Extendability of Simplicial Maps is Undecidable

We present a short proof of the adek–Král–Matouek–Vokřínek–Wagner result from title (in following form due to Filakovský–Wagner–Zhechev). For any fixed even l there is no algorithm recognizing extendability identity map $$S^l$$ PL $$X\rightarrow S^l$$ given 2l-dimensional simplicial complex X containing subdivision as subcomplex. also exhibit gap in Filakovský–Wagner–Zhechev that embeddability complexes undecidable codimension $$>1$$ .