The functor of singular chains detects weak homotopy equivalences
| Autor: | Mahmoud Zeinalian, Felix Wierstra, Manuel Rivera |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Physics
Fundamental group Pure mathematics Connected space Functor Applied Mathematics General Mathematics Homotopy Mathematics::Rings and Algebras Algebraic topology Hopf algebra Mathematics::Algebraic Topology Bialgebra Mathematics::K-Theory and Homology Mathematics::Quantum Algebra Mathematics::Category Theory Mathematics - Quantum Algebra FOS: Mathematics Algebraic Topology (math.AT) Quantum Algebra (math.QA) Mathematics - Algebraic Topology Pointed space |
| Zdroj: | Proceedings of the American Mathematical Society |
| Popis: | 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. Comment: This version contains minor changes based on a referee report mostly concerning presentation and typos. We have also added new acknowledgements |
| Databáze: | OpenAIRE |
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