Extendability of simplicial maps is undecidable
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We present a short proof of the Čadek-Krčál-Matoušek-Vokřínek-Wagner result from the title (in the following form due to Filakovský-Wagner-Zhechev). For any fixed integer l>1 there is no algorithm recognizing the extendability of the identity map of S^l∨ S^l to a PL map X→ S^l∨ S^l of given 2l-dimensional simplicial complex X containing a subdivision of S^l∨ 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.
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