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.
ASJC Scopus subject areas
- Applied Mathematics