The coalgebra of chains and the fundamental group

Abstract: Rational homotopy theory tells us that simply connected spaces, up to rational homotopy equivalence, may be classified algebraically by means of rational cocommutative coalgebras (Quillen) or in the finite type case by rational dg commutative algebras (Sullivan). Goerss and Mandell proved versions of these results for fields of arbitrary characteristic by means of simplicial cocommutative coalgebras and E-infinity algebras, respectively. The algebraic structures in these settings are considered up to quasi-isomorphism. In this talk, I will describe how to extend these results to spaces with arbitrary fundamental group.The key new observation is that the homotopy cocommutative coalgebraic structure of the chains on a space determines the fundamental group in complete generality. The corresponding algebraic notion of weak equivalence between coalgebras is drawn from Koszul duality. The end goal of this program is to completely understand homotopy types in terms of algebraic “chain level” structure. This is joint work with M. Zeinalian and F. Wierstra.

algebraic topology

Audience: researchers in the topic

Online algebraic topology seminar

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