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Rational homotopy equivalences and singular chains

Manuel Rivera, Felix Wierstra and Mahmoud Zeinalian

Algebraic & Geometric Topology 21 (2021) 1535–1552

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 π1–rational homotopy equivalences. We compare these two notions and show that π1–rational homotopy equivalences correspond to maps that induce Ω–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 Ω–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.

rational homotopy theory, cobar construction, fundamental group
Mathematical Subject Classification 2010
Primary: 55P62, 57T30
Secondary: 55P60
Received: 16 January 2020
Revised: 17 June 2020
Accepted: 30 June 2020
Published: 11 August 2021
Manuel Rivera
Department of Mathematics
Purdue University
West Lafayette, IN
United States
Felix Wierstra
Department of Mathematics
Stockholm University
Mahmoud Zeinalian
Department of Mathematics
City University of New York, Lehman College
Bronx, NY
United States