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2020

Extendability of simplicial maps is undecidable

Skopenkov, A.

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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 $>1$.

Extendability of simplicial maps is undecidable · Around

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Similar

A. B. Skopenkov. On the deleted product criterion for embeddability in ℝ m {\mathbb{R}}^{m} , Proc. Amer. Math. Soc. 1998. 126:8. P. 2467-2476
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R. Karasev and A. Skopenkov. Some ‘converses’ to intrinsic linking theorems, arXiv:2008.02523
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* J. Matoušek. Using the Borsuk-Ulam theorem: Lectures on topological methods in combinatorics and geometry. Springer Verlag, 2008
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J. Matoušek, M. Tancer, U. Wagner. Hardness of embedding simplicial complexes in ℝ d {\mathbb{R}}^{d} , J. Eur. Math. Soc. 13:2 (2011), 259–295. arXiv:0807.0336
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A. Nabutovsky, S. Weinberger . Algorithmic aspects of homeomorphism problems. arXiv:math/9707232
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* A. Skopenkov. Algebraic Topology From Algorithmic Standpoint, draft of a book, mostly in Russian, http://www.mccme.ru/circles/oim/algor.pdf
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* A. Skopenkov. On different reliability standards in current mathematical research, arXiv:2101.03745. More often updated version: https://www.mccme.ru/circles/oim/rese_inte.pdf
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Then

M. Čadek, M. Krčál. J. Matoušek, L. Vokřínek, U. Wagner. Extendability of continuous maps is undecidable, Discr. and Comp. Geom. 51 (2014) 24–66. arXiv:1302.2370
Original
2014
Later among the works it cites.
M. Filakovský, U. Wagner, S. Zhechev. Embeddability of simplicial complexes is undecidable. Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, https://epubs.siam.org/doi/pdf/10.1137/1.9781611975994.47
2020
Closest in time.
* A. Skopenkov. Invariants of graph drawings in the plane. Arnold Math. J., 6 (2020) 21–55; full version: arXiv:1805.10237
Original
2020
Closest in time.
* A. Skopenkov, Algebraic Topology From Geometric Viewpoint (in Russian), MCCME, Moscow, 2020 (2nd edition). Part of the book: http://www.mccme.ru/circles/oim/obstruct.pdf
2020
Closest in time.

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