Extendability of simplicial maps is undecidable

TL;DR

This paper proves that determining whether a simplicial map can be extended is algorithmically undecidable for certain fixed dimensions, highlighting fundamental limits in computational topology.

Contribution

It provides a concise proof of the undecidability of extendability of simplicial maps and identifies a gap in previous proofs regarding embeddability undecidability.

Findings

Extendability of simplicial maps is undecidable for fixed even dimensions.

There is no algorithm to recognize extendability of the identity map of spheres.

A gap is identified in the proof of embeddability undecidability in higher codimension.

Abstract

We present a short proof of the \v{C}adek-Kr\v{c}\'al-Matou\v{s}ek-Vok\v{r}\'inek-Wagner result from the title (in the following form due to Filakovsk\'y-Wagner-Zhechev). For any fixed even $l$ there is no algorithm recognizing the extendability of the identity map of $S^l$ to a PL map $X\to S^l$ of given $2l$-dimensional simplicial complex $X$ containing a subdivision of $S^l$ as a given subcomplex. We also exhibit a gap in the Filakovsk\'y-Wagner-Zhechev proof that embeddability of complexes is undecidable in codimension $>1$.