Homotopy completion and topological Quillen homology of structured ring spectra

Details Homotopy completion and topological Quillen homology of structured ring spectra Homotopy completion and topological Quillen homology of structured ring spectra

2011-02-07

arxiv.org/abs/1102.1234v2

Mathematics

Algebraic Topology

This paper is a significant revision of the earlier manuscript which was titled "On Quillen homology and a homotopy completion

Scientific paper

Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads (e.g., structured ring spectra) in the category of modules over a commutative ring spectrum. We prove a strong convergence theorem that for 0-connected algebras and modules over a (-1)-connected operad, the homotopy completion tower interpolates (in a strong sense) between topological Quillen homology and the identity functor. By systematically exploiting the strong convergence properties of the homotopy completion tower, we prove a selection of theorems concerning the topological Quillen homology of algebras and modules over operads. This includes a finiteness theorem relating finiteness properties of topological Quillen homology groups and homotopy groups; it can be thought of as a spectral algebra analog of Serre's finiteness theorem for spaces and H.R. Miller's boundedness result for simplicial commutative rings (but in reverse form). We prove a Hurewicz theorem for topological Quillen homology that provides conditions under which the first non-trivial homotopy group agrees via the Hurewicz map with the first non-trivial topological Quillen homology group. We also prove a relative Hurewicz theorem (resp. Whitehead theorem) that provides conditions under which topological Quillen homology detects n-connected maps (resp. weak equivalences). We also prove a rigidification theorem and use this to describe completion with respect to topological Quillen homology (or TQ-completion) of algebras and left modules over operads (e.g., structured ring spectra). This TQ-completion construction can be thought of as a spectral algebra analog of Sullivan's localization and completion of spaces, Bousfield-Kan's completion of spaces with respect to homology, and Arone-Kankaanrinta's localization of spaces with respect to stable homotopy.

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