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Singular cochains and rational homotopy type

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Abstract

Rational homotopy types of simply connected topological spaces have been classified by weak equivalence classes of commutative cochain algebras (Sullivan) and by isomorphism classes of minimal commutative A ∞-algebras (Kadeishvili).

We classify rational homotopy types of the space X by using the (noncommutative) singular cochain complex C*(X, Q), with additional structure given by the homotopies introduced by Baues, {E 1,k } and {F p,q}. We show that if we modify the resulting B ∞-algebra structure on this algebra by requiring that its bar construction be a Hopf algebra up to a homotopy, then weak equivalence classes of such algebras classify rational homotopy types.

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  1. Georgian Technical University, Tbilisi, Georgia

    Z. Kharebava

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 43, Topology and Its Applications, 2006.

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Kharebava, Z. Singular cochains and rational homotopy type. J Math Sci 152, 330–371 (2008). https://doi.org/10.1007/s10958-008-9069-4

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