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#### The functor of singular chains detects weak homotopy equivalences

##### External Resource

https://doi.org/10.1090/proc/14555

(Publisher version)

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##### Fulltext (public)

arXiv:1808.10237.pdf

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##### Citation

Rivera, M., Wierstra, F., & Zeinalian, M. (2019). The functor of singular chains
detects weak homotopy equivalences.* Proceedings of the American Mathematical Society,* *147*(11), 4987-4998. doi:10.1090/proc/14555.

Cite as: https://hdl.handle.net/21.11116/0000-0005-0A6A-A

##### 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.

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.