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

## Topic

## Details

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.

**Scientific, Seminar**

**March 22, 2011**

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