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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The functor of singular chains detects weak homotopy equivalences
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by Manuel Rivera, Felix Wierstra and Mahmoud Zeinalian PDF
Proc. Amer. Math. Soc. 147 (2019), 4987-4998 Request permission

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
  • Email: manuelr@math.miami.edu
  • Felix Wierstra
  • Affiliation: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
  • MR Author ID: 1295763
  • Email: felix.wierstra@gmail.com
  • 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
  • Email: mahmoud.zeinalian@lehman.cuny.edu
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4987-4998
  • MSC (2010): Primary 55P10, 57T30; Secondary 55P35
  • DOI: https://doi.org/10.1090/proc/14555
  • MathSciNet review: 4011530