000189592 001__ 189592 000189592 005__ 20190316235720.0 000189592 022__ $$a1465-3060 000189592 02470 $$2ISI$$a000322344000003 000189592 0247_ $$2doi$$a10.2140/gt.2013.17.1325 000189592 037__ $$aARTICLE 000189592 245__ $$aHomotopy completion and topological Quillen homology of structured ring spectra 000189592 269__ $$a2013 000189592 260__ $$bGeometry & Topology Publications$$c2013$$aCoventry 000189592 300__ $$a92 000189592 336__ $$aJournal Articles 000189592 520__ $$aWorking in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum (eg structured ring spectra). We prove a strong convergence theorem that shows 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 strong convergence, we prove several theorems concerning the topological Quillen homology of algebras and modules over operads. These include a theorem relating finiteness properties of topological Quillen homology groups and homotopy groups that can be thought of as a spectral algebra analog of Serre's finiteness theorem for spaces and HR Miller's boundedness result for simplicial commutative rings (but in reverse form). We also prove absolute and relative Hurewicz Theorems and a corresponding Whitehead Theorem for topological Quillen homology. Furthermore, we prove a rigidification theorem, which we use to describe completion with respect to topological Quillen homology (or TQ-completion). The TQ-completion construction can be thought of as a spectral algebra analog of Sullivan's localization and completion of spaces, Bousfield and Kan's completion of spaces with respect to homology and Carlsson's and Arone and Kankaanrinta's completion and localization of spaces with respect to stable homotopy. We prove analogous results for algebras and left modules over operads in unbounded chain complexes. 000189592 700__ $$uPurdue Univ, Dept Math, W Lafayette, IN 47907 USA$$aHarper, John E. 000189592 700__ $$0240499$$g105396$$aHess, Kathryn 000189592 773__ $$j17$$tGeometry & Topology$$k3$$q1325-1416 000189592 8564_ $$uhttps://infoscience.epfl.ch/record/189592/files/HtpyCompTower_published.pdf$$zPostprint$$s650798$$yPostprint 000189592 909C0 $$xU10968$$0252139$$pUPHESS 000189592 909CO $$ooai:infoscience.tind.io:189592$$qGLOBAL_SET$$pSV$$particle 000189592 917Z8 $$x105396 000189592 937__ $$aEPFL-ARTICLE-189592 000189592 973__ $$rREVIEWED$$sPUBLISHED$$aEPFL 000189592 980__ $$aARTICLE