### Abstract

The normalized singular chains of a path connected pointed space X may be considered as a connected E∞-coalgebra C*(X) with the property that the 0th 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 → Y is a weak homotopy equivalence if and only if C*(f) : C*(X) → C*(Y ) is an Ω-quasi-isomorphism, i.e., a quasi-isomorphism of dg algebras after applying the cobar functor Ω 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.

Original language | English (US) |
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Pages (from-to) | 4987-4998 |

Number of pages | 12 |

Journal | Proceedings of the American Mathematical Society |

Volume | 147 |

Issue number | 11 |

DOIs | |

State | Published - Jan 1 2019 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*147*(11), 4987-4998. https://doi.org/10.1090/proc/14555