-------------------------- 6 new papers this month, by Chacholski-Pitsch-Scherer, Ching, DavisDaniel, Dugger, Flores-Scherer, and May-Sigurdsson. Mark Hovey New papers appearing on hopf between 1/10/05 and 2/5/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Chacholski-Pitsch-Scherer/hopullbacks Title: Homotopy pull-back squares up to localization Authors: Wojciech Chacholski, Wolfgang Pitsch, Jerome Scherer AMS classification numbers: Primary 55P60, 55R70; Secondary 55U35, 18G55 ArXiv submission number: math.AT/0501250 Abstract: We characterize the class of homotopy pull-back squares by means of elementary closure properties. The so called Puppe theorem which identifies the homotopy fiber of certain maps constructed as homotopy colimits is a straightforward consequence. Likewise we characterize the class of squares which are homotopy pull-backs ``up to Bousfield localization". This yields a generalization of Puppe's theorem which allows to identify the homotopy type of the localized homotopy fiber. When the localization functor is homological localization this is one of the key ingredients in the group completion theorem. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Ching/operad_bar Bar constructions for topological operads and the Goodwillie derivatives of the identity Michael Ching Massachusetts Institute of Technology Includes 19 PS figures with filenames *.pstex Abstract: We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the `Lie' operad structure on the homology groups of those derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over ooperads) and show that a based space naturally gives rise to a right module over the operad formed by the derivatives of the identity. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/enhfps2 Title: Homotopy fixed points for L_K(n)(E_n ^ X) using the continuous action Author: Daniel Davis Address: Purdue University, Department of Mathematics, 150 N. University Street, West Lafayette, IN 47907-2067 Abstract: Let G be a closed subgroup of G_n, the extended Morava stabilizer group. Let E_n be the Lubin-Tate spectrum, let X be an arbitrary spectrum with trivial G-action, and define E^(X) to be L_K(n)(E_n ^ X). We prove that E^(X) is a continuous G-spectrum with a G-homotopy fixed point spectrum, defined with respect to the continuous action. Also, we construct a descent spectral sequence whose abutment is the homotopy groups of the G-homotopy fixed point spectrum of E^(X). We show that the homotopy fixed points of E^(X) come from the K(n)-localization of the homotopy fixed points of the spectrum (F_n ^ X). 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger/spenrich Spectral enrichments of model categories Daniel Dugger Abstract: We prove that every stable, combinatorial model category has a natural enrichment by symmetric spectra (really a natural equivalence class of enrichments). This in some sense generalizes the simplicial enrichment of model categories provided by the Dwyer-Kan hammock localization. As a particular application, we associate to every object in a stable, combinatorial model category a certain "homotopy endomorphism ring spectrum". The homotopy type of this ring spectrum is preserved by Quillen equivalences, and so serves as an invariant of model categories. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Flores-Scherer/cwandfusion Title: Cellularization of classifying spaces and fusion properties of finite groups Authors: Ramon J. Flores, Jerome Scherer AMS classification numbers: Primary 55P60, 20D200; Secondary 55R37, 55Q05 ArXiv submission number: math.AT/0501442 Abstract: One way to understand the mod p homotopy theory of classifying spaces of finite groups is to compute their B\Z/p-cellularization. In the easiest cases this is a classifying space of a finite group (always a finite p-group). If not, we show that it has infinitely many non-trivial homotopy groups. Moreover they are either p-torsion free or else infinitely many of them contain p-torsion. By means of techniques related to fusion systems we exhibit concrete examples where p-torsion appears. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/May-Sigurdsson/MSMaster Parametrized homotopy theory J. P. May and J. Sigurdsson University of Chicago, University of Notre Dame Primary 19D99, 55N20, 55P42; Secondary 19L99, 55N22, 55T25 Abstract: We provide rigorous modern foundations for parametrized (equivariant, stable) homotopy theory in this four part monograph. In Part I, we give preliminaries on the necessary point-set topology, on base change and other relevant functors, and on generalizations of various standard results to the context of proper actions of non-compact Lie groups. In Part II, we give a leisurely development of the homotopy theory of ex-spaces that emphasizes several issues of independent interest. It includes much new material on the general theory of topologically enriched model categories. The essential point is to resolve problems in the homotopy theory of ex-spaces that have no nonparametrized counterparts. In contrast to previously encountered situations, model theoretic techniques are intrinsically insufficient for this purpose. Instead, a rather intricate blend of model theory and classical homotopy theory is required. In Part III, we develop the homotopy theory of parametrized spectra. We work equivariantly and with highly structured smash products and function spectra. The treatment is based on equivariant orthogonal spectra, which are simpler for the purpose than alternative kinds of spectra. Again, there are many difficulties that have no nonparametrized counterparts and cannot be dealt with model theoretically. In Part IV, we give a fiberwise duality theorem that allows fiberwise recognition of dualizable and invertible parametrized spectra. This allows application of the formal theory of duality in symmetric monoidal categories to the construction and analysis of transfer maps. A construction of fiberwise bundles of spectra, which are like bundles of tangents along fibers but with spectra replacing spaces as fibers, plays a central role. Using it, we obtain a simple conceptual proof of a generalized Wirthmuller isomorphism theorem that calculates the right adjoint to base change along an equivariant bundle with manifold fibers in terms of a shift of the left adjoint. Due to the generality of our bundle theoretic context, the Adams isomorphism theorem relating orbit and fixed point spectra is a direct consequence. -------------