Topological André-Quillen homology for cellular commutative S-algebras

Baker, Andrew; Gilmour, Helen; Reinhard, Philipp

doi:10.1007/s12188-008-0005-9

Cited by 10 publications

(22 citation statements)

References 17 publications

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“…Then by Theorem 2.2, assuming that it is not trivial, the element βP p n−1 u n−1 is of order p; we let f n : S 2p n −3 −→ R n−1 be a representative of this homotopy class. Thus as in [3,10] we can form the pushout diagram of commutative S-algebras…”

Section: Construction Of the R Nmentioning

confidence: 99%

“…Suppose that we have constructed R 0 −→ R m−1 so that π k R m−1 is torsion free for k m − 2 and the natural map induces an isomorphism Q ⊗ π * R 0 ∼ = Q ⊗ π * R m−1 . Now following [3,5] we attach m-cells minimally to kill the torsion of π m−1 R m−1 . In fact, following a suggestion of Tyler Lawson, we can do slightly more: factoring the attaching maps through Moore spectra of the form S m−1 ∪ p r D m , we can define R m using the pushout diagram…”

Section: Killing the Torsionmentioning

confidence: 99%

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BP: close encounters of the $$E_\infty $$ E ∞ kind

Baker

2013

J. Homotopy Relat. Struct.

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Inspired by Stewart Priddy's cellular model for the p-local Brown-Peterson spectrum BP , we give a construction of a p-local E∞ ring spectrum R which is a close approximation to BP . Indeed we can show that if BP admits an E∞ structure then these are weakly equivalent as E∞ ring spectra. Our inductive cellular construction makes use of power operations on homotopy groups to define homotopy classes which are then killed by attaching E∞ cells.Date: 29/07/2013 version 6 (to appear in Journal of Homotopy and Related Structures) arXiv:1204.4878. 2010 Mathematics Subject Classification. Primary 55N20; Secondary 55N22 55S12 55S15.Key words and phrases. E∞ ring spectrum; Brown-Peterson spectrum; power operation. The author would like to thank Bob Bruner, Mike Mandell, Peter May, Birgit Richter, John Rognes and Markus Szymik; special thanks are due to Tyler Lawson who pointed out the usefulness of coning off Moore spectra rather than spheres. Finally, we note that this paper would not exist without the inspiration provided by Stewart Priddy's elegant cellular construction of the Brown-Peterson spectrum.

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Section: Construction Of the R Nmentioning

confidence: 99%

Section: Killing the Torsionmentioning

confidence: 99%

BP: close encounters of the $$E_\infty $$ E ∞ kind

Baker

2013

J. Homotopy Relat. Struct.

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“…In this paper we are primarily interested in the topological analog of Quillen homology, called topological Quillen homology, for (generalized) algebraic structures on spectra. The topological analog for commutative ring spectra, called topological André-Quillen homology, was originally studied by Basterra [6]; see also Baker-Gilmour-Reinhard [4], Baker-Richter [5], Basterra-Mandell [7,8], Goerss-Hopkins [25], Lazarev [46], Mandell [52], Richter [62], Rognes [63,64] and Schwede [65,67].…”

Section: Introductionmentioning

confidence: 99%

Homotopy completion and topological Quillen homology of structured ring spectra

Harper

Hess

2013

Geom. Topol.

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Abstract. Working 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 (e.g., structured ring spectra). 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 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 H.R. 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 TQcompletion 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 Carlsson's and Arone-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.

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“…In the p-local connective setting, we will use ideas on minimal atomic S-modules and commutative S-algebras which may be found in [5,6,15]. In particular, the notions of nuclear CW S-modules and commutative S-algebras will play a central rôle in our work.…”

Section: Background Materials On S-modules and Commutative S-algebrasmentioning

confidence: 99%

Characteristics for ℰ_{∞} ring spectra

Baker¹

2018

An Alpine Bouquet of Algebraic Topology

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We introduce a notion of characteristic for connective p-local E∞ ring spectra and study some basic properties. Apart from examples already pointed out by Markus Szymik, we investigate some examples built from Hopf invariant 1 elements in the stable homotopy groups of spheres and make a series of conjectures about spectra for which they may be characteristics; these appear to involve hard questions in stable homotopy theory.Date: 01/06/2017 version 5 arXiv:1405.3695 .

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Topological André-Quillen homology for cellular commutative S-algebras

Cited by 10 publications

References 17 publications

BP: close encounters of the $$E_\infty $$ E ∞ kind

BP: close encounters of the $$E_\infty $$ E ∞ kind

Homotopy completion and topological Quillen homology of structured ring spectra

Characteristics for ℰ_{∞} ring spectra

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