------------------------------ 4 new papers this month, from Bendersky-Churchill, Hovey, Naumann, and Zivaljevic. Mark Hovey New papers appearing on hopf between 12/14/04 and 1/10/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bendersky-Churchill/NormalForms Title: A spectral sequence approach to normal forms. Authors: Martin Bendersky & Richard C. Churchill Address: CUNY/Hunter College, Graduate Center New York, NY 10021 AMS Classification: 55T05, 34C20 Abstract: The theory of normal forms has been around since Poincare's time. An incomplete list of applications are to vector fields, Hamiltonians at equilibria, differential equations and singularity theory. In general one tries to modify a given element in a Lie algebra into a particularly useful form. The algorithm that performs the conversion (the normal form algorithm) can be a formidable computation. In this paper we generalize the notion of normal form to that of an initially linear group representation. In this general setting we are able to interpret the normal form algorithm as a calculation of a particularly simple spectral sequence. As a consequence we show that various vector spaces that appear in the process of carrying out the normal form algorithm are invariants of the orbit of the group representation. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/prod-spec-seq The generalized homology of products Mark Hovey Wesleyan University We construct a spectral sequence that computes the E-homology of a product of spectra. The E_{2}-term of this spectral sequence consists of the right derived functors of product in the category of E_{*}E-comodules, and the spectral sequence always converges (with a horizontal vanishing line at E_{infty}) when E is the Johnson-Wilson theory E(n) and each factor of the product is L_{n}-local. We are able to prove some results about the E_{2}-term of this spectral sequence; in particular, we show that the E(n)-homology of a product of E(n)-module spectra X^{\alpha} is just the comodule product of the E(n)_{*}X^{\alpha}. This spectral sequence is relevant to the chromatic splitting conjecture. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Naumann/comodlandweber Comodule categories and the geometry of the stack of formal groups N. Naumann We generalise recent results of M. Hovey and N. Strickland on comodule categories for Landweber exact algebras using the formalism of algebraic stacks. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Zivaljevic/synergia Title: Equipartitions of measures in R^4 Author: Rade Zivaljevic AMS Class.: 52A39; 52C35; 55S40; 57R22; 57R91; 68P30 arXiv:math.CO/0412483 v1 December 2004 Address: Mathematical Institute SANU, Knez Mihailova 35/1, p.o. box 367 11001 Belgrade Serbia and Montenegro A measure in R^4 admits an equipartition by 4 hyperplanes, provided it is symmetric with respect to a 2-dimensional, affine subspace L of R^4. The computation is based on the Koschorke's exact singularity sequence for groups of normal bordisms and the remarkable properties of the essentially unique, balanced binary Gray code in dimension 4. --------------- -------------------------- 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. ------------- --------------------------------- 13 new papers this month, by Angeltveit-Rognes, Arkowitz-Oshima-Strom, Arkowitz-Stanley-Strom, Barker-Snaith, Broto-Castellana-Grodal-Levi-Oliver, Broto-Levi-Oliver, Iwase-Stanley-Strom, Jardine, Levi-Oliver, Lupton-SmithSB, Nendorf-Scoville-Strom, and Rognes (2). Mark Hovey New papers appearing on hopf between 2/5/05 and 3/5/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Angeltveit-Rognes/vigleik Title: Hopf algebra structure on topological Hochschild homology Author(s): Vigleik Angeltveit and John Rognes Author's e-mail address: Abstract: The topological Hochschild homology THH(R) of a commutative S-algebra (E-infinity ring spectrum) R naturally has the structure of a Hopf algebra over R, in the homotopy category. We show that under a flatness assumption this makes the Bokstedt spectral sequence converging to the mod p homology of THH(R) into a Hopf algebra spectral sequence. We then apply this additional structure to study some interesting examples, including the commutative S-algebras ku, ko, tmf, ju and j, and to calculate the homotopy groups of THH(ku) and THH(ko) after smashing with suitable finite complexes. This is part of a program to make systematic computations of the algebraic K-theory of S-algebras, using topological cyclic homology. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Oshima-Strom/Equiv Homotopy classes of self-maps and induced homomorphisms of homotopy groups Martin Arkowitz Hideaki Oshima Jeffrey Strom For a based space X, we consider the group of all self homotopy classes of $X$ such that which induce the identity on homotopy groups in dimensions 1 through n, and the group of all homotopy classes which loop to the identity. Analogously, we study the semigroups defined by replacing `identity' by `0' above. There is a chain of containments of these groups and semigroups, and we discuss examples for which the containment is proper. We then obtain various conditions on X which ensure that these groups are equal, or when the semigroups are equal. When X is a group-like space, we derive lower bounds for the order of these groups and their localizations. In the last section we make specific calculations for certain low dimensional Lie groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Stanley-Strom/Length1 The A-category and A-cone length of a map Martin Arkowitz Donald Stanley Jeffrey Strom For any collection A of spaces we define two numerical invariants of maps: A-category of f and the A-cone length of f. These invariants are defined axiomatically, and our first results give equivalent constructive definitions in terms of mapping cone decompositions. We show that if A is the collection of all spaces, then the A-category of f is the category of f as defined by Fadell and Husseini and the A-category of f is the cone length of f as defined by Marcum. By specializing to the unique maps from and to a one-point space, we obtain four invariants of spaces. Each of these four invariants has its own axiomatic and constructive definitions. We compare them similar invariants defined by Scheerer and Tanr\'e. We conclude by giving lower bounds for these invariants in terms of cohomology. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Barker-Snaith/psi3triangle3 \psi^3 as an upper triangular matrix Jonathan Barker and Victor Snaith 55S25 (Primary) 55P42 (Secondary) math.AT/0502472 Jonathan Barker Building 54 (School of Mathematics) University of Southampton Highfield Southampton SO17 1BJ UK Victor Snaith Department of Pure Mathematics University of Sheffield Hicks Building Hounsfield Road Sheffield S3 7RH UK In the 2-local stable homotopy category the group of left-bu-module automorphisms of bu\wedge bo which induce the identity on mod 2 homology is isomorphic to the group of infinite upper triangular matrices with entries in the 2-adic integers. We identify the conjugacy class of the matrix corresponding to 1\wedge\psi^3, where \psi^3 is the Adams operation. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Castellana-Grodal-Levi-Oliver/bcglo2 EXTENSIONS OF p-LOCAL FINITE GROUPS C. Broto, N. Castellana, J. Grodal, R. Levi, and B. Oliver A $p$-local finite group consists of a finite $p$-group $S$, together with a pair of categories which encode ``conjugacy'' relations among subgroups of $S$, and which are modelled on the fusion in a Sylow $p$-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as $p$-completed classifying spaces of finite groups. In this paper, we study and classify extensions of $p$-local finite groups, and also compute the fundamental group of the classifying space of a $p$-local finite group. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Levi-Oliver/blo4 A GEOMETRIC CONSTRUCTION OF SATURATED FUSION SYSTEMS Carles Broto, Ran Levi, and Bob Oliver A saturated fusion system consists of a finite $p$-group $S$, together with a category which encodes ``conjugacy'' relations among subgroups of $S$, and which satisfies certain axioms which are motivated by properties of the fusion in a Sylow $p$-subgroup of a finite group. We describe here new ways of constructing abstract saturated fusion systems, first as fusion systems of spaces with certain properties, and then via certain graphs. Subject class: Primary 55R35. Secondary 55R40, 20D20 Keywords: classifying space, $p$-completion, finite groups, fusion. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Iwase-Stanley-Strom/GaneaCond Implications of the Ganea Condition Norio Iwase Donald Stanley Jeffrey Strom Suppose the spaces X and X x A have the same Lusternik-Schnirelmann category: cat(X x A) = cat(X). Then there is a strict inequality cat(X x (A \halfsmash B)) < cat (X) + cat(A \halfsmash B) for every space B, provided the connectivity of A is large enough (depending only on X). This is applied to give a partial verification of a conjecture of Iwase on the category of products of spaces with spheres. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/diagrams Author: J.F. Jardine Author's mailing address: Department of Mathematics University of Western Ontario London, ON N6A 5B7 Canada Suppose that A is a small presheaf of categories enriched in simplicial sets on a small Grothendieck site. It is shown that the homotopy theory of enriched A-diagrams taking values in simplicial sets can be identified with the homotopy theory of simplicial presheaves fibred over the diagonalized nerve dBA of A. One can also identify the set [*,dBA] of morphisms in the simplicial presheaf homotopy category with path components of the category of A-torsors, suitably defined. These statements are special cases of localized results which hold when the corresponding localized model structures are proper. Examples of the latter include the motivic homotopy category of Morel and Voevodsky, and so these results lead to a theory of motivic A-torsors which is classifiable up to equivalence by a family of morphisms in the motivic homotopy category. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Levi-Oliver/sol-corr Correction to: CONSTRUCTION OF 2-LOCAL FINITE GROUPS OF A TYPE STUDIED BY SOLOMON AND BENSON by Ran Levi and Bob Oliver A $p$-local finite group is an algebraic structure with a classifying space which has many of the properties of $p$-completed classifying spaces of finite groups. In our paper \cite{Sol}, we constructed a family of 2-local finite groups which are ``exotic'' in the following sense: they are based on certain fusion systems over the Sylow 2-subgroup of $\Spin_7(q)$ ($q$ an odd prime power) shown by Solomon not to occur as the 2-fusion in any actual finite group. As predicted by Benson, the classifying spaces of these 2-local finite groups are very closely related to the Dwyer-Wilkerson space $BDI(4)$. An error in our paper \cite{Sol} was pointed out to us by Andy Chermak, and we correct that error here. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton-SmithSB/pi_one(map) Title: Rank of the fundamental group of a component of a function space Authors: Gregory Lupton and Samuel Bruce Smith ArXive: math.AT/0502311 MSC-class: 55Q52; 55P15 We compute the rank of the fundamental group of an arbitrary connected component of the space map(X, Y) for X and Y nilpotent CW complexes with X finite. For the general component corresponding to a homotopy class f : X --> Y, we give a formula directly computable from the Sullivan model for f. For the component of the constant map, our formula expresses the rank in terms of classical invariants of X and Y. Among other applications and calculations, we obtain the following: Let G be a compact simple Lie group with maximal torus T^n. Then the fundamental group of map(S^2, G/T^n; f) is a finite group if and only if f: S^2 --> G/T^n is essential. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Nendorf-Scoville-Strom/Seq1 Categorical Sequences Rob Nendorf Nick Scoville Jeffrey Strom We define and study the categorical sequence of a space, which is a new formalism that streamlines the computation of the Lusternik-Schnirelmann category of a space X by induction on its CW skelta. The k-th term in the categorical sequence of a CW complex X, is the least integer n for which the n-skeleton of X has L-S category at least k. We show that the categorical sequence of X is a well-defined homotopy invariant. We prove that the sequence is `superadditive' which is one of three keys to the power of categorical sequences. In addition to this formula, we provide formulas relating the categorical sequences of spaces and some of their algebraic invariants, including their cohomology algebras and their rational models; we also find relations between the categorical sequences of the spaces in a fibration sequence and give a preliminary result on the categorical sequence of a product of two spaces in the rational case. We completely characterize the sequences which can arise as categorical sequences of formal rational spaces. The most important of the many examples that we offer is a simple proof of a theorem of Ghienne: if X is a member of the Mislin genus of the Lie group Sp(3), then cat(X) = cat(Sp(3)) (which is known to be 5). 12. http://hopf.math.purdue.edu/cgi-bin/generate?/Rognes/dualizable Title: Stably dualizable groups Author: John Rognes Abstract: We extend the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by (Dwyer and) J.R. Klein and the p-complete study for p-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the E-local stable homotopy category, for any spectrum E. The principal new examples occur in the K(n)-local category, where the Eilenberg-Mac Lane spaces G = K(Z/p, q) are stably dualizable for all 0 <= q <= n. We show how to associate to each E-locally stably dualizable group G a stably defined representation sphere S^{adG}, called the dualizing spectrum, which is dualizable and invertible in the E-local category. Each stably dualizable group is Atiyah-Poincare self-dual in the E-local category, up to a shift by S^{adG}. There are dimension-shifting norm- and transfer maps for spectra with G-action, again with a shift given by S^{adG}. The stably dualizable group G also admits a kind of framed bordism class [G] in the homotopy of L_E S, in degree dim_E(G) = [S^{adG}] of the Pic_E-graded homotopy groups of the E-localized sphere spectrum. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/Rognes/galois Title: Galois extensions of structured ring spectra Author: John Rognes Abstract: We introduce the notion of a Galois extension of commutative S-algebras (E-infinity ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological K-theory, Lubin-Tate spectra and cochain S-algebras. We establish the main theorem of Galois theory in this generality. Its proof involves the notions of separable (and etale) extensions of commutative S-algebras, and the Goerss-Hopkins-Miller theory for E-infinity mapping spaces. We show that the global sphere spectrum~S is separably closed (using Minkowski's discriminant theorem), and we estimate the separable closure of its localization with respect to each of the Morava K-theories. We also define Hopf-Galois extensions of commutative S-algebras, and study the complex cobordism spectrum MU as a common integral model for all of the local Lubin-Tate Galois extensions. ------------ ------------------------------------------- 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 ---------------- --------------------- 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 Università 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. ---------------- ------------------------------------- There are 11 new papers this time, from Behrens (3), Behrens-Lawson, Chebolu, DavisDaniel, Hovey, Lueck, Morava, Neusel, and Neusel-Wisniewski. Mark Hovey New papers appearing on hopf between 7/1/05 and 8/1/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens/rootkin/rootkin Title: Some root invariants at the prime 2 Author(s): Mark Behrens Abstract: The first part of this paper consists of lecture notes which summarize the machinery of filtered root invariants. A conceptual notion of "homotopy Greek letter element" is also introduced, and evidence is presented that it may be related to the root invariant. In the second part we compute some low dimensional root invariants of v_1-periodic elements at the prime 2. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens/rootpub/rootpub Title: Root invariants in the Adams spectral sequence Author(s): Mark Behrens Abstract: Let E be a ring spectrum for which the E-Adams spectral sequence converges. We define a variant of Mahowald's root invariant called the `filtered root invariant' which takes values in the E_1 term of the E-Adams spectral sequence. The main theorems of this paper concern when these filtered root invariants detect the actual root invariant, and explain a relationship between filtered root invariants and differentials and compositions in the E-Adams spectral sequence. These theorems are compared to some known computations of root invariants at the prime 2. We use the filtered root invariants to compute some low dimensional root invariants of v_1-periodic elements at the prime 3. We also compute the root invariants of some infinite v_1-periodic families of elements at the prime 3. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens/K2S/K2S Title: A modular description of the K(2)-local sphere at the prime 3 Author(s): Mark Behrens Abstract: Using degree N isogenies of elliptic curves, we produce a spectrum Q(N). This spectrum is built out of spectra related to tmf. At p=3 we show that the K(2)-local sphere is built out of Q(2) and its K(2)-local Spanier-Whitehead dual. This gives a conceptual reinterpretation a resolution of Goerss, Henn, Mahowald, and Rezk. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Behrens-Lawson/dense Title: Isogenies of elliptic curves and the Morava stabilizer group Authors: Mark Behrens and Tyler Lawson Abstract: Let MS_2 be the p-primary second Morava stabilizer group, C a supersingular elliptic curve over \br{FF}_p, O the ring of endomorphisms of C, and \ell a topological generator of Z_p^x (respectively Z_2^x/{+-1} if p = 2). We show that for p > 2 the group \Gamma \subseteq O[1/\ell]^x of quasi-endomorphisms of degree a power of \ell is dense in MS_2. For p = 2, we show that \Gamma is dense in an index 2 subgroup of MS_2. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu/KS Title: Krull-Schmidt decompositions for thick subcategories Author: Sunil Chebolu AMS classifictaion numbers: Primary: 55p42; Secondary: 18E30 Address: Department of Mathematics, University of Western Ontario, London, ON, N6A 5B7 Abstract: Following Krause, we prove Krull-Schmidt theorems for thick subcategories of various triangulated categories: derived categories of rings, noetherian stable homotopy categories, stable module categories over Hopf algebras, and the stable homotopy category of spectra. In all these categories, it is shown that the thick ideals of small objects decompose uniquely into indecomposable thick ideals. Some consequences of these decomposition results are also discussed. In particular, it is shown that all these decompositions respect $K$-theory 6. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/p1v5ams Title: Homotopy fixed points for L_K(n)(E_n ^ X) using the continuous action (Revised version) 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). 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/freyd On Freyd's generating hypothesis Mark Hovey We revisit Freyd's generating hypothesis in stable homotopy theory. We derive new equivalent forms of the generating hypothesis and some new consequences of it. A surprising one is that $I$, the Brown-Comenetz dual of the sphere and the source of many counterexamples in stable homotopy, is the cofiber of a self map of a wedge of spheres. We also show that a consequence of the generating hypothesis, that the homotopy of a finite spectrum that is not a wedge of spheres can never be finitely generated as a module over $\pi_{*}S$, is in fact true for finite torsion spectra. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Lueck/lueck_tkcsr Title: Rational Computations of the Topological K-Theory of Classifying Spaces of Discrete Groups Author: Wolfgang Lueck AMS Classification Numbers: 55N15 Address: Wolfgang Lueck Mathematisches Institut der Westfaelischen Wilhelms-Universitaet Einsteinstr. 62 48149 Muenster Germany xxx-archive: KT/0507237 Abstract: We compute rationally the topological (complex) K-theory of the classifying space BG of a discrete group provided that G has a cocompact G-CW-model for its classifying space for proper G-actions. For instance word-hyperbolic groups and cocompact discrete subgroups of connected Lie groups satisfy this assumption. The answer is given in terms of the group cohomology of G and of the centralizers of finite cyclic subgroups of prime power order. We also analyze the multiplicative structure. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Morava/Rosendal Title: Toward a fundamental groupoid for tensor triangulated categories Author: Jack Morava AMS classification: 11G, 19F, 57R, 81T Abstract: Notes for a talk at the conference on arithmetic of structured ring spectra; Rosendal, Norway, August 19 - 28 2005: This very speculative talk suggests that a theory of fundamental groupoids for tensor triangulated categories could be used to describe the ring of integers as the singular fiber in a family of ring-spectra parametrized by a structure space for the stable homotopy category, and that Bousfield localization might be part of a theory of `nearby' cycles for stacks or orbifolds. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/survey Degree bounds--an invitation to postmodern invariant theory Mara D. Neusel Abstract: This is a survey article on degree bounds in invariant theory of finite groups. A finite subgroup $G$ of the general linear group $\GL(n\, \F)$ over some field $\F$ acts via matrix multiplication on the vector space $V=\F^n$. This induces an action of $G$ on the polynomials $\F[x_1\commadots x_n]$ in $n$ variables. The polynomials $\F[x_1\commadots x_n]^G\subseteq \F[x_1\commadots x_n]$ invariant under this action form a subring. This ring is our center of study. In particular we will discuss how to calculate this ring. In this context degree bounds are central, and we want to present the known results. We also sketch the techniques that are used to obtain good bounds and describe open questions. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Wisniewski/piotr Connected Hopf algebras with Dixmier bases and infinite primary decomposition Mara D. Neusel, Piotr Wisniewski Abstract: In this paper we show the existence of invariant primary decompositions in the categories of modules and rings over a Hopf algebra of Dixmier type. --------------- ------------------------------------ There are 6 new papers this time, from Bergner (2), Chebolu, Hornbostel-Naumann, Lueck-Reich, and Notbohm. Mark Hovey New papers appearing on hopf between 8/4/05 and 9/5/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bergner/MultiSort Title: Rigidification of algebras over multi-sorted algebraic theories Author: Julia E. Bergner AMS Classification: 18C10, 18G30, 18E35, 55P48 arXiv submission number: math.AT/0508152 Author's address: Kansas State University 138 Cardwell Hall Manhattan, KS 66506 Abstract: We define the notion of a multi-sorted algebraic theory, which is a generalization of an algebraic theory in which the objects are of different ``sorts." We prove a rigidification result for simplicial algebras over these theories, showing that there is a Quillen equivalence between a model category structure on the category of strict algebras over a multi-sorted theory and an appropriate model category structure on the category of functors from a multi-sorted theory to the category of simplicial sets. In the latter model structure, the fibrant objects are homotopy algebras over that theory. Our two main examples of strict algebras are operads in the category of simplicial sets and simplicial categories with a given set of objects. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bergner/SimplicialMonoids Title: Simplicial monoids and Segal categories Author: Julia E. Bergner AMS Classification: 18G30, 18E35, 18C10, 55U40 arXiv submission number: math.AT/0508416 Author's address: Kansas State University 138 Cardwell Hall Manhattan, KS 66506 Abstract: Much research has been done on structures equivalent to topological or simplicial groups. In this paper, we consider instead simplicial monoids. In particular, we show that the usual model category structure on the category of simplicial monoids is Quillen equivalent to an appropriate model category structure on the category of simplicial spaces with a single point in degree zero. In this second model structure, the fibrant objects are reduced Segal categories. We then generalize the proof to relate simplicial categories with a fixed object set to Segal categories with the same fixed set in degree zero. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu/chromatic Title: Refining thick subcategory theorems Author: Sunil Chebolu AMS classification numbers: Primary: 55P42, 18G55, 19A99 Address: Department of Mathematics, University of Western Ontario, London, ON, N6A 5B7 Abstract: We use a $K$-theory recipe of Thomason to obtain classifications of triangulated subcategories via refining some standard thick subcategory theorems. We apply this recipe to the full subcategories of finite objects in the derived categories of rings and the stable homotopy category of spectra. This gives, in the derived categories, a complete classification of the triangulated subcategories of perfect complexes over some noetherian rings. In the homotopy category of spectra we obtain only a partial classification of the triangulated subcategories of the finite $p$-local spectra. We use this partial classification to study the lattice of triangulated subcategories. This study gives some new evidence to a conjecture of Adams that the thick subcategory $\C_2$ can be generated by iterated cofiberings of the Smith-Toda complex. We also discuss various consequences of these classifications theorems. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Hornbostel-Naumann/f-invofbeta Title: Beta-elements and divided congruences Authors: Jens Hornbostel, Niko Naumann Abstract: The f-invariant is an injective homomorphism from the 2-line of the Adams-Novikov spectral sequence to a group which is closely related to divided congruences of elliptic modular forms. We compute the f-invariant for two infinite families of beta-elements and explain the relation of the arithmetic of divided congruences with the Kervaire invariant one problem. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Lueck-Reich/lueck+reich0805 Title of Paper: Detecting K-theory by cyclic homology Author(s): Wolfgang Lueck and Holger Reich AMS Classification number: 19D55 xxx_archive: math.KT/0509002 Addresses of Authors: Mathematisches Institut Westfaelische Wilhelms-Universitaet Einsteinstr. 62 48149 Muenster Germany Text of Abstract (try for 20 lines or less) We discuss which part of the rationalized algebraic K-theory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Notbohm/cmcomplex Title: Cohen-Macaulay and Gorenstein complexes from a topological point of view Author: Dietrich Notbohm AMS Classification numbers: 13F55, 55R35 Address of Author: Dept. of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, England Abstract: The main invariant to study the combinatorics of a simplicial complex $K$ is the associated face ring or Stanley-Reisner algebra. Reisner respectively Stanley explained in which sense Cohen-Macaulay and Gorenstein properties of the face ring are reflected by geometric and/or combinatoric properties of the simplicial complex. We give a new proof for these result by homotopy theoretic methods and constructions. Our approach is based on ideas used very successfully in the analysis of the homotopy theory of classifying spaces. ------------------ --------------------------------- There are 8 new papers this time, from Arkowitz-Lupton, DavisDaniel(2), DavisD-Sun, Felix-Lupton, Henn, Kreck-Lueck, and Lupton-Phillips-Schochet-SmithSB. Mark Hovey New papers appearing on hopf between 9/5/05 and 10/1/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Lupton/ArkLupActions Homotopy Actions, Cyclic Maps and their Duals Martin Arkowitz and Gregory Lupton MSC 2000 55Q05, 55M30, 55P30 Abstract: An action of A on X is a map F: AxX to X such that F|_X = id: X to X. The restriction F|_A: A to X of an action is called a cyclic map. Special cases of these notions include group actions and the Gottlieb groups of a space, each of which has been studied extensively. We prove some general results about actions and their Eckmann-Hilton duals. For instance, we classify the actions on an H-space that are compatible with the H-structure. As a corollary, we prove that if any two actions F and F' of A on X have cyclic maps f and f' with Omega(f) = Omega(f'), then Omega(F) and Omega(F') give the same action of Omega(A) on Omega(X). We introduce a new notion of the category of a map g and prove that g is cocyclic if and only if the category is less than or equal to 1. From this we conclude that if g is cocyclic, then the Berstein-Ganea category of g is <= 1. We also briefly discuss the relationship between a map being cyclic and its cocategory being <= 1. Note: Appeared in Homology, Homotopy and Applications, vol. 7(1) (2005), 169-184. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/galois Title: Rognes's theory of Galois extensions and the continuous action of G_n on E_n Author: Daniel G. Davis Address: Purdue University Abstract: Let us take for granted that $L_{K(n)}S^0 \rightarrow E_n$ is some kind of a G_n-Galois extension. Of course, this is in the setting of continuous G_n-spectra. How much structure does this continuous G-Galois extension have? How much structure does one want to build into this notion to obtain useful conclusions? If the author's conjecture that "E_n/I, for a cofinal collection of I's, is a discrete G_n-symmetric ring spectrum" is true, what additional structure does this give the continuous G_n-Galois extension? Is it useful or merely beautiful? This paper is an exploration of how to answer these questions. This inactive manuscript arose as a letter to John Rognes, whom he thanks for a helpful conversation in Rosendal. This paper was written before John's preprints (the initial version and the final one) on Galois extensions were available. The author thanks Paul Goerss for his encouragement. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/subhg Title: Attempting to construct homotopy orbits for profinite groups Author: Daniel G. Davis Address: Purdue University Abstract: This note gives a heuristic argument for how one might like to define X_{hG}, for G profinite; it represents a first step in attempting to do this. The argument is not shown to work, and though the heuristic seems plausible, the author does not know how to complete the critical Definition 4.2. Also, the proof of Theorem 5.2 is incomplete. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD-Sun/DavisSun A number-theoretic approach to homotopy exponents of SU(n) Donald M. Davis and Zhi-Wei Sun AMS Classifications: 55Q52, 57T20, 11A07, 11B65, 11S05 Abstract: We use methods of combinatorial number theory to prove that, for all n and p, some homotopy group pi_i(SU(n)) contains an element of order p^{n-1+ord_p([n/p]!)}, where ord_p(m) denotes the exponent of p in m. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Felix-Lupton/FelLupEval Title: Evaluation Maps in Rational Homotopy Authors: Yves Felix and Gregory Lupton AMS MSC2000: 55P62, 55Q05 arXiv: math.AT/0509632 Abstract: Let E be an H-space acting on a based space X. Then we refer to ev: E -> X, the map obtained by acting on the base point of X, as a ``generalized evaluation map." We establish several fundamental results about the rational homotopy behaviour of a generalized evaluation map, all of which apply to the usual evaluation map Map(X, X;1) -> X. With mild hypotheses on X, we show that a generalized evaluation map ev factors, up to rational homotopy, through a map Gamma_ev: S_ev -> X where S_ev is a (relatively small) finite product of odd-dimensional spheres and the map induced by Gamma_ev on rational homotopy groups is injective. This result has strong consequences: if the image in rational homotopy groups of ev is trivial, then the generalized evaluation map is null-homotopic after rationalization; unless X satisfies a very strong splitting condition, any generalized evaluation map induces the trivial homomorphism in rational cohomology; the map Gamma_ev is rationally a homotopy monomorphism and a generalized evaluation map may be written as a composition of a homotopy epimorphism and this homotopy monomorphism. We include illustrative examples and prove numerous subsidiary results of interest. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Henn/kn-res-ded Title: On finite resolutions of K(n)-local spheres Author: Hans-Werner Henn Abstract: For odd primes p we construct finite resolutions of the trivial module Z_p for the n-th Morava stabilizer group by (direct summands of) permutation modules with respect to finite p-subgroups. Furthermore we discuss the problem of realizing these resolutions by finite resolutions of the K(n)-local sphere via spectra which are (direct summands of) wedges of homotopy fixed point spectra for the action of these finite p-subgroups on the Lubin-Tate spectrum E_n. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Kreck-Lueck/kreck+lueck0905 Title of Paper: Topological rigidity for non-aspherical manifolds Author(s): Matthias Kreck and Wolfgang Lueck AMS Classification number: 57N99, 57R67. xxx_archive: math.GT/0509238 The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate when a non-aspherical oriented connected closed manifold M is topological rigid in the following sense. If f: N ---> M is an orientation preserving homotopy equivalence with a closed oriented manifold as target, then there is an orientation preserving homeomorphism h: N ---> M such that h and f induce up to conjugation the same maps on the fundamental groups. We call such manifolds Borel manifolds. We give partial answers to this questions for S^k x S^d, for sphere bundles over aspherical closed manifolds of dimension less or equal to 3 and for 3-manifolds with torsionfree fundamental groups. We show that this rigidity is inherited under connected sums in dimensions greater or equal to 5. We also classify manifolds of dimension 5 or 6 whose fundamental group is the one of a surface and whose second homotopy group is trivial. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton-Phillips-Schochet-SmithSB/RationalTaylor Title: Banach Algebras and Rational Homotopy Theory Authors: Gregory Lupton, N.Christopher Phillips, Claude L.~Schochet and Samuel B. Smith AMS MSC (2000): 46J05, 46L85, 55P62, 54C35, 55P15, 55P45 arXiv number: math.AT/0509269 Abstract: Let A be a unital commutative Banach algebra with maximal ideal space Max(A). We determine the rational H-type of $GL_n (A)$, the group of invertible $n \times n$ matrices with coefficients in A in terms of the rational cohomology of Max(A). We also address an old problem of J. L. Taylor. Let $Lc_n (A)$ denote the space of ``last columns'' of $GL_n (A).$ We construct a natural isomorphism \[ {\check{H}}^s (Max(A); Q) \cong \pi_{2 n - 1 - s} (Lc_n (A)) \otimes Q \] for $n > (1/2) s + 1$ which shows that the rational cohomology groups of Max(A) are determined by a topological invariant associated to A. As part of our analysis, we determine the rational H-type of certain gauge groups F(X,G) for G a Lie group or, more generally, a rational H-space. ----------------- --------------------------------------- There are 7 new papers this time, from Bartels-Reich, Bousfield, Fausk-Isaksen (2), Neusel, Neusel-Sezer, and Siebenmann. Mark Hovey New papers appearing on hopf between 10/1/05 and 11/11/05 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bartels-Reich/erb Title: Coefficients for the Farrell-Jones Conjecture Authors: Arthur Bartels, Holger Reich Author's e-mail address: bartelsa@math.uni-muenster.de, reichh@math.uni-muenster.de Abstract: We introduce the Farrell-Jones Conjecture with coefficients in an additive category with G-action. This is a variant of the Farrell-Jones Conjecture about the algebraic K- or L-Theory of a group ring RG. It allows to treat twisted group rings and crossed product rings. The conjecture with coefficients is stronger than the original conjecture but it has better inheritance properties. Since known proofs using controlled algebra carry over to the set-up with coefficients we obtain new results about the original Farrell-Jones Conjecture. The conjecture with coefficients implies the fibered version of the Farrell-Jones Conjecture. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bousfield/kunneth Title: Kunneth theorems and unstable operations in 2-adic KO-cohomology Author: A.K. Bousfield E-mail: bous@uic.edu AMS classifications: 55N15,55S25,55U25 Abstract: We develop Kunneth theorems and obtain detailed results on unstable operations in 2-adic KO-cohomology and, more generally, in united 2-adic K-cohomology. These results are needed for work on the K-localizations of H-spaces at the prime 2 and should be of independent interest. Our proofs of relations for unstable operations rely on Atiyah's Real K-theory and on a convenient mod 2 simplification of 2-adic KO-cohomology. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Fausk-Isaksen/filtered Title: Model structures on pro-categories Authors: Halvard Fausk, Daniel C. Isaksen E-mail: fausk@math.uio.no, isaksen@math.wayne.edu AMS Classification: 55U35 Primary ; Secondary 55P91, 18G55 Abstract: We introduce a notion of a filtered model structure and use this notion to produce various model structures on pro-categories. This framework generalizes several known examples. We give several examples, including a homotopy theory for $G$-spaces, where $G$ is a pro-finite group. The class of weak equivalences is an approximation to the class of underlying weak equivalences. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Fausk-Isaksen/t-model Title: T-model structures Authors: Halvard Fausk and Daniel C. Isaksen E-mail: fausk@math.uio.no, isaksen@math.wayne.edu AMS Classification: Primary 55P42; Secondary 18E30, 55U35 Abstract: For every stable model category $\mathcal{M}$ with a certain extra structure, we produce an associated model structure on the pro-category pro-$\mathcal{M}$ and a spectral sequence, analogous to the Atiyah-Hirzebruch spectral sequence, with reasonably good convergence properties for computing in the homotopy category of pro-$\mathcal{M}$. Our motivating example is the category of pro-spectra. The extra structure referred to above is a t-model structure. This is a rigidification of the usual notion of a t-structure on a triangulated category. A t-model structure is a proper simplicial stable model category $\mathcal{M}$ with a t-structure on its homotopy category together with an additional factorization axiom. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel/piotr Connected Hopf algebras with Dixmier bases and infinite primary decomposition Mara D. Neusel Mara.D.Neusel@ttu.edu Abstract: In this paper we study the existence of invariant primary decompositions for algebras and modules over Hopf algebras. This is an update of the previous preprint of Neusel-Wisniewski of the same title. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Sezer/noether The Noether map AUTHORS: Mara D. Neusel (Texas Tech University), M\"ufit Sezer (Bo\u gazici \"Universitesi) EMAILS: mara.d.neusel@ttu.edu mufit.sezer@boun.edu.tr ABSTRACT: Let $\rho: G\hra GL(n\/,\ \F)$ be a faithful representation of a finite group $G$. In this paper we study the image of the associated Noether map \[ \eta_G^G: \F[V(G)]^G \longrightarrow \F[V]^G\/. \] It turns out that the image of the Noether map characterizes the ring of invariants in the sense that its integral closure $\overline{\Im(\eta_G^G)} =\F[V]^G$. This is true without any restrictions on the group, representation, or ground field. Furthermore, we show that the Noether map is surjective, i.e., its image integrally closed, if $V=\F^n$ is a projective $\F G$-module. Moreover, we show that the converse of this statement is true if $G$ is a $p$-group and $\F$ has characteristic $p$, or if $\rho$ is a permutation representation. We apply these results and obtain upper bounds on the Noether number and the Cohen-Macaulay defect of $\F[V]^G$. We illustrate our results with several examples. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Siebenmann/Schoen-02Sept2005 The Osgood-Schoenflies Theorem Revisited by Laurent Siebenmann Math'ematique, B^at. 425, Universit'e de Paris-Sud, 91405-Orsay, France http://topo.math.u-psud.fr/~lcs/contact This retrospective article presents an elementary, and hopefully direct and clear, geo- metric proof of what is usually called the (classical planar) Schoenflies Theorem; it is stated as (ST) in x4 below _ with mention of its early history, including W.F. Osgood's rarely cited contributions. This (ST) is essentially the fact _ surprising in view of known fractal curves _ that every compact subset of the cartesian plane R2 that is homeomorphic to the circle S1, is necessarily the frontier in R2 of a set homeomorphic to the 2-disk. Beware that the `Generalized Schoenflies theorem' of B. Mazur [Maz] and M. Brown [Brow1] _ proved five decades later and valid in all dimensions _ does not imply (ST) since it assumes a condition of flatness (or local flatness [Brow2]). ------------------