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Rational Homotopy Type and Computability

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Abstract

Given a simplicial pair (X, A), a simplicial complex Y, and a map \(f:A \rightarrow Y\), does f have an extension to X? We show that for a fixed Y, this question is algorithmically decidable for all X, A, and f if Y has the rational homotopy type of an H-space. As a corollary, many questions related to bundle structures over a finite complex are likely decidable. Conversely, for all other Y, the question is at least as hard as certain special cases of Hilbert’s tenth problem which are known or suspected to be undecidable.

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Notes

  1. This is the triviality problem for group presentations, translated into topological language. This work was extended by Adian and others to show that many other properties of nonabelian group presentations are likewise undecidable.

  2. The results can plausibly be extended to nilpotent spaces.

  3. It’s worth pointing out that this fits into a larger family of localizations of spaces, another of which is used in the proof of Lemma 3.3.

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Acknowledgements

I would like to thank Shmuel Weinberger for explaining some facts about H-spaces, and Marek Filakovský, Lukáš Vokřínek, and Uli Wagner for other useful conversations and encouragement. I would also like to thank the two referees for their careful reading and their many corrections and suggestions which have greatly improved the paper. The second referee, in particular, caught a major error which was present in previous versions. I was partially supported by NSF Grant DMS-2001042.

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  1. Department of Mathematics, UCSB, Santa Barbara, CA, USA

    Fedor Manin

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  1. Fedor Manin
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Correspondence to Fedor Manin.

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Communicated by Herbert Edelsbrunner.

In memory of Edgar H. Brown, 1926–2021.

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Manin, F. Rational Homotopy Type and Computability. Found Comput Math 23, 1817–1849 (2023). https://doi.org/10.1007/s10208-022-09582-8

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