PIMS Voyageur Colloquium: John Harper (University of Western Ontario)
- Date: 03/22/2011
University of Calgary
Homotopy completion, homology completion, and a finiteness theorem for operadic algebras.
Abstract:
Quillen introduced a notion of homology in terms of derivedabelianization.
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---inparticular, 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. Wewill
illustrate the results in the special case of commutativedifferential
graded algebras and André-Quillen homology. Many of the results
described are joint with K. Hess.
