The functor of singular chains detects weak homotopy equivalences

2018/08/30 by Rivera, Manuel, Wierstra, Felix, Zeinalian, Mahmoud
#Algebraic Topology (math.AT) #FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1808.10237

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 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→ Y is a weak homotopy equivalence if and only if C_*(f): C_*(X)→ C_*(Y) is an \mathbfΩ-quasi-isomorphism, i.e. a quasi-isomorphism of dg algebras after applying the cobar functor \mathbfΩ 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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