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The functor of singular chains detects weak homotopy equivalences

Authors: Manuel Rivera, Felix Wierstra and Mahmoud Zeinalian
Journal: Proc. Amer. Math. Soc. 147 (2019), 4987-4998
MSC (2010): Primary 55P10, 57T30; Secondary 55P35
Published electronically: July 30, 2019
MathSciNet review: 4011530
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Abstract: The normalized singular chains of a path connected pointed space $X$ may be considered as a connected $E_{\infty }$-coalgebra $\mathbf {C}_*(X)$ with the property that the $0$th homology of its cobar construction, which is naturally a cocommutative bialgebra, has an antipode; i.e., it is a cocommutative Hopf algebra. We prove that a continuous map of path connected pointed spaces $f: X\to Y$ is a weak homotopy equivalence if and only if $\mathbf {C}_*(f): \mathbf {C}_*(X)\to \mathbf {C}_*(Y)$ is an $\mathbf {\Omega }$-quasi-isomorphism, i.e., a quasi-isomorphism of dg algebras after applying the cobar functor $\mathbf {\Omega }$ to the underlying dg coassociative coalgebras. The proof is based on combining a classical theorem of Whitehead together with the observation that the fundamental group functor and the data of a local system over a space may be described functorially from the algebraic structure of the singular chains.

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Additional Information

Manuel Rivera
Affiliation: Department of Mathematics, University of Miami, 1365 Memorial Drive, Coral Gables, Florida 33146; Departamento de Matemáticas, Cinvestav, Av. Instituto Politécnico Nacional 2508, Col. San Pedro Zacatenco, México, D.F. CP 07360, México
MR Author ID: 1022985

Felix Wierstra
Affiliation: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
MR Author ID: 1295763

Mahmoud Zeinalian
Affiliation: Department of Mathematics, City University of New York, Lehman College, 250 Bedford Park Boulevard W, Bronx, New York 10468
MR Author ID: 773273

Received by editor(s): August 30, 2018
Received by editor(s) in revised form: January 10, 2019
Published electronically: July 30, 2019
Additional Notes: The second author was partially supported by the grant GA CR No. P201/12/G028.
Communicated by: Mark Behrens
Article copyright: © Copyright 2019 American Mathematical Society