4 new papers this month, from Bousfield, Castellana-Crespo-Scherer, IsaksenD, and Kitchloo-Morava. Mark Hovey New papers appearing on hopf between 4/5/04 and 5/4/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bousfield/vper On the 2-primary v1-periodic homotopy groups of spaces A.K. Bousfield bous@uic.edu AMS Classification Numbers: 55Q51(Primary),55N15,55P60,55S25,57T20 We develop foundations of a general approach for calculating p-primary v1-periodic homotopy groups of spaces using their p-adic KO-cohomologies and K-cohomolgies with particular attention to the case p = 2. As a main application, we derive a method for calculating v1-periodic homotopy groups of simply-connected compact Lie groups using their complex, real, and quaternionic representation theories. This method has been applied very effectively by D.M. Davis in recent work. We rely heavily on the p-primary v1-stabilization functor Phi from spaces to spectra. Roughly speaking, we obtain the p-primary v1-periodic homotopy of a space X from the p-adic KO-cohomology of Phi X, which we obtain from the p-adic KO-cohomology and K-cohomology of X by a v1-stabilization process under suitable conditions. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Castellana-Crespo-Scherer/DeconstructH Title: Deconstructing Hopf spaces Authors: Natalia Castellana, Juan A. Crespo, Jerome Scherer email: natalia@mat.uab.es, JuanAlfonso.Crespo@uab.es, jscherer@mat.uab.es AMS classification number: 55P45; 55S10; 55P60; 55P47; 55S45 Abstract: We characterize Hopf spaces with finitely generated cohomology as algebra over the Steenrod algebra. We ``deconstruct" the original space into an H-space Y with finite mod p cohomology and a finite number of p-torsion Eilenberg-Mac Lane spaces. One reconstructs X from Y by taking extensions by principal H-fibrations. We give a precise description of homotopy commutative H-spaces in this setting and give a criterion to recognize connected covers of H-spaces with finite mod p cohomology. The key observation is that the module of indecomposables lies in some stage of the Krull filtration of the category of unstable modules over the Steenrod algebra. We compare this algebraic condition with a topological one, namely that some iterated loop space of X is BZ/p-local. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/IsaksenD/completion Author: Daniel C. Isaksen Author's e-mail address: isaksen@math.wayne.edu Author's mailing address: Department of Mathematics \\ Wayne State University \\ Detroit, MI 48202 Included ps or eps files: None AMS classification number: 55P60, 55N10 (Primary); 18G55, 55U35 (Secondary) Abstract: For every ring R, we present a pair of model structures on the category of pro-spaces. In the first, the weak equivalences are detected by cohomology with coefficients in R. In the second, the weak equivalences are detected by cohomology with coefficients in all R-modules (or equivalently by pro-homology with coefficients in R). In the second model structure, fibrant replacement is essentially just the Bousfield-Kan R-tower. When R = Z/p, the first homotopy category is equivalent to a homotopy theory defined by Morel but has some convenient categorical advantages. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Morava/Thomprospectra2 THOM PROSPECTRA FOR LOOP GROUP REPRESENTATIONS NITU KITCHLOO, JACK MORAVA We construct an S^1-equivariant prospectrum that models the Atiyah dual of a free loop space of a manifold. By applying a suitably completed S^1-equivariant K-theory to the Atiyah dual, we show how to recover the Witten genus of a manifold. The main technical tool is a Tits building for the loop group. We use the Tits building to construct a dualizing spectrum for the loop group and relate it to work of Freed, Hopkins and Teleman. ------------------