Quillen homology

General Introduction
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Paper References
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
arXiv:1102.1234 On Quillen homology and a homotopy completion tower for algebras over operads from arXiv Front: math.AT by John E. Harper, Kathryn Hess We describe and study a (homotopy) completion tower for algebras and left modules over operads in symmetric spectra. We prove that a weak equivalence on topological Quillen homology induces a weak equivalence on homotopy completion, and that for $0$connected algebras and modules over a $1$connected operad, the homotopy completion tower interpolates between topological Quillen homology and the identity functor. By an explicit calculation of its layers, we show that the homotopy completion tower is the precise analogin the context of algebras and modules over operadsof the Goodwillie tower of the identity functor.
As easy consequences of the strong convergence properties of the homotopy completion tower, we prove a Whitehead theorem and a Hurewicz theorem for topological Quillen homology. We also prove a relative Hurewicz theorem that provides conditions under which topological Quillen homology detects $n$connected maps. We prove a finiteness theorem relating finiteness properties of topological Quillen homology groups and homotopy groups; this result should be thought of as an algebras over operads in spectra analog of Serre's finiteness theorem for the homotopy groups of spheres. We describe a rigidification of the derived cosimplicial resolution with respect to topological Quillen homology, and use this to define Quillen homology completionin the sense of BousfieldKanfor algebras and modules over operads. We also prove analogous results for algebras and modules over operads in unbounded chain complexes.
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