Extendability of Simplicial Maps is Undecidable

Abstract

We present a short proof of the adek–Král–Matouek–Vokřínek–Wagner result from the title (in the following form due to Filakovský–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 \(X\rightarrow S^l\) of given 2l-dimensional simplicial complex X containing a subdivision of \(S^l\) as a given subcomplex. We also exhibit a gap in the Filakovský–Wagner–Zhechev proof that embeddability of complexes is undecidable in codimension \(>1\).