---------------------------------- There are 7 new papers this time, from Castellana-Crespo-Scherer, Clement-Scherer, Colman, Maltsiniotis, Matthey-Pitsch-Scherer, Nakagawa, and Ruiz. Mark Hovey New papers appearing on hopf between 12/7/06 and 1/1/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Castellana-Crespo-Scherer/covers Title: On the cohomology of highly connected covers of finite complexes Authors: Natalia Castellana, Juan A. Crespo, and Jerome Scherer Abstract: Relying on the computation of the Andre-Quillen homology groups for unstable Hopf algebras, we prove that the mod p cohomology of the n-connected cover of a finite H-space is always finitely generated as algebra over the Steenrod algebra. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Clement-Scherer/exponent Title: Homology exponents for H-spaces Authors: Alain Clement and Jerome Scherer Abstract: We say that a space X admits a homology exponent if there exists an exponent for the torsion subgroup of the integral homology. Our main result states that if an H-space of finite type admits a homology exponent, then either it is, up to 2-completion, a product of spaces of the form BZ/2^r, S^1, K(Z,2), and K(Z,3), or it has infinitely many non-trivial homotopy groups and k-invariants. We then show with the same methods that simply connected H-spaces whose mod 2 cohomology is finitely generated as an algebra over the Steenrod algebra do not have homology exponents, except products of mod 2 finite H-spaces with copies of K(Z,2) and K(Z,3). 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Colman/ColmanHTG Title of Paper: On the homotopy type of Lie groupoids Author: Hellen Colman Address of Author: Department of Mathematics, Wilbur Wright College, 4300 N. Narragansett Avenue, Chicago, IL 60634 USA Text of Abstract: We propose a notion of groupoid homotopy for generalized maps. This notion of groupoid homotopy generalizes the notions of natural transformation and strict homotopy for functors. The groupoid homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the orbifold homotopy between orbifold maps induced by the groupoid homotopy. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Maltsiniotis/adjquill Title: Le theoreme de Quillen, d'adjonction des foncteurs derives, revisite Author: Georges MALTSINIOTIS English Translation: math.AT/0611952 Address: Université Paris 7 Denis Diderot Case Postale 7012 2, place Jussieu F-75251 PARIS CEDEX 05 Abstract: The aim of this paper is to present a very simple original, purely formal, proof of Quillen's adjunction theorem for derived functors, and of some more recent variations and generalizations of this theorem. This is obtained by proving an abstract adjunction theorem for "absolute" derived functors. In contrast with all known proofs, the explicit construction of the derived functors is not used. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Matthey-Pitsch-Scherer/Blochlift Title: Generalized orientations and the Bloch invariant Authors: Michel Matthey, Wolfgang Pitsch, and Jerome Scherer Abstract: For compact hyperbolic 3-manifolds we lift the Bloch invariant defined by Neumann and Yang to an integral class in K_3(C). Applying the Borel and the Bloch regulators, one gets back the volume and the Chern-Simons invariant of the manifold. We also discuss the non-compact case, in which there appears a Z/2-ambiguity. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Nakagawa/cohomologyE8 Title: The integral cohomology ring of E_8/T^1 E_7 Author: Masaki Nakagawa Address of author: Department of General Education, Takamatsu National College of Technology, 355 Chokushi-cho, Takamatsu, 761-8058, Japan Abstract: The generalized flag manifolds are homogeneous spaces of the form G/C, where G is a compact connected Lie group and C is the centralizer of a torus in G. These homogeneous spaces play an important role in algebraic topology, algebraic geometry and differential geometry. In this paper, using the Borel presentation and a method due to Toda, we determine the integral cohomology ring of a certain generalized flag manifold which is a quotient space of the exceptional Lie group E8. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Ruiz/ar-gl-rv Title: Exotic normal fusion subsystems of General Linear Groups. Author: Albert Ruiz Institution: Departament de Matematiques, Universitat Autonoma de Barcelona, 08193 Cerdanyola del Valles, Spain. Abstract: We classify the saturated fusion subsystems of index prime to $p$ of the general linear group over $F_q$ over a Sylow $p$-subgroup, where $q$ is a prime power prime to an odd prime $p$. In this classification we get some of the exotic $p$-local finite groups discovered by C. Broto and J. Moller as saturated fusion subsystems of the general linear group. --------------------------------------------------------------------- There are 4 new papers this time, from Bartels-Lueck-Reich, Davis-Dula-Mahowald, and Vespa (2). In adddition, there are 3 updates of papers recently posted to Hopf; I will just list these rather than including the abstracts again. They are Benson-Chebolu-Christensen-Minac/GH-pgroup-new Chebolu-Christensen-Minac/GH-Stmod Kuhn/primitives Mark Hovey New papers appearing on hopf between 1/1/07 and 2/3/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bartels-Lueck-Reich/blr-hyperbolic Title: The K-theoretic Farrell-Jones Conjecture for hyperbolic groups Authors: Arthur Bartels, Wolfgang Lueck, Holger Reich Abstract: We prove the K-theoretic Farrell-Jones Conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Davis-Dula-Mahowald/imms2 Immersions of RP^{2^e-1} Donald M. Davis, Giora Dula, and Mark Mahowald Abstract: We prove that RP^{2^e-1} cannot be immersed in R^{2^{e+1}-e-8} provided e>6. If e>13, this is 2 better than previously known immersions. Our method is primarily an induction on geometric dimension, incorporating also sections obtained from the Radon-Hurwitz theorem. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Vespa/Fquad Generic representations of orthogonal groups: the functor category Fquad Christine Vespa In this paper, we define the functor category Fquad associated to vector spaces over the field with two elements equipped with a quadratic form. We show the existence of a fully-faithful, exact functor from F to Fquad, which preserves simple objects, where F is the category of functors from the category of finite dimensional vector spaces over the field with two elements to the category of all vector spaces. We define a subcategory Fquad, which is equivalent to the product of the categories of modules over the orthogonal groups; the inclusion is a fully-faithful functor which preserves simple objects. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Vespa/mixtes Generic representations of orthogonal groups: the mixed functors Christine Vespa In previous work, we defined the category of functors Fquad, associated to vector spaces over the field with two elements equipped with a nondegenerate quadratic form. In this paper, we define a special family of objects in the category Fquad, named the mixed functors. We give the complete decompositions of two elements of this family that give rise to two new infinite families of simple objects in the category Fquad. --------------- I am running late this month. There are 3 new papers this time, fron BrownR, Muro, and SmithL. Mark Hovey New papers appearing on hopf between 2/3/07 and 3/19/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/glob-gpds2 AUTHOR: Ronald Brown AUTHOR ADDRESS: School of Computer Science, University of Wales, Dean St., Bangor, Gwynedd, LL57 1UT, UK; TITLE: A new higher homotopy groupoid: the fundamental globular omega-groupoid of a filtered space MSC Classification:18D10, 18G30, 18G50, 20L05, 55N10, 55N25. KEY WORDS: filtered space, higher homotopy van Kampen theorem, cubical singular complex, free globular groupoid xxxLANL archive: math.AT/0702677 2 eps files, 19 pages ABSTRACT: We show that the graded set of filter homotopy classes rel vertices of maps from the n-globe to a filtered space may be given the structure of globular omega--groupoid. The proofs use an analogous fundamental cubical omega--groupoid due to the author and Philip Higgins. This method also relates the construction to the fundamental crossed complex of a filtered space, and this relation allows the proof that the crossed complex associated to the free globular omega-groupoid on one element of dimension n is the fundamental crossed complex of the n-globe. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Muro/tcwm6 Title: A triangulated category without models Author(s): Fernando Muro Author's mailing address: Universitat de Barcelona, Facultat de Matemàtiques, Departament d'Àlgebra i Geometria, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain AMS classification number: 18E30, 55P42, 16E40 Abstract: We exhibit a triangulated category which is neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/SmithL/13a50 (This abstract was written by Mark) Reference list on Invariant Theory Author: Larry Smith AMS Code: 13A50 Invariant Theory Address: Mathematisches Institut Bunsenstrasse 3--5 D 37073 Goettingen Federal republic of germany Abstract: This is a list of references in invariant theory. The .tex file is included so that one can import references into a document. The journals.tex file includes macros for journals. -------------------------------------------------- 6 new papers this month, from Benson, Chebolu-Christensen-Minac, DavisD-Mahowald, Muro-Schwede-Strickland, Oliver-Ventura, and Panin-Pimenov-Roendigs. Mark Hovey New papers appearing on hopf between 3/19/07 and 4/19/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Benson/loops An algebraic model for chains on $\Omega BG\phat$ Dave Benson Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE Abstract: We provide an interpretation of the homology of the loop space on the $p$-completion of the classifying space of a finite group in terms of representation theory, and demonstrate how to compute it. We then give the following reformulation. If $f$ is an idempotent in $kG$ such that $f.kG$ is the projective cover of the trivial module $k$, and $e=1-f$, then we exhibit isomorphisms for $n\ge 2$: H_n(\Omega BG\phat;k) \cong \Tor_{n-1}^{e.kG.e}(kG.e,e.kG) H^n(\Omega BG\phat;k) \cong \Ext^{n-1}_{e.kG.e}(e.kG,e.kG). Further algebraic structure is examined, such as products and coproducts, restriction and Steenrod operations. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu-Christensen-Minac/ghostnumber TITLE: Ghosts in modular representation theory AUTHORS: Sunil K. Chebolu, J. Daniel Christensen, and Jan Minac Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada AMS Subject classsification: Primary 20C20, 20J06; Secondary 55P42 ABSTRACT: A \emph{ghost} over a finite group $G$ is a map between modular representations of $G$ which is invisible in Tate cohomology. Motivated by the failure of the \emph{generating hypothesis}---the statement that ghosts between finite-dimensional $G$-representations factor through a projective---we define the \emph{compact ghost number} of $kG$ to be the smallest integer $l$ such that the composition of any $l$ ghosts between finite-dimensional $G$-representations factors through a projective. In this paper we study ghosts and the compact ghost numbers of $p$-groups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation $k$, holds for all $p$-groups. We do this by proving that a map between finite-dimensional $G$-representations is a ghost if and only if it is a \emph{dual ghost}. We then compute the compact ghost numbers of all cyclic $p$-groups and all abelian $2$-groups with $C_2$ as a summand. We obtain bounds on the compact ghost numbers for abelian $p$-groups and for all $2$-groups which have a cyclic subgroup of index $2$. Using these bounds we determine the finite abelian groups which have compact ghost number at most $2$. %Finally, using universal ghosts, we establish various sets of conditions which %guarantee the existence of a non-trivial ghost out of a $G$-representation. Our methods involve techniques from group theory, representation theory, triangulated category theory, and constructions motivated from homotopy theory. COMMENTS: This version replaces an earlier one with file name ghost.tex. This is a substantial improvement with many new results and major reorganisation of the paper. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD-Mahowald/overlook Nonimmersions of RP^n implied by tmf, revisited Donald M. Davis and Mark Mahowald In a 2002 paper, the authors and Bruner used the new spectrum tmf to obtain some new nonimmersions of real projective spaces. In this note, we complete/correct two oversights in that paper. The first is to note that in that paper a general nonimmersion result was stated which yielded new nonimmersions for RP^n with n as small as 48, and yet it was stated there that the first new result occurred when n=1536. Here we give a simple proof of those overlooked results. Secondly, we fill in a gap in the proof of the 2002 paper. There it was claimed that an axial map f must satisfy f^*(X)=X_1+X_2. We realized recently that this is not clear. However, here we show that it is true up multiplication by a unit in the appropriate ring, and so we retrieve all the nonimmersion results claimed in the original paper. Finally, we present a complete determination of tmf^{8*}(RP^\infty\times RP^\infty) and tmf^*(CP^\infty\times CP^\infty) in positive dimensions. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Muro-Schwede-Strickland/tcwm17 Author(s): Fernando Muro, Stefan Schwede, Neil Strickland Abstract: We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Oliver-Ventura/ov2 Saturated fusion systems over $2$-groups Bob Oliver & Joana Ventura AMS classification: Primary 20D20. Secondary 20D45, 20D08 Abstract: We develop methods for listing, for a given 2-group $S$, all nonconstrained centerfree saturated fusion systems over $S$. These are the saturated fusion systems which could, potentially, include minimal examples of exotic fusion systems: fusion systems not arising from any finite group. To test our methods, we carry out this program over four concrete examples: two of order $2^7$ and two of order $2^{10}$. Our long term goal is to make a wider, more systematic search for exotic fusion systems over 2-groups of small order. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Panin-Pimenov-Roendigs/BGL-post Author: Ivan Panin Author2: Konstantin Pimenov Author3: Oliver Roendigs Title: On Voevodsky's algebraic K-theory spectrum BGL Under a certain normalization assumption we prove that the Voevodsky's spectrum BGL which represents algebraic $K$-theory is unique over the integers. Following an idea of Voevodsky, we equip the spectrum BGL with the structure of a commutative ring spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over the integers We pull this structure back to get a distinguished monoidal structure on BGL for an arbitrary Noetherian base scheme. ------------------------------------------------------------ 4 new papers this month, from Barge-Lannes, Biedermann, Bubenik, and Devinatz. Mark Hovey New papers appearing on hopf between 4/19/07 and 5/14/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Barge-Lannes/SMB Title: Suites de Sturm, indice de Maslov et p\'e riodicit\'e de Bott Authors: Jean Barge and Jean Lannes Abstract: This memoir presents a reworking of a very classical subject; it is related to works of many people, especially: Richard W. Sharpe, Max Karoubi, Andrew Ranicki, Fran\c{c}ois Latour... We explain in particular how the usual theory of Sturm sequences is linked to the fundamental theorem of hermitian K-theory (due to Karoubi) and to Bott periodicity. Keywords: Sturm sequences, Maslov index, Bott periodicity, hermitian K-theory. AMS classification: 19G38, 19C99. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Biedermann/L-stable L-stable functors by Georg Biedermann We generalize and greatly simplify the approach of Lydakis and Dundas-R\"ondigs-{\O}stv{\ae}r to construct an L-stable model structure for small functors from a closed symmetric monoidal model category V to a V-model category M, where L is a small cofibrant object of V. For the special case V=M=S_* pointed simplicial sets and L=S^1 this is the classical case of linear functors and has been described as the first stage of the Goodwillie tower of a homotopy functor. We show, that our various model structures are compatible with a closed symmetric monoidal product on small functors. We compare them with other L-stabilizations described by Hovey, Jardine and others. This gives a particularly easy construction of the classical and the motivic stable homotopy category with the correct smash product. We establish the monoid axiom under certain conditions. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Bubenik/sep Author: Peter Bubenik Title: Separated Lie models and the homotopy Lie algebra AMS classification number: Primary 55P62; Secondary 17B55 to appear in the Journal of Pure and Applied Algebra Abstract: The homotopy Lie algebra of a simply connected topological space, X, is given by the rational homotopy groups on the loop space of X. Following Quillen, there is a connected differential graded free Lie algebra (dgL) called a Lie model, which determines the rational homotopy type of X, and whose homology is isomorphic to the homotopy Lie algebra. We show that such a Lie model can be replaced with one that has a special property we call separated. The homology of a separated dgL has a particular form which lends itself to calculations. We give connections to the radical of the homotopy Lie algebra and the Avramov-Felix conjecture. Examples that are worked out in detail include wedges of spheres on any "thickness" and connected sums of products of spheres. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Devinatz/towardsfiniteness Title: Towards the finiteness of the homotopy groups of the K(n)-localization of S^0. Author: Ethan S. Devinatz Abstract: Let G be a closed subgroup of the nth Morava stabilizer group S_n, n>1, and let E_n^{hG} denote the continuous homotopy fixed point spectrum of Devinatz and Hopkins. If G=, the subgroup topologically generated by an element z in the p-Sylow subgroup S_n^0 of S_n, and z is non-torsion in the quotient of S_n^0 by its center, we prove that the E_n^{h}-homology of any K(n-2)-acyclic finite spectrum annihilated by p is of essentially finite rank. (The definition of essentially finite rank is given in the paper.) We also show that the units in the coefficient ring of E_n which are fixed by z are just the units in the Witt vectors with coefficients in the field of p^n elements. If n=2 and p>3, we show that, if G is a closed subgroup of S_n^0 not contained in the center, then G contains an open subnormal subgroup U such that the mod(p) homotopy of E_n^{hV} is of essentially finite rank, where V is the product of U with the units in the field of p elements. --------------------------------------------------------------------------------------- 4 new papers this month, from Arone-Dwyer-Lesh, Bendersky-DavisD, Karoubi, and Wuethrich. Mark Hovey New papers appearing on hopf between 5/14/07 and 6/8/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arone-Dwyer-Lesh/LoopStructuresTaylorTowers Title Loop structures in Taylor towers Authors G. Z. Arone, W. G. Dwyer, K. Lesh Kerchof Hall, U. of Virginia, P.O. Box 400137, Charlottesville VA 22904 USA Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 Department of Mathematics, Union College, Schenectady, NY 12308 Abstract We study spaces of natural transformations between homogeneous functors in Goodwillie's calculus of homotopy functors and in Weiss's orthogonal calculus. We give a description of such spaces of natural transformations in terms of the homotopy fixed point construction. Our main application is a delooping theorem for connecting maps in the Goodwillie tower of the identity and in the Weiss tower of BU(V). The interest in such deloopings stems from conjectures made by the first and the third author in a 2007 paper that these towers provide a source of contracting homotopies for certain projective chain complexes of spectra. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bendersky-DavisD/DW2 v1-periodic homotopy groups of the Dwyer-Wilkerson space Martin Bendersky Donald M. Davis Abstract: The Dwyer-Wilkerson space DI(4) is the only exotic 2-compact group. We compute its v1-periodic homotopy groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Karoubi/Karoubi Cochaines quasi-commutatives en Topologie Algebrique Max Karoubi Abstract : We describe a new category of "quasi-commutative" DGA's , called D*, where the product is "almost" commutative : it is commutative on a subcomplex of C = D* tensor D* (with some axioms). To each simplicial set (or even ringed space) we associate a quasi-commutative DGA, from which we recover the homotopy type and are able to describe an explicit procedure to "compute" homotopy groups and cohomology operations. The basic idea of the construction is to use difference calculus, instead of differential calculus as in Sullivan's theory. This paper is an extension of ideas posted in the Archives a few years ago under the title "Methodes quantiques en Topologie Algebrique". However, the point of view is simpler and the proofs are now complete. It is going to appear in the Quarterly Journal of Pure and Applied Math. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Wuethrich/thickenings_final Title: Infinitesimal thickenings of Morava K-theories (final version) Author: Samuel Wuethrich AMS classification number: 55P42, 55P43; 55U20, 55N22 arXive submission number: math.AT/0607110 Comments: 25 pages. Final version, to appear in J. Pure Appl. Algebra. Contents of former section 5 mostly rewritten and reorganized into two sections; some minor corrections and changes Abstract: A. Baker has constructed certain sequences of cohomology theories which interpolate between the Johnson-Wilson and the Morava K-theories. We realize the representing sequences of spectra as sequences of MU-algebras. Starting with the fact that the spectra representing the Johnson-Wilson and the Morava K-theories admit such structures, we construct the sequences by inductively forming singular extensions. Our methods apply to other pairs of MU-algebras as well. ------------------------------------------------------------- 5 new papers this month, from Bisson-Tsemo, ChornyB, and Neusel(3). Mark Hovey New papers appearing on hopf between 6/8/07 and 8/13/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bisson-Tsemo/AlgGeomOps Title: Extended powers and Steenrod operations in algebraic geometry (Preliminary Draft, July 2007 version) Authors: Terrence Bisson and Aristide Tsemo Abstract: Steenrod operations have been defined by Voedvodsky in motivic cohomology in order to show the Milnor and Bloch-Kato conjectures. These operations have also been constructed by Brosnan for Chow rings. The purpose of this paper is to provide a setting for the construction of the Steenrod operations in algebraic geometry, for generalized cohomology theories whose formal group law has order two. We adapt the methods used by Bisson-Joyal in studying Steenrod and Dyer-Lashof operations in unoriented cobordism and mod 2 cohomology. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/ChornyB/BrownRep Title: Brown representability for space-valued functors Author(s): Boris Chorny Abstract: 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 weakly 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. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/hanoi Inseparable Extensions of Algebras over the Steenrod Algebra with Applications to Modular Invariant Theory of Finite Groups II author: Mara D. Neusel abstract: We continue our study of the homological properties of the purely inseparable extensions of integrally closed unstable Noetherian integral domains over the Steenrod algebra. It turns out that the projective dimension of an algebra is a lower bound for the projective dimension of its inseparable closure. Furthermore, its depth is an upper bound for the depth of its inseparable closure. Moreover, both algebras have the same global dimension. We apply these results to invariant theory. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/schmid Degree bounds and the regular representation author: Mara D. Neusel abstract: Let rho : G --> GL(n , F) be a faithful representation of a finite group G. Denote by beta(F[V]^G) the maximal degree of an F-algebra generator of the ring of polynomial invariants F[V]^G in a minimal generating set. We prove the old conjecture that in the nonmodular case beta(F[V]^G)<= beta(F[FG]^G), where FG is the regular representation. Along the way we show that rings of permutation invariants that are Cohen-Macaulay always satisfy Noether's bound. Furthermore, we show that rings of invariants of sums of permutation representations that are Cohen-Macaulay are generated by polarizations. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/unstable The Unstable Parts Functor and Injective Objects author: Mara D. Neusel abstract: The unstable part functor Un assigns to an arbitrary module over the Steenrod algebra the largest unstable submodule. We start by showing some general properties of this functor. Then we study the functor Un S^{-1} obtained from Un by precomposition with a localization. We show that Un S^{-1} is an exact functor from the category of unstable noetherian modules over some unstable noetherian algebra to itself. Along the lines we describe the injective objects in this category. ---------------- ----------------------------------------------- 7 new papers this month, from Ausoni-Rognes, DavisD, Gonzalez-Wilson, Kitchloo-Wilson(2), Neusel, and Neusel-Sezer. Mark Hovey New papers appearing on hopf between 8/13/07 and 9/24/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Ausoni-Rognes/ausoni-rognes-kku Title: Rational algebraic K-theory of topological K-theory. Authors: Christian Ausoni and John Rognes. MSC-class: 19D55; 55N99 arXiv:0708.2160v1 [math.KT] Christian Ausoni Mathematical Institute University of Bonn John Rognes Department of Mathematics University of Oslo Abstract: We show that after rationalization there is a homotopy fiber sequence BBU -> K(ku) -> K(Z). We interpret this as a correspondence between the virtual 2-vector bundles over a space X and their associated anomaly bundles over the free loop space LX. We also rationally compute K(KU) by using the localization sequence, and K(MU) by a method that applies to all connective S-algebras. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD/pcpt Homotopy type and v1-periodic homotopy groups of p-compact groups Donald M. Davis Lehigh University, Bethlehem, PA 18015 Abstract: We determine the v1-periodic homotopy groups of all irreducible p-compact groups. In the most difficult, modular, cases, we follow a direct path from their associated invariant polynomials to these homotopy groups. We show that, with several exceptions, every irreducible p-compact group is a product of explicit spherically-resolved spaces which occur also as factors of p-completed Lie groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Gonzalez-Wilson/products The ${BP}$-theory of two-fold products of projective spaces Jes\'us Gonz\'alez Departamento de Matem\'aticas Centro de Investigaci\'on y de Estudios Avanzados del IPN W. Stephen Wilson Department of Mathematics Johns Hopkins University We compute the BP (co)homology of the product of two (stunted) projective spaces. The behavior under maps (particularly of the Tor term) is studied. This is used extensively by Kitchloo and Wilson in their work on non-immersions. Additional work with lens spaces is also included. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Wilson/nonimmersions1 The second real Johnson-Wilson theory and non-immersions of $RP^n$ Nitu Kitchloo Department of Mathematics University of California, San Diego (UCSD) W. Stephen Wilson Department of Mathematics Johns Hopkins University Hu and Kriz construct the real Johnson-Wilson spectrum, $ER(n)$, which is $2^{n+2}(2^n-1)$ periodic, from the $2(2^n-1)$ periodic spectrum $E(n)$. $ER(1)$ is just $KO_{(2)}$ and $E(1)$ is just $KU_{(2)}$. We compute $ER(n)^*(RP^\infty)$ and set up a Bockstein spectral sequence to compute $ER(n)^*(-)$ from $E(n)^*(-)$. We combine these to compute $ER(2)^*(RP^{2n})$ and use this to get new non-immersions for real projective spaces. Our lowest dimensional new example is an improvement of 2 for $RP^{48}$. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Wilson/nonimmersions2 The second real Johnson-Wilson theory and non-immersions of $RP^n$, Part II. Nitu Kitchloo Department of Mathematics University of California, San Diego (UCSD) W. Stephen Wilson Department of Mathematics Johns Hopkins University This paper is a continuation of the study begun in the paper with the same name. We analyze $ER(2)^{16*+8}(RP^{2n})$ and compute $ER(2)^*(RP^{16K+1})$ and use these to prove more non-immersion theorems for $RP^n$ including many in fairly low dimensions. In particular, we get 12 new non-immersion results for $RP^n$ where $n \le 192$, the range included in Don Davis's tables. These complement the 10 already found in part I. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/schmidappendix title: Degree Bounds and the Regular Representation: Appendix author: Mara D. Neusel subjclass[2000]: Primary 13A50 keywords: Invariant Theory of Finite Permutation Groups, Permutation Representation, Cohen-Macaulay, Gorenstein, Complete Intersection, Hypersurface, Polynomial Algebra, Pseudo-Reflection abstract: Let G be a matrix group consisting of permutation matrices. Let F and K be two different fields. We show that if the polynomial invariants F[V]^G and K[V]^G are both Cohen-Macaulay, then they are simultaneously Gorenstein, complete intersections, hypersurfaces, resp. polynomial. Thus Cohen-Macaulay rings of permutation invariants are polynomial exactly when G is generated by pseudo-reflections. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Sezer/psquare title: the invariants of modular indecomposable representations of Z_{p^2} authors: Mara D. Neusel and M\"ufit Sezer abstract: We consider the invariant ring for an indecomposable representation of a cyclic group of order p^2 over a field F of characteristic p. We describe a set of F-algebra generators of this ring of invariants, and thus derive an upper bound for the largest degree of an element in a minimal generating set for the ring of invariants. This bound, as a polynomial in p, is of degree two. -------------- ------------------------------------- 6 new papers this time, from Chebolu-Christensen-Minac, Elmendorf-Mandell, Flores-Foote, Gray-Theriault, Kahn-Maltsiniotis, and Stacey-Whitehouse. Mark Hovey New papers appearing on hopf between 9/24/07 and 11/29/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu-Christensen-Minac/GH-periodic TITLE: Freyd's generating hypothesis for groups with periodic cohomology. AUTHORS: Sunil K. Chebolu, J. Daniel Christensen, and Jan Minac Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada ABSTRACT: Let $G$ be a finite group and let $k$ be a field whose characteristic $p$ divides the order of $G$. Freyd's generating hypothesis for the stable module category of $G$ is the statement that a map between finite-dimensional $kG$-modules in the thick subcategory generated by $k$ factors through a projective if the induced map on Tate cohomology is trivial. We show that if $G$ has periodic cohomology then the generating hypothesis holds if and only if the Sylow $p$-subgroup of $G$ is $C_2$ or $C_3$. We also give some other conditions that are equivalent to the GH for groups with periodic cohomology. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Elmendorf-Mandell/EM2v5 A. D. Elmendorf and M. A. Mandell Multiplicative structure in infinite loop space theory We extend the K-theory functor constructed in our previous paper (Rings, modules, and algebras in infinite loop space theory, Advances in Mathematics 205 (2006) 163-228) to the bicomplete symmetric monoidal closed category of based (symmetric) multicategories, to which our previous source category of permutative categories and lax morphisms maps fully and faithfully. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Flores-Foote/Flores-Foote Title: Strongly closed subgroups and the cellular structure of classifying spaces Authors: Ram\'on J. Flores and Richard M. Foote Abstract: In this paper we give a complete classification of the finite groups that contain a strongly closed $p$-subgroup, generalizing previous work of the second author to the case of an odd prime. We use this result to also obtain a description of the BZ/p-cellularization (in the sense of Dror-Farjoun) of all the classifying spaces of finite groups. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Gray-Theriault/Gray-Theriault An elementary construction of Anick's fibration Brayton Gray Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago Chicago, IL 60607-7045 Stephen Theriault Department of Mathematical Sciences University of Aberdeen Aberdeen, AB24 3UE, United Kingdom Cohen, Moore, and Neisendorfer's work on the odd primary homotopy theory of spheres and Moore spaces, as well as the first author's work on the secondary suspension, predicted the existence of a p-local fibration S^2n-1 --> T --> \Omega S^2n+1 whose connecting map is degree p^r. In a long and complex monograph, Anick constructed such a fibration for p>=5 and r>=1. Using new methods we give a much more conceptual construction which is also valid for p=3 and r>=1. We go on to establish several properties of the space T. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Kahn-Maltsiniotis/StrDer Structures de dérivabilité Bruno KAHN & Georges MALTSINIOTIS Institut de Mathématiques de Jussieu We introduce a very general framework in which Quillen's theorems of existence, composition and adjunction for derived functors can be proved. We thus generalize and unify previous results by Dwyer, Hirschhorn, Kan and Smith, obtained in their formalism of "homotopical categories", and by Radulescu-Banu in the context of Cisinski's "derivable categories". 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Stacey-Whitehouse/hopf Title: The Hunting of the Hopf Ring Authors: Andrew Stacey and Sarah Whitehouse Addresses of authors: Andrew Stacey Institutt for matematiske fag NTNU 7491 Trondheim Norway Sarah Whitehouse Department of Pure Mathematics University of Sheffield Sheffield S3 7RH UK Abstract: We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, E^*. Our description is as a graded and completed version of a Tall-Wraith monoid. The E^*-cohomology of a space X is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the Tall-Wraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another. -----------------