A central problem in algebraic topology is to determine how much information about the category of topological spaces considered up to a specified notion of weak equivalence is preserved by a particular functorial invariant. In this talk, I will discuss a version of this problem for spaces up to “pi_1-R-equivalence”, for a commutative ring R, by means of simplicial cocommutative R-coalgebras. A continuous map of spaces is called a pi_1-R-equivalence if it induces an isomorphism on fundamental groups and an R-homology equivalence between universal covers. I will describe a corresponding notion of weak equivalence for simplicial R-coalgebras using ideas from Koszul duality theory. One of our main results is that, when R is an algebraically closed field, the simplicial chains functor embeds fully and faithfully the homotopy theory of spaces up to pi_1-R-equivalence into the corresponding homotopy theory for simplicial R-coalgebras. The key observation for constructing an appropriate homotopy theory for simplicial coalgebras is an extension to non-simply connected spaces of a classical theorem of F. Adams describing an algebraic model for the based loop space of a space X in terms of the differential graded coalgebra of chains on X. A surprising conceptual consequence of this observation is that the algebraic structure of the chains on a space, considered up to the appropriate algebraic notion of weak equivalence, completely determines the fundamental group. This is based on a series of joint projects with F. Wierstra, M. Zeinalian, and G. Raptis.

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