Homology of E<sub>n</sub>ring spectra and iterated THH

Basterra, Maria; Mandell, Michael A.

doi:10.2140/agt.2011.11.939

Cited by 18 publications

(14 citation statements)

References 21 publications

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“…Computationally, applying the universal coefficient spectral sequences of [10, IV §4], we obtain the following universal coefficient spectral sequences for Quillen homology and cohomology. For computing H Cn * (A; H), the main result (Theorem 1.3) of [3] provides an iterative method. For A a cofibrant E n R-algebra lying over H, let N be a cofibrant non-unital E n H-algebra with a weak equivalence KN → H ∧ R A of E n H-algebras lying over H. By definition then H Cn * (A; H) is π * QN .…”

Section: Properties Of Quillen Homology and Cohomologymentioning

confidence: 99%

The multiplication on BP

Basterra

Mandell

2012

Journal of Topology

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Section: Properties Of Quillen Homology and Cohomologymentioning

confidence: 99%

The multiplication on BP

Basterra

Mandell

2012

Journal of Topology

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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%

“…8) with left adjoints on top; the latter adjunction is the composition of the former adjunctions. For each p ≥ 0, define the evaluation functor Ev p : SymSeq−→Mod R objectwise by Ev p (A) := A[p], and for each finite group G, consider the forgetful functor SymSeq G −→SymSeq.…”

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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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“…Ho.C/. Since the publication of the present preprint, a topological version of this result has been obtained by Basterra and Mandell [5], independently from our work, by relying, at some point, on the relationship between topological little n-cubes operads and iterated loop spaces (see Boardman and Vogt [11] and May [39]). …”

Section: Introductionmentioning

confidence: 62%

Iterated bar complexes ofE–infinity algebras and homology theories

Fresse¹

2011

Algebr. Geom. Topol.

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We proved in a previous article that the bar complex of an E 1 -algebra inherits a natural E 1 -algebra structure. As a consequence, a well-defined iterated bar construction B n .A/ can be associated to any algebra over an E 1 -operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A.The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E 1 -algebras. We use this effective definition to prove that the n-fold bar construction admits an extension to categories of algebras over E n -operads.Then we prove that the n-fold bar complex determines the homology theory associated to the category of algebras over an E n -operad. In the case n D 1, we obtain an isomorphism between the homology of an infinite bar construction and the usual -homology with trivial coefficients. 57T30; 55P48, 18G55, 55P35 IntroductionThe standard reduced bar complex B.A/ is basically defined as a functor from the category of associative differential graded algebras (associative dg-algebras for short) to the category of differential graded modules (dg-modules for short). In the case of a commutative dg-algebra, the bar complex B.A/ inherits a natural multiplicative structure and still forms a commutative dg-algebra. This observation implies that an iterated bar complex B n .A/ is naturally associated to any commutative dg-algebra A, for every n 2 N . In this paper, we use techniques of modules over operads to study extensions of iterated bar complexes to algebras over E n -operads. Our main result asserts that the n-fold bar complex B n .A/ determines the homology theory associated to an E n -operad.For the purpose of this work, we take the category of dg-modules as a base category and we assume that all operads belong to this category. An E n -operad refers to a dg-operad equivalent to the chain operad of little n-cubes. Many models of E n -operads come in nested sequences ( )such that E D colim n E n is an E 1 -operad, an operad equivalent in the homotopy category of dg-operads to the operad of commutative algebras C. Recall that an E 1 -operad is equivalent to the operad of associative algebras A and forms, in another usual terminology, an A 1 -operad. The structure of an algebra over an A 1 -operad includes a product W A˝A ! A and a full set of homotopies that make this product associative. The structure of an algebra over an E 1 -operad includes a product W A˝A ! A and a full set of homotopies that make this product associative and commutative. The intermediate structure of an algebra over an E n -operad includes a product W A˝A ! A which is fully homotopy associative, but homotopy commutative up to some degree only.Throughout the paper, we use the letter C to denote the base category of dg-modules and the notation P C , where P is any operad, refers to the category of P-algebras in dg-modules. The category of commutative dg-algebras, identified with the category of algebras over the commutative operad C, is denoted by ...

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Homology of E_nring spectra and iterated THH

Cited by 18 publications

References 21 publications

The multiplication on BP

The multiplication on BP

Homotopy completion and topological Quillen homology of structured ring spectra

Iterated bar complexes ofE–infinity algebras and homology theories

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Homology of Enring spectra and iterated THH

Cited by 18 publications

References 21 publications

The multiplication on BP

The multiplication on BP

Homotopy completion and topological Quillen homology of structured ring spectra

Iterated bar complexes ofE–infinity algebras and homology theories

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Homology of E_nring spectra and iterated THH