Abstract
Abstract We show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.
Highlights
We investigate the notion of homotopy epimorphism in the category of differential graded algebras, possibly noncommutative
Our second main result is a characterization of k-acyclic maps f : X → Y as those maps for which the map of dg algebras C∗(GX, k), → C∗(GY, k) is a homological epimorphism
In the case when π1(X) is finite we show that the dg algebra C∗(GXp+, Fp) is the derived localization of C∗(GX, k) at a certain idempotent of Fp[π1(X)]
Summary
Let A, B and C be graded algebras with A being flat over k. The map f induces a map on the associated graded to these filtrations and, since B, B , C, C are cofibrant as left A-modules, we conclude that the E1-terms of the corresponding spectral sequences are isomorphic and the desired conclusion follows. Assuming that the unit maps A → B and A → C are injections and that B and C are flat as left A modules (disregarding the differential) allows to identify the associated graded of the appropriate filtration of B ∗A C. The conclusion of Corollary 2.3 holds under the assumption that the unit maps A → B and A → C are cofibrations of right dg A-modules
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