PIMS Voyageur Colloquium: John Harper (University of Western Ontario)

Abstract:

Quillen introduced a notion of homology in terms of derived
abelianization. Working in the contexts of operadic algebras in chain complexes and spectra, we prove a finiteness theorem relating finiteness properties of Quillen homology groups and homotopy groups---this result should be thought of as a Quillen homology analog of Serre's finiteness theorem for the homotopy groups of spheres. We describe a rigidification of the derived cosimplicial resolution with respect to Quillen homology, and use this to define homology completion---in the sense of Bousfield-Kan---for operadic algebras. We prove that under appropriate connectivity conditions, the coaugmentation into homology completion is a weak equivalence---in
particular, such operadic algebras can be recovered from their Quillen homology. This talk will focus on the chain complex version of these results, beginning with a short introduction to Quillen's notion of derived abelianization, and followed by a sketch of the proofs with an emphasis on several of the more conceptual homotopical arguments. We
will illustrate the results in the special case of commutative
differential graded algebras and André-Quillen homology. Many of the results described are joint with K. Hess.