TY - JOUR

T1 - Rational homotopy equivalences and singular chains

AU - Rivera, Manuel

AU - Wierstra, Felix

AU - Zeinalian, Mahmoud

N1 - Funding Information:
Rivera acknowledges the support of the grant Fordecyt 265667 and the excellent working conditions of Centro de colaboración Samuel Gitler in Mexico City. Wierstra and Zeinalian would like to thank the Max Planck Institute for Mathematics, where they first met and their collaboration started, for the hospitality and support during their stays.
Publisher Copyright:
© 2021, Mathematical Science Publishers. All rights reserved.

PY - 2021

Y1 - 2021

N2 - Bousfield and Kan’s Q–completion and fiberwise Q–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.

AB - Bousfield and Kan’s Q–completion and fiberwise Q–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.

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U2 - 10.2140/agt.2021.21.1535

DO - 10.2140/agt.2021.21.1535

M3 - Article

AN - SCOPUS:85113805667

VL - 21

SP - 1535

EP - 1552

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

SN - 1472-2747

IS - 3

ER -