--------------------- There are 7 new papers this time, from Bendersky-DavisD, Elmendorf-Mandell, Goerss-Hopkins, Murillo-Buijs (2), Rezk, and Vistoli. Mark Hovey New papers appearing on hopf between 5/7/05 and 7/1/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bendersky-DavisD/sgdp2 Stable geometric dimension of vector bundles over odd-dimensional real projective spaces Martin Bendersky, Hunter College, CUNY 10021, Donald M. Davis, Lehigh University, Bethlehem, Pa. 18015 55S40, 55R50, 55T15 Abstract: In a recent paper, the geometric dimension of all stable vector bundles over real projective space P^n was determined if n is even and sufficiently large with respect to the order 2^e of the bundle. Here we perform a similar determination when n is odd and e>6. The work is more delicate since P^n does not admit a v1-map when n is odd. There are a few extreme cases which we are unable to settle precisely. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Elmendorf-Mandell/RMA2 Rings, modules, and algebras in infinite loop space theory A. D. Elmendorf and M. A. Mandell Subject classes: Primary 19D23; Secondary 55P43, 18D10 xxx-LANL identifier: math.KT/0403403 Addresses: A. D. Elmendorf Dept. of Mathematics Purdue University Calumet Hammond, IN 46323 M. A. Mandell (current) DPMMS CMS University of Cambridge Cambridge CB3 0WB England M. A. Mandell (effective Fall 2005) Department of Mathematics Indiana University Bloomington, IN 47405 This is a major revision of a previous submission of the same name. We have completely rewritten sections 5 -- 7, giving a new construction of the first part of our functor. The main abstract is as follows: We give a new construction of the algebraic $K$-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and output. This requires us to define multiplicative structure on the category of small permutative categories. The framework we use is the concept of multicategory (elsewhere also called colored operad), a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction. Our method ends up in the Hovey-Shipley-Smith category of symmetric spectra, with an intermediate stop at a category of functors out of a particular wreath product. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Goerss-Hopkins/obstruct Title: Moduli spaces for Structured Ring Spectra Authors: P.G. Goerss and M.J. Hopkins Abstract: In this document we make good on all the assertions we made in the previous paper ``Moduli spaces of commutative ring spectra'' wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of commutative ring spectra. In particular, we develop a theory of moduli spaces of algebra structures on spectra, and give a decomposition of the moduli space as a tower of fibrations wherein the successive fibers can be calculated using Andre'-Quillen cohomology. By examining the obstructions to lifting a basepoint up the tower, we then produce successively defined obstructions to the realizing an algebra structure. A point worth emphasizing is that the moduli problems here begin with algebra: for example, we may have a homology theory E and a commutative ring A in the category comodules associated to E and we wish to discuss the homotopy type of the space of all commutative (in the strict sense) ring spectra X so that the E-homology of X is A as a commutative ring. We do not, a priori, assume that this moduli space is non-empty, or even that there is a spectrum whose E-homology is A. For a variety of applications we are not simply interested in this absolute problem, but in a relative version as well. Fortunately, Andre'-Quillen cohomology is inherently relative and the theory adapts well to this case. The main idea, which goes back to Dwyer, Kan, and Stover, is to try to construct a simplicial ring spectrum, whose geometric realization will realize A. Then we use the new simplicial direction and apply Postnikov tower techniques to get the decomposition of the moduli space. Making this work requires a certain amount of technical detail. In particular, we need to be very careful with resolution model categories and their localizations at a homology theory. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Murillo-Buijs/mapping Title of Paper: Basic constructions in rational homotopy theory of function spaces Author(s) Aniceto Murillo and Urtzi Buijs AMS Classification numbers 55P62 Addresses of Authors Departamento de Algebra, Geometria y Topologia Universidad de Malaga, AP. 59, 29080 Malaga SPAIN Text of Abstract Via the Bousfield-Gugenheim realization functor, and starting from the Brown-Szczarba approach to the Haefliger model of a function space, we give a functorial framework to describe basic objects and maps concerning the rational homotopy type of function spaces and its path components. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Murillo-Buijs/lie_algebra Title of Paper: The rational homotopy Lie algebra of function spaces Author(s) Aniceto Murillo and Urtzi Buijs AMS Classification numbers 55P62 Addresses of Authors Departamento de Algebra, Geometria y Topologia Universidad de Malaga, AP. 59, 29080 Malaga SPAIN Text of Abstract We give a full and explicit description of the rational homotopy Lie algebra of function spaces (free or pointed) 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Rezk/rezk-units-and-logs Title: The units of a ring spectrum and a logarithmic cohomology operation Author: Charles Rezk Abstract: We construct a ``logarithmic'' cohomology operation on Morava E-theory, which is a homomorphism defined on the multiplicative group of invertible elements in the ring E^0(K) of a space K. We obtain a formula for this map in terms of the action of Hecke operators on Morava E-theory. Our formula is closely related to that for an Euler factor of the Hecke L-function of an automorphic form. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Vistoli/PGL_p On the cohomology and the Chow ring of the classifying space of PGL_p Angelo Vistoli Dipartimento di Matematica Universitŕ di Bologna Piazza di Porta San Donato 5 40014 Bologna Italy arXive submission number: math.AG/0505052 Abstract: We investigate the integral cohomology ring and the Chow ring of the classifying space of the complex projective linear group PGL_p, when p is an odd prime. In particular, we determine their additive structures completely, and we reduce the problem of determining their multiplicative structures to a problem in invariant theory. ----------------