The functor of singular chains detects weak homotopy equivalences

TL;DR

This paper shows that the functor of normalized singular chains can detect weak homotopy equivalences between path connected spaces by analyzing the algebraic structure of their associated $E_{ extinfty}$-coalgebras and the cobar construction.

Contribution

It establishes a new criterion for weak homotopy equivalences using the algebraic structure of singular chains and the cobar functor, linking topological and algebraic properties.

Findings

Weak homotopy equivalences correspond to $ ext{Ω}$-quasi-isomorphisms of singular chain coalgebras.

The algebraic structure of singular chains encodes fundamental homotopy information.

The proof combines classical Whitehead theorem with algebraic descriptions of fundamental groups.

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 $0^{\text{th}}$ 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.