------------------------------------------- I seem to have forgot to send this out in April. My apologies. There are 6 new papers this month, by Arone-Lesh, Bergner, DavisD-Potocka, Lawson, Lueck, and Strohm. Mark Hovey New papers appearing on hopf between 3/5/05 and 5/7/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arone-Lesh/arone-lesh-filtered-spectra Title: Filtered spectra arising from permutative categories Authors: Gregory Arone University of Virginia Kathryn Lesh Union College Abstract: Given a special Gamma-category C satisfying some mild hypotheses, we construct a sequence of spectra interpolating between the spectrum associated to C and the Eilenberg-Mac Lane spectrum HZ. Examples of categories to which our construction applies are: the category of finite sets, the category of finite-dimensional vector spaces, and the category of finitely-generated free modules over a reasonable ring. In the case of finite sets, our construction recovers the filtration of HZ by symmetric powers of the sphere spectrum. In the case of finite-dimensional complex vector spaces, we obtain an apparently new sequence of spectra, A_{m}, that interpolate between bu and HZ. We think of A_{m} as a ``bu-analogue'' of the m'th symmetric power and describe far-reaching formal similarities between the two sequences of spectra. For instance, in both cases the m'th subquotient is contractible unless m is a power of a prime, and in v_{k}-periodic homotopy the filtration has only k+2 nontrivial terms. There is an intriguing relationship between the bu-analogues of symmetric powers and Weiss's orthogonal calculus, parallel to the not yet completely understood relationship between the symmetric powers of spheres and the Goodwillie calculus of homotopy functors. We conjecture that the sequence {A_{m}}, when rewritten in a suitable chain complex form, gives rise to a minimal projective resolution of the connected cover of $bu$. This conjecture is the bu-analogue of a theorem of Kuhn and Priddy about the symmetric power filtration. The calculus of functors provides substantial supporting evidence for the conjecture. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bergner/ThreeModels Title: Three models for the homotopy theory of homotopy theories Author: Julia E. Bergner AMS classification number: Primary: 55U35; Secondary 18G30, 18E35 arXiv submission number: math.AT/0504334 Abstract: Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the ``homotopy theory" of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a ``homotopy theory of homotopy theories." In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with their respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk's complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD-Potocka/sun2long 2-primary v1-periodic homotopy groups of SU(n) revisited Donald M. Davis, Lehigh University, Bethlehem, PA 18015, Katarzyna Potocka, Ramapo College of New Jersey, Mahwah, NJ 07430 Abstract In 1991, Bendersky and Davis used the BP-based unstable Novikov spectral sequence to study the 2-primary v1-periodic homotopy groups of SU(n). Here we use a K-theoretic approach to add more detail to those results. In particular, whereas only the order of the groups v1^{-1} pi_{2k-1}(SU(n)) was determined in the 1991 paper, here we determine the number of summands in these groups and much information about the orders of those summands. In addition, we give explicit conditions for certain differentials and extensions in a spectral sequence, which affect the homotopy groups. Finally, we give complete results for v1^{-1} pi_*(SU(n)) for n < 14. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Lawson/lawson_productformula Title: The product formula in unitary deformation $K$-theory Author: Tyler Lawson MSC classification: 19D23; 19L41; 20C99 PaperID: math.KT/0503468 Address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 Abstract: We prove that "unitary deformation K-theory" takes products of finitely generated groups to coproducts of algebra spectra over ku, the connective K-theory spectrum. Additionally, we give spectral sequences for computing the homotopy groups of the unitary deformation K-theory of a group G and the cofiber of the Bott map in terms of PU(n)-equivariant K-theory and homology of spaces of G-representations. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Lueck/lueck_burnside0504 Title of Paper: The Burnside Ring and Equivariant Cohomotopy for Infinite Groups Author: Wolfgang Lueck AMS Classification numbers: 55P91, 19A22. math.AT/0504051 Fachbereich Mathematik Universitaet Muenster Einsteinstr. 62 48149 Muenster Germany Abstract: After we have given a survey on the Burnside ring of a finite group, we discuss and analyze various extensions of this notion to infinite (discrete) groups. The first three are the finite-$G$-set-version, the inverse-limit-version and the covariant Burnside group. The most sophisticated one is the fourth definition as the equivariant zero-th cohomotopy of the classifying space for proper actions. In order to make sense of this definition we define equivariant cohomotopy groups of finite proper equivariant CW-complexes in terms of maps between the sphere bundles associated to equivariant vector bundles. We show that this yields an equivariant cohomology theory with a multiplicative structure. We formulate a version of the Segal Conjecture for infinite groups. All this is analogous and related to the question what are the possible extensions of the notion of the representation ring of a finite group to an infinite group. Here possible candidates are projective class groups, Swan groups and the equivariant topological K-theory of the classifying space for proper actions. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Strohm/diploma_main Title: The Proportionality Principle of Simplicial Volume Authors: Clara Strohm (=Clara Löh) Address: Einsteinstr. 62, 48143 Münster, Germany MSC: 57R19, 55N35 Abstract: The simplicial volume is a homotopy invariant of oriented closed connected manifolds measuring the efficiency of representing the fundamental class by singular chains with real coefficients. Despite of its topological nature, the simplicial volume is linked to Riemannian geometry in various ways, e.g., by the proportionality principle. The proportionality principle of simplicial volume states that the simplicial volume and the Riemannian volume are proportional for oriented closed connected Riemannian manifolds sharing the same universal Riemannian covering. Thurston indicated a proof of the proportionality principle using his (smooth) measure homology. It is the purpose of this diploma thesis to provide a full proof of the proportionality principle based on Thurston's approach. In particular, it is shown that (smooth) measure homology and singular homology are isometrically isomorphic for all smooth manifolds. This implies that the simplicial volume indeed can be computed in terms of measure homology. Included eps files: fg.eps, dragon_schoon.eps ----------------