Cosimplicial resolutions and homotopy spectral sequences in model categories

We develop a general theory of cosimplicial resolutions, homotopy spectral sequences, and completions for objects in model categories, extending work of Bousfield-Kan and Bendersky-Thompson for ordinary spaces. This is based on a generalized cosimplicial version of the Dwyer-Kan-Stover theory of resolution model categories, and we are able to construct our homotopy spectral sequences and completions using very flexible weak resolutions in the spirit of relative homological algebra. We deduce that our completion functors have triple structures and preserve certain fiber squares up to homotopy. We also deduce that the Bendersky-Thompson completions over connective ring spectra are equivalent to Bousfield-Kan completions over solid rings. The present work allows us to show, in a subsequent paper, that the homotopy spectral sequences over arbitrary ring spectra have well-behaved composition pairings.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper29.abs.htm

On the comparison of stable and unstable P-completion

In this note we show that a p-complete nilpotent space X has a p-complete suspension spectrum if and only if its homotopy groups pi X-* are bounded p-torsion. In contrast, if pi X-* is not all bounded p-torsion, we locate uncountable rational vector spaces in the integral homology and in the stable homotopy groups of X. To prove this, we establish a homological criterion for p-completeness of connective spectra. Moreover, we illustrate our results by studying the stable homotopy groups of K(Z(p), n) via Goodwillie calculus

Automorphisms of p-compact groups and their root data

We construct a model for the space of automorphisms of a connected p-compact group in terms of the space of automorphisms of its maximal torus normalizer and its root datum. As a consequence we show that any homomorphism to the outer automorphism group of a p-compact group can be lifted to a group action, analogous to a classical theorem of de Siebenthal for compact Lie groups. The model of this paper is used in a crucial way in our paper ``The classification of 2-compact groups'', where we prove the conjectured classification of 2-compact groups and determine their automorphism spaces.Comment: 24 pages. Introduction restructured and title changed (from "Automorphisms of root data, maximal torus normalizers, and p-compact groups"). Various other adjustments mad

Exploring the reactivity of 2-trichloromethylbenzoxazoles for access to substituted benzoxazoles

The reactivity of 2-trichloromethylbenzoxazoles towards various nucleophiles, under metal free or iron-catalyzed conditions, for the synthesis of substituted benzoxazoles is described. These methods allow for selective substitution at either the 2- or 2’- position of the benzoxazoles using the same starting materials / reagents. This approach allows for the controlled synthesis of a variety of key derivatives from a single 2-trichloromethylbenzoxazole starting material

Spaces with Noetherian cohomology

Is the cohomology of the classifying space of a p-compact group, with Noetherian twisted coefficients, a Noetherian module? This note provides, over the ring of p-adic integers, such a generalization to p-compact groups of the Evens-Venkov Theorem. We consider the cohomology of a space with coefficients in a module, and we compare Noetherianity over the field with p elements, with Noetherianity over the p-adic integers, in the case when the fundamental group is a finite p-group.Comment: 12 page

BV-structures on the homology of the framed long knot space

We introduce BV-algebra structures on the homology of the space of framed long knots in $\mathbb{R}^n$ in two ways. The first one is given in a similar fashion to Chas-Sullivan's string topology. The second one is defined on the Hochschild homology associated with a cyclic, multiplicative operad of graded modules. The latter can be applied to Bousfield-Salvatore spectral sequence converging to the homology of the space of framed long knots. Conjecturally these two structures coincide with each other.Comment: 13 pages, 3 figures, to appear in Journal of Homotopy and Related Structure

Brown representability for space-valued functors

In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify small contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every small contravariant functor from spaces to spaces which takes coproducts to products up to homotopy and takes homotopy pushouts to homotopy pullbacks is naturally weekly equivalent to a representable functor. The second representability theorem states: every contravariant continuous functor from the category of finite simplicial sets to simplicial sets taking homotopy pushouts to homotopy pullbacks is equivalent to the restriction of a representable functor. This theorem may be considered as a contravariant analog of Goodwillie's classification of linear functors.Comment: 19 pages, final version, accepted by the Israel Journal of Mathematic

La vie et la mort en peinture

L'abstraction visuelle ne constituerait pas d'abord une réflexion sur la nature du beau en soi, mais une approche cognitive renvoyant aux épistémologies et aux idéologies des époques où elle se manifeste. L'étude de différents discours tenus sur l'abstraction picturale au XIXe et au XXe siècles permet de suivre les valeurs attribuées à cette notion, notamment autour de l'opposition entre le vitalisme et la mort à partir des réflexions d'A. Riegl et W. Worringer. La différenciation entre l'abstrait et le concret, le sujet et l'objet se voit ainsi constamment relancée, dans la possibilité de leur réversibilité.Visual abstraction is not, at least not in the first place, a reflection upon the nature of beauty as such, but rather a cognitive approach to the world that bears witness to the epistemologies and ideologies of those periods of history where it appeared. The study of some of the discourses that have been held about pictural abstraction during the 19th and 20th centuries, notably the opposition between vitalism and death founded on the works of A. Riegl and W. Worringer, allows us to understand the various values that have been given to the notion. The distinction between abstraction and concreteness, subject and object, can here be seen, in regard to the possibility of their reversibility, as an ever-open question