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Rational homotopy
equivalences and singular chains
Manuel Rivera, Felix Wierstra and Mahmoud Zeinalian |
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Algebraic & Geometric Topology 21 (2021) 1535–1552
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Abstract |
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Bousfield and Kan’s –completion and fiberwise –completion of spaces lead to two different approaches to the rational homotopy theory of nonsimply connected spaces. In the first approach, a map is a weak equivalence if it induces an isomorphism on rational homology. In the second, a map of path-connected pointed spaces is a weak equivalence if it induces an isomorphism between fundamental groups and higher rationalized homotopy groups; we call these maps –rational homotopy equivalences. We compare these two notions and show that –rational homotopy equivalences correspond to maps that induce –quasi-isomorphisms on the rational singular chains, ie maps that induce a quasi-isomorphism after applying the cobar functor to the dg coassociative coalgebra of rational singular chains. This implies that both notions of rational homotopy equivalence can be deduced from the rational singular chains by using different algebraic notions of weak equivalences: quasi-isomorphisms and –quasi-isomorphisms. We further show that, in the second approach, there are no dg coalgebra models of the chains that are both strictly cocommutative and coassociative. |
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Keywords
rational homotopy theory, cobar construction, fundamental
group
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Mathematical Subject Classification 2010
Primary: 55P62, 57T30
Secondary: 55P60
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References |
Publication
Received: 16 January 2020
Revised: 17 June 2020
Accepted: 30 June 2020
Published: 11 August 2021
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Authors |
| Manuel Rivera | |
| Department of Mathematics Purdue University West Lafayette, IN United States |
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| Felix Wierstra | |
| Department of Mathematics Stockholm University Stockholm Sweden |
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| Mahmoud Zeinalian | |
| Department of Mathematics City University of New York, Lehman College Bronx, NY United States |
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