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

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

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

Empty spaces and the value of symbols: Estonia's 'war of monuments' from another angle

Taking as its point of departure the recent heightened discussion surrounding publicly sited monuments in Estonia, this article investigates the issue from the perspective of the country's eastern border city of Narva, focusing especially upon the restoration in 2000 of a 'Swedish Lion' monument to mark the 300th anniversary of Sweden's victory over Russia at the first Battle of Narva. This commemoration is characterised here as a successful local negotiation of a potentially divisive past, as are subsequent commemorations of the Russian conquest of Narva in 1704. A recent proposal to erect a statue of Peter the Great in the city, however, briefly threatened to open a new front in Estonia's ongoing 'war of monuments'. Through a discussion of these episodes, the article seeks to link the Narva case to broader conceptual issues of identity politics, nationalism and post-communist transition

Specialization among amphipods: the invasiveGammarus tigrinushas narrower niche space compared to native gammarids

Human-mediated invasions of nonindigenous species are modifying global biodiversity. Despite significant interest in the topic, niche separation and specialization of invasive and closely related native sympatric species are not well understood. It is expected that combined use of various methods may reveal different aspects of niche space and provide stronger evidence for niche partitioning as compared to a single method. We applied the species marginality index (OMI) and species distribution modeling (SDM) in the northern Baltic Proper to determine (1) if environmental niche spaces at habitat scale differ between taxonomically and functionally closely related invasive and native gammarid species, and (2) whether the observed pattern relates to the species distribution overlap. Both methods agreed in notably narrower and more segregated realized niche of invasive Gammarus tigrinus compared to the studied native gammarids. Among native species, the distribution of G. zaddachi overlapped the most with G. tigrinus. Our results confirm that widespread colonization does not require a wide niche of the colonizer, but may rather be a function of other biological traits and/or the saturation of the recipient ecosystem. The niche divergence and wider environmental niche space of native species are likely to safeguard their existence in habitats less suitable for G. tigrinus