeddability

## TL;DR

This paper investigates the algorithmic decidability of immersibility and embeddability of manifolds in Euclidean spaces, revealing conditions under which these problems are decidable or undecidable.

## Contribution

It provides a comprehensive analysis of the algorithmic aspects of immersion and embeddability theory, establishing new decidability and undecidability results.

## Key findings

- Decidability of immersibility depends on dimensions and category (Diff or PL).
- Embeddability of manifolds with boundary is undecidable under specific dimension constraints.
- The paper links immersion theory with computational complexity in topology.

## 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 $11m \geq 10n+1$.