Rational homotopy type and computability
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866912076281151488 |
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| author | Manin, Fedor |
| author_facet | Manin, Fedor |
| contents | Given a simplicial pair $(X,A)$, a simplicial complex $Y$, and a map $f:A \to Y$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_10632 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Rational homotopy type and computability Manin, Fedor Algebraic Topology Computational Geometry 55Q05, 03D35, 55-08, 55P62 Given a simplicial pair $(X,A)$, a simplicial complex $Y$, and a map $f:A \to Y$, 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. |
| title | Rational homotopy type and computability |
| topic | Algebraic Topology Computational Geometry 55Q05, 03D35, 55-08, 55P62 |
| url | https://arxiv.org/abs/2007.10632 |