There are 8 new papers this time, from Arone-Lesh, BrownR, Gutierrez, Inoue-Yagita, Klein-Williams, Korbas, Lockridge, and Ziemianski Mark Hovey New papers appearing on hopf between 11/11/05 and 1/4/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arone-Lesh/arone-lesh-press 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 of the sphere 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. This is a revision of a preprint previously submitted to Hopf. The paper has been accepted for publication in Journal für die reine und angewandte Mathematik (Crelle's Journal). 2. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/PB-jordan Title: Groupoids, the Phragmen-Brouwer Property, and the Jordan Curve Theorem Author: Ronald Brown Author's e-mail address: r.brown at bangor.ac.uk Author's mailing address: Mathematics Department, School of Informatics, University of Wales, Bangor, Gwynedd LL57 1UT, UK Author's web site: www.bangor.ac.uk/r.brown Preprint: University of Wales Math Preprint 06.01 Abstract: We publicise a proof of the Jordan Curve Theorem which relates it to the Phragmen-Brouwer Property, and whose proof uses the van Kampen theorem for the fundamental groupoid on a set of base points. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Gutierrez/hlems Homological localizations of Eilenberg-Mac Lane spectra Javier J. Guti\'errez We discuss the Bousfield localization $L_E X$ for any spectrum $E$ and any $HR$-module $X$, where $R$ is a ring with unit. Due to the splitting property of $HR$-modules, it is enough to study the localization of Eilenberg--Mac\,Lane spectra. Using general results about stable $f$-localizations, we give a method to compute the localization of an Eilenberg--Mac\,Lane spectrum $L_E HG$ for any spectrum $E$ and any abelian group $G$. We describe $L_E HG$ explicitly when $G$ is one of the following: finitely generated abelian groups, $p$-adic integers, Pr\"ufer groups, and subrings of the rationals. The results depend basically on the $E$-acyclicity patterns of the spectrum $H\Q$ and the spectrum $H\Z/p$ for each prime $p$. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Inoue-Yagita/bpso Title: The complex cobordism of BSOn Authors: K.Inoue and N.Yagita Abstract: In this paper, we compute MU(BSO(2n)) and show that it is generated as an MU-algebra by Conner-Floyd Chern classes and one 2n-dimensional element. For the case BO(m) are still studied by W.S.Wilson. We get the result by using (equivariant) stratification methods introduced to compute Chow rings by Guillot, Molina, Vessozi and Vistoli. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Klein-Williams/int-theoryI Homotopical intersection theory, I. by John R. Klein and E. Bruce Williams Abstract: We give a new approach to intersection theory. Our ``cycles'' are closed manifolds mapping into compact manifolds and our ``intersections'' are elements of a homotopy group of a certain Thom space. The results are then applied in various contexts, including fixed point, linking and disjunction problems. Our main theorems resemble those of Hatcher and Quinn. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Korbas/cuplength Title: Bounds for the cup-length of Poincare spaces and their applications Author: Julius Korbas AMS Classification numbers: Primary: 57N65; 55M30 Secondary: 53C30 Author's addresses: Department of Algebra, Geometry, and Mathematical Education, Faculty of Mathematics, Physics, and Informatics, Comenius University, Mlynska dolina, SK-842 48 Bratislava 4, Slovakia or Mathematical Institute, Slovak Academy of Sciences, Stefanikova 49, SK-814 73 Bratislava 1, Slovakia Abstract: Our main result offers a new (quite systematic) way of deriving bounds for the cup-length of Poincare spaces over fields; we outline a general research program based on this result. For the oriented Grassmann manifolds, already a limited realization of the program leads, in many cases, to the exact values of the cup-length and to interesting information on the Lyusternik-Shnirel'man category. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Lockridge/gh Title: The generating hypothesis in the derived category of R-modules. Author: Keir H. Lockridge Abstract: In this paper, we prove a version of Freyd's generating hypothesis for triangulated categories: if D is a cocomplete triangulated category and S is an object in D whose endomorphism ring is graded commutative and concentrated in degree zero, then S generates (in the sense of Freyd) the thick subcategory determined by S if and only if the endomorphism ring of S is von Neumann regular. As a corollary, we obtain that the generating hypothesis is true in the derived category of a commutative ring R if and only if R is von Neumann regular. We also investigate alternative formulations of the generating hypothesis in the derived category. Finally, we give a characterization of the Noetherian stable homotopy categories in which the generating hypothesis is true. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Ziemianski/SpinHtpReps TITLE: Homotopy representations of Spin(7) and SO(7) at prime 2 AUTHOR: Krzysztof Ziemianski ADDRESS: Faculty of Mathematics, Informatics and Mechanics Warsaw University Banacha 2 02-097 Warszawa POLAND ABSTRACT: A homotopy (complex) representation of a compact connected Lie group L at prime p is a map from BL into the p-completion of the classifying space of the unitary group. In this paper we give a partial classification of homotopy representations of SO(7) and Spin(7) at prime 2. Motiviation for considering this problem is twofold: first, one may hope that it would help to understand maps between classifying spaces. Secondly, construction of a homotopy representation of Spin(7) is a crucial step in the construction of a faithful representation of the 2-compact group DI(4). ------------------ There are 4 new papers this time, from Biedermann-Chorny-Roendigs, Bubenik-Worytkiewicz, DavisD-Theriault, and Fresse. Mark Hovey New papers appearing on hopf between 1/4/06 and 2/8/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Biedermann-Chorny-Roendigs/biedermann-chorny-roendigs Title: Goodwillie's calculus and model categories Author(s): Georg Biedermann, Boris Chorny, Oliver Roendigs Abstract: The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for classification of polynomial and homogeneous functors. Finally we show that the $n$-th derivative induces a Quillen map between the $n$-homogeneous model structure on small functors from pointed simplicial sets to spectra and the category of spectra with $\Sigma_n$-action. We consider also a finitary version of the $n$-homogeneous model structure and the $n$-homogeneous model structure on functors from pointed finite simplicial sets to spectra. In these two cases the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T. G. Goodwillie. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bubenik-Worytkiewicz/lps title: A model category for local po-spaces author: Peter Bubenik author: Krzysztof Worytkiewicz to appear in: Homology, Homotopy and Applications abstract: Locally partial-ordered spaces (local po-spaces) have been used to model concurrent systems. We provide equivalences for these spaces by constructing a model category containing the category of local po-spaces. We show the category of simplicial presheaves on local po-spaces can be given Jardine's model structure, in which we identify the weak equivalences between local po-spaces. In the process we give an equivalence between the category of sheaves on a local po-space and the category of {\'e}tale bundles over a local po-space. Finally we describe a localization that should provide a good framework for studying concurrent systems. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD-Theriault/theri6 Odd-primary homotopy exponents of compact simple Lie groups Donald M. Davis and Stephen D. Theriault We note that a recent result of the second author yields upper bounds for odd-primary homotopy exponents of compact simple Lie groups which are often quite close to the lower bounds obtained from v1-periodic homotopy theory. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Fresse/Bar-StructureUniqueness Title: The bar construction of an $E$-infinity algebra Author: Benoit Fresse Abstract: We consider the classical reduced bar construction of associative algebras B(A). If the product of A is commutative, then B(A) can be equipped with the classical shuffle product, so that B(A) is still a commutative algebra. This assertion can be generalized for algebras which are commutative up to homotopy. Namely, one observes that the bar construction of an E-infinite algebra B(A) can be endowed with the structure of an E-infinite algebra. The purpose of this article is to give an existence and uniqueness theorem for this claim. We would like to insist on the uniqueness property: our statement makes the construction of $E$-infinite structures easier and more flexible. Therefore, the proof of our existence theorem differs from other constructions of the literature. In addition, the uniqueness property allows to give easily a homotopy interpretation of the bar construction. ----------------- There are 4 new papers this time, from BrownR, DavisDaniel, DavisD, and Hovey. Mark Hovey New papers appearing on hopf between 2/8/06 and 3/1/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/bedlewo Title: Three themes in the work of Charles Ehresmann: Local-to-global; Groupoids; Higher dimensions. Author: Ronald Brown AMS classification number: 01A60,53C29,81Q70,22A22,55P15 Expansion of an invited talk given to the 7th Conference on the Geometry and Topology of Manifolds: The Mathematical Legacy of Charles Ehresmann, Bedlewo 8.05.2005-15.05.2005 (Poland). Abstract: This paper illustrates the themes of the title in terms of: van Kampen type theorems for the fundamental groupoid; holonomy and monodromy groupoids; and higher homotopy groupoids. Interaction with work of the writer is explored. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/cplx2 Title: The E_2-term of the descent spectral sequence for continuous G-spectra Author: Daniel G. Davis Author's address: Purdue University Abstract: Let {X_i} be a tower of discrete G-spectra, each of which is fibrant as a spectrum, so that X=holim_i X_i is a continuous G-spectrum, with homotopy fixed point spectrum X^{hG}. The E_2-term of the descent spectral sequence for \pi_*(X^{hG}) cannot always be expressed as continuous cohomology. However, we show that the E_2-term is always built out of a certain complex of spectra, that, in the context of abelian groups, is used to compute the continuous cochain cohomology of G with coefficients in lim_i M_i, where {M_i} is a tower of discrete G-modules. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD/CPcrabb4 Some new immersion results for complex projective space Donald M. Davis Lehigh University, Bethlehem, PA 18015 Abstract: We prove the following two new optimal immersion results for complex projective space. First, if n equiv 3 mod 8 but n not equiv 3 mod 64, and alpha(n)=7, then CP^n can be immersed in R^{4n-14}. Second, if n is even and alpha(n)=3, then CP^n can be immersed in R^{4n-4}. Here alpha(n) denotes the number of 1's in the binary expansion of n. The first contradicts a result of Crabb, who said that such an immersion does not exist, apparently due to an arithmetic mistake. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/injective-comod Injective comodules and Landweber exact homology theories Mark Hovey Wesleyan University Middletown, CT We classify the indecomposable injective E(n)_{*}E(n)-comodules, where $E(n)$ is the Johnson-Wilson homology theory. They are suspensions of the J_{n,r}, where J_{n,r} is the E(n)-homology of the rth monochromatic piece M_{r} E(r) of E(r) and $0\leq r\leq n$. The endomorphism ring of J_{n,r} is the ring of operations in the completed E(r) theory; this ring of operations is not really known so far as I know, though it is closely related to the stabilizer group S_r. An interesting byproduct of this study is the isomorphism E^{*}(X) = \Hom_{E(n)_{*}} (E(n)_{*}M_{n}X, K) where E is completed E(n) theory and K is the n-fold desuspension of E(n)_{*}/I_{n}^{\infty}). ----------------- There are 4 new papers this time, from Blanc-Johnson-Turner, Clarke-Crossley-Whitehouse, Muro-Tonks, Ziemianski. I also wanted to say that in my paper of last time, there is an isomorphism between the completed E(n)-cohomology of X and Hom from the E(n)-homology of M_n X to an appropriate module. This isomorphism was known before to Greenlees, Hopkins, Sadofsky, and others, though it does not appear to be in print. The version of the paper now on the archive reflects that. Mark Hovey New papers appearing on hopf between 3/1/06 and 4/7/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Johnson-Turner/rdpa Title: On Realizing Diagrams of Pi-algebras Authors: David Blanc, Mark W. Johnson, and James M. Turner Abstract: Given a diagram of Pi-algebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulated in terms of generalized Pi-algebras. This extends a program begun by Dwyer, Kan, and Stover to study the realization of a single Pi-algebra. In particular, we explicitly analyze the simple case of a single map, and provide a detailed example, illustrating the connections to higher homotopy operations. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Clarke-Crossley-Whitehouse/ccwDiscrete The discrete module category for the ring of K-theory operations Francis Clarke, Martin Crossley, Sarah Whitehouse We study the category of discrete modules over the ring of degree zero stable operations in p-local complex K-theory. We show that the p-local K-homology of any space or spectrum is such a module, and that this category is isomorphic to a category defined by Bousfield and used in his work on the K-local stable homotopy category (Amer. J. Math., 1985). We also provide an alternative characterisation of discrete modules as locally finitely generated modules. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Muro-Tonks/1tK3 Title:The 1-type of a Waldhausen K-theory spectrum Authors: Fernando Muro and Andrew Tonks Abstract: We give a small functorial algebraic model for the 2-stage Postnikov section of the K-theory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Ziemianski/DI4Rep TITLE: A faithful unitary representation of the 2-compact group DI(4) AUTHOR: Krzysztof Ziemianski ABSTRACT: We construct a monomorphism from the $2$-compact group $DI(4)$ into a $2$-compact unitary group. ---------------- ------------------------------ There are 7 new papers this time, from Kuhn, Neusel-Sezer (2), Pengelley-Williams, RadulescuBanu, and SanchezGarcia (2) Mark Hovey New papers appearing on hopf between 4/7/06 and 5/3/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/autgrp Title: The nilpotent filtration and the action of automorphisms on the cohomology of finite $p$--groups Author: Nicholas J. Kuhn AMS classification number: 20J06 Abstract: We study H^*(P), the mod p cohomology of a finite p--group P, viewed as an Out(P)--module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if e_S in Z/p[Out(P)] is a primitive idempotent associated to an irreducible Z/p[Out(P)]--module S, then the Krull dimension of e_SH^*(P) equals the rank of P. The rank is an upper bound by Quillen's work, and the conjecture can be viewed as the statement that every irreducible Z/p[Out(P)]--module occurs as a composition factor in H^*(P) with similar frequency. In summary, our results are as follows. A strong form of the conjecture is true when p is odd. The situation is much more complex when p=2, but is reduced to a question about 2--central groups (groups in which all elements of order 2 are central), making it easy to verify the conjecture for many finite 2--groups, including all groups of order 128, and all groups that can be written as the product of groups of order 64 or less. Featured is the nilpotent filtration of the category of unstable A--modules. Also featured are unstable algebras of cohomology primitives associated to central group extensions. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Sezer/noether-map-I The Noether Map I Mara D Neusel and M"ufit Sezer Abstract: Let $\rho: G\hookrightarrow GL(n, F)$ 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. Moreover, we show that the extension $Im (\eta_G^G) \subseteq F[V]^G$ is a finite $p$-root extension. Furthermore, we show that the Noether map is surjective, i.e., its image integrally closed, if $V=F^n$ is a projective $FG$-module. We apply these results and obtain upper bounds on the degrees of a minimal generating set of $F[V]^G$ and the Cohen-Macaulay defect of $F[V]^G$. We illustrate our results with several examples. Note that this paper together with noether-map-II contain stronger results than the authors' previous paper Neusel-Sezer/noether. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Neusel-Sezer/noether-map-II The Noether Map II Mara D Neusel and M"ufit Sezer Abstract: Let $\rho: G\hookrightarrow GL(n, F)$ be a faithful representation of a finite group G. In this paper we proceed with the study of the image of the associated Noether map \[ \eta_G^G: F[V(G)]^G \longrightarrow F[V]^G. \] In [Noether Map I] it has been shown that the Noether map is surjective if $V$ is a projective $FG$-module. This paper deals with the converse. The converse is in general not true: we illustrate this with an example. However, for $p$-groups (where $p$ is the characteristic of the ground field $F$) as well as for permutation representations of any group the surjectivity of the Noether map implies the projectivity of $V$. Note that this paper together with noether-map-I contain stronger results than the authors' previous paper Neusel-Sezer/noether. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Pengelley-Williams/peng-will-oddkam The odd-primary Kudo-Araki-May algebra of algebraic Steenrod operations, and invariant theory David J. Pengelley and Frank Williams New Mexico State University Las Cruces, NM 88003 Primary 16W22; Secondary 16W30, 16W50, 55S10, 55S12, 55S99, 57T05. We describe bialgebras of lower-indexed algebraic Steenrod operations over the field with p elements, p an odd prime. These go beyond the operations that can act nontrivially in topology, and their duals are closely related to algebras of polynomial invariants under subgroups of the general linear groups that contain the unipotent upper triangular groups. There are significant differences between these algebras and the analogous one for p=2 , in particular in the nature and consequences of the defining Adem relations. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/RadulescuBanu/cofib-cat Title: Cofibrations in Homotopy Theory Author: Andrei Radulescu-Banu Author's mailing address: 86 Cedar St, Lexington, MA 02421 Abstract: We define Anderson-Brown-Cisisnski (ABC) cofibration categories, and construct homotopy colimits of diagrams of objects in ABC cofibraction categories. Homotopy colimits for Quillen model categories are obtained as a particular case. We attach to each ABC cofibration category a right derivator. A dual theory is developed for homotopy limits in ABC fibration categories and for left derivators. These constructions provide a natural framework for 'doing homotopy theory' in ABC (co)fibration categories. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/SanchezGarcia/bredon Title: Bredon homology and equivariant K-homology of SL(3,Z) Author: Ruben Sanchez-Garcia Author's address: Department of Pure Maths, Hicks Building University of Sheffield Sheffield S3 7RH, United Kingdom Included ps or eps files: SouleFundamentalDomainLabelled.eps TruncatedCube.eps AMS classification number: 19L47, 55N91 (Primary); 19K99, 46L80 (Secondary) Other useful information: arXiv:math.KT/0601587 Abstract: We obtain the equivariant K-homology of the classifying space \underline{E}SL(3,Z) from the computation of its Bredon homology with respect to finite subgroups and coefficients in the representation ring. We also obtain the corresponding results for GL(3,Z). Our calculations give therefore the topological side of the Baum-Connes conjecture for these groups. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/SanchezGarcia/coxeter Title: Equivariant K-homology for some Coxeter groups Author: Ruben Sanchez-Garcia Author's address: Department of Pure Maths, Hicks Building University of Sheffield Sheffield S3 7RH, United Kingdom Included eps files: hexagon3.eps hexagon4.eps interval.eps tessellation0.eps trianglesd2.eps AMS classification number: 19L47, 55N91 (Primary); 19K99, 46L80 (Secondary) Other useful information: arXiv:math.KT/0604402 Abstract: We obtain the equivariant K-homology of the classifying space \underline{E}W for W a right-angled or, more generally, an even Coxeter group. The key result is a formula for the relative Bredon homology of \underline{E}W in terms of Coxeter cells. Our calculations amount to the K-theory of the reduced C^*-algebra of W, via the Baum-Connes assembly map. --------------- There are 7 new papers this time, from Bartels-Rosenthal, Chermak-Oliver-Shpectorov, Dwyer-Wilkerson, Jardine (2), Stacey-Whitehouse, and Wilkerson. Mark Hovey New papers appearing on hopf between 5/3/06 and 6/5/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bartels-Rosenthal/asymptotic Authors: Arthur Bartels, David Rosenthal arXiv submission number: math.KT/0605088 Abstract: It is proved that the assembly maps in algebraic K- and L-theory with respect to the family of finite subgroups is injective for groups with finite asymptotic dimension that admit a finite model for the classifying space for proper actions. The result also applies to certain groups that admit only a finite dimensional model for this space. In particular, it applies to discrete subgroups of virtually connected Lie groups. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chermak-Oliver-Shpectorov/fundsol The simple connectivity of $B\Sol(q)$ by Andrew Chermak, Bob Oliver, and Sergey Shpectorov Andrew Chermak Kansas State University Bob Oliver LAGA, Institut Galil\'ee Sergey Shpectorov University of Birmingham Abstract: A $p$-local finite group is an algebraic structure which includes two categories, a fusion system and a linking system, which mimic the fusion and linking categories of a finite group over one of its Sylow subgroups. The $p$-completion of the geometric realization of the linking system is the classifying space of the finite group. In this paper, we study the geometric realization, \emph{without} completion, of linking systems of certain exotic 2-local finite groups whose existence was predicted by Solomon and Benson, and prove that they are all simply connected. The file "Co3graph.eps" must be included with the dvi file. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Wilkerson/GorensteinCoinvariants POINCAR'E DUALITY AND STEINBERG'S THEOREM ON RINGS OF COINVARIANTS W. G. DWYER AND C. W. WILKERSON In this note we use elementary methods to prove Steinberg's result for fields of characteristic 0 or of characteristic prime to the order of W . This gives a new proof even in the characteristic zero case. 1.1. Theorem. Let k be a field, V an r-dimensional k-vector space, and W a finite subgroup of Aut k(V ). Let S = S[V #] be the symmetric algebra on V # the k-dual of V, and R = S^W the ring of invariants of under the natural action of W on S. Define P* to be the quotient algebra S i\tensor_R k. If the characteristic of k is zero or prime to the order of W and P* satisfies Poincar'e duality, then R is isomorphic to a polynomial algebra on r generators. Steinberg [9] has shown that R is polynomial if k is the field of complex numbers and the quotient algebra P* = S\tensor_R k satisfies Poincar'e duality (1.3). Steinberg's result was extended by Kane [3, 4] to other fields of characteristic zero, and by T.-C. Lin [5] to the case in which k is a finite field of characteristic prime to the order of W . The current proof is independent of previous methods. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/coc-cat3 Title: Cocycle categories Author: J.F. Jardine arXive submission number: math.AT/0605198 Abstract: A cocycle category H(X,Y) is defined for objects X and Y in a model category, and it is shown that the set of morphisms [X,Y] is isomorphic to the set of path components of H(X,Y) provided the ambient model category is right proper and satisfies the extra condition that weak equivalences are closed under finite products. Various applications of this result are displayed, including the homotopy classification of torsors, abelian cohomology groups, group extensions and gerbes. The older classification results have simple new proofs involving canonically defined cocycles. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/gerbes6 Title: Homotopy classification of gerbes Author: J.F. Jardine arXive submission number: math.AT/0605200 Abstract: Gerbes are locally connected presheaves of groupoids. They are classified up to local weak equivalence by path components in a 2-cocycle category taking values in all sheaves of groups, their isomorphisms and homotopies. If F is a full presheaf of sheaves of groups, isomorphisms and homotopies, then [*,BF] is isomorphic to equivalence classes of gerbes locally equivalent to groups appearing in F. Giraud's non-abelian cohomology object of equivalence classes of gerbes with band L is isomorphic to morphisms in the homotopy category from the point * to the homotopy fibre over L for a map defined on BF and taking values in the classifying space for the stack completion of the fundamental groupoid of F. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Stacey-Whitehouse/deloopv2 Title: Stable and Unstable Operations in mod p Cohomology Theories Authors: Andrew Stacey and Sarah Whitehouse Other useful information: math.AT/0605471 Abstract: We consider operations between two multiplicative, complex orientable cohomology theories. Under suitable hypotheses, we construct a map from unstable to stable operations, left-inverse to the usual map from stable to unstable operations. The main example is where the target theory is one of the Morava K-theories in which case our map is closely related to the Bousfield-Kuhn functor. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Wilkerson/newfred Loop Spaces and Finiteness Clarence W. Wilkerson Purdue University This expository note began as comments on a shorter note of F.R. Cohen \cite{cohen}. Cohen's paper is an elegant application of powerful recent results in unstable homotopy theory to a problem of interest to analysts. {\sl {\bf Theorem :} (F.\,R. Cohen,\cite{cohen}) Let $X$ be a simply connected finite complex which is not contractible and let $\Omega^j_0X$ be the component of the constant map in the $j$-th pointed loop space of $X$. If $j \geq 2$, then the Lusternik-Schnirlman category of $\Omega^j_0X$ is not finite. }\\ This note includes a rederivation of the above theorem using H-space methods of W. Browder from the 60's, \cite{Browder-loop}, \cite{Browder-Torsion}. The aim is to reduce the prerequisites for Cohen's theorem to those available after a second course in algebraic topology. We end with a discussion of recent work of Lannes-Schwartz on various notions of finiteness properties and the behavior under looping. The common theme is extensive use of the action of the Steenrod algebra on the cohomology of a topological space. --------------- ----- There are 7 new papers this time, from Biedermann, Blanc, Oliver-Ventura, Shipley, Stacey-Whitehouse, Wuethrich, and YauD. Mark Hovey New papers appearing on hopf between 6/5/06 and 7/8/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Biedermann/presh-n-types Title: On the homotopy theory of n-types Author: Georg Biedermann Mail address: Dep. of Mathematics, Middlesex College, UWO, London, Ontario, N5X 2W8, Canada Abstract: We achieve a classification of n-types of simplicial presheaves in terms of (n-1)-types of presheaves of groupoids enriched in simplicial sets. This can be viewed as a different description of the homotopy theory of higher hyperstacks. As a special case we obtain a good substitute for the homotopy theory of (weak) higher groupoids. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc/comp Title: Comparing homotopy categories Author: David Blanc Address: Department of Mathematics University of Haifa 31905 Haifa Israel Abstract: Given a suitable functor T:C -> D between model categories, we define a long exact sequence relating the homotopy groups of any X in C with those of TX, and use this to describe an obstruction theory for lifting an object G in D to C. Examples include finding spaces with given homology or homotopy groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Oliver-Ventura/ov1 Extensions of linking systems with $p$-group kernel Bob Oliver and Joana Ventura LAGA Departamento de Matem\'atica Institut Galil\'ee Instituto Superior T\'ecnico Av. J-B Cl\'ement Av. Rovisco Pais 93430 Villetaneuse, France 1049--001 Lisboa, Portugal bobol@math.univ-paris13.fr jventura@math.ist.utl.pt Subject class: Primary 55R35. Secondary 55R40, 20D20 Keywords: Classifying space, $p$-completion, finite groups, fusion. Abstract: We study extensions of $p$-local finite groups where the kernel is a $p$-group. In particular, we construct examples of saturated fusion systems $\calf$ which do not come from finite groups, but which have normal $p$-subgroups $A\nsg\calf$ such that $\calf/A$ is the fusion system of a finite group. One of the tools used to do this is the concept of a ``transporter system'', which is modelled on the transporter category of a finite group, and is more general than a linking system. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Shipley/zdga17 Title: HZ-algebra spectra are differential graded algebras Author: Brooke Shipley Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZ-algebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZ-algebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that basically any rational stable model category is Quillen equivalent to modules over a differential graded Q-algebra (with many objects). 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Stacey-Whitehouse/deloopv2 Title: Stable and Unstable Operations in mod p Cohomology Theories Authors: Andrew Stacey and Sarah Whitehouse Abstract: We consider operations between two multiplicative, complex orientable cohomology theories. Under suitable hypotheses, we construct a map from unstable to stable operations, left-inverse to the usual map from stable to unstable operations. The main example is where the target theory is one of the Morava K-theories in which case our map is closely related to the Bousfield-Kuhn functor. Resubmitted to correct font generation problem with the conversion to postscript and PDF. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Wuethrich/thickenings Title: Infinitesimal thickenings of Morava K-theories Author: Samuel Wuethrich 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. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/GD2 Title: Gerstenhaber structure and Deligne's conjecture for Loday algebras Author: Donald Yau Abstract: A method for establishing a Gerstenhaber algebra structure on the cohomology of Loday-type algebras is presented. This method is then applied to dendriform dialgebras and three types of trialgebras introduced by Loday and Ronco. Along the way, our results are combined with a result of McClure-Smith to prove an analogue of Deligne's conjecture for Loday algebras. ----------------- ----------------------------------- There are 7 new papers this time, from Arone-Lambrechts-Volic, Broto-Levi-Oliver, Dwyer-Wilkerson, Gillespie, Naumann, Ulrich-Wilkerson, and YauD. Mark Hovey New papers appearing on hopf between 7/8/06 and 8/4/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Arone-Lambrechts-Volic/CalculusFormalityEmbeddings Title: Calculus of functors, operad formality, and rational homology of embedding spaces Authors: Gregory Arone, Department of Mathematics, University of Virginia, Charlottesville, VA, USA. Pascal Lambrechts Institut Math\'{e}matique, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium Ismar Voli\'c Department of Mathematics, University of Virginia, Charlottesville, VA, USA Abstract: Let M be a smooth manifold and V a Euclidean space. Let Ebar(M,V) be the homotopy fiber of the map from Emb(M,V) to Imm(M,V). This paper is about the rational homology of Ebar(M,V). We study it by applying embedding calculus and orthogonal calculus to the bi-functor (M,V) |--> HQ /\Ebar(M,V)_+. Our main theorem states that if the dimension of V is more than twice the embedding dimension of M, the Taylor tower in the sense of orthogonal calculus (henceforward called ``the orthogonal tower'') of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at E^1. In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich's theorem on the formality of the little balls operad. We write explicit formulas for the layers in the orthogonal tower of the functor HQ /\Ebar(M,V)_+. The formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homotopy type of M. This, together with our rational splitting theorem, implies that under the above assumption on codimension, the rational homology groups of Ebar(M,V) are determined by the rational homotopy type of M. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Levi-Oliver/blo3 Title of paper: Discrete models for the $p$-local homotopy theory of compact Lie groups and $p$-compact groups Authors: Carles Broto, Ran Levi, Bob Oliver AMS Classification: Primary 55R35. Secondary 55R40, 57T10 Addresses of authors: Departament de Matem\`atiques Universitat Aut\`onoma de Barcelona E--08193 Bellaterra, Spain Department of Mathematical Sciences University of Aberdeen, Meston Building 339 Aberdeen AB24 3UE, U.K. LAGA, Institut Galil\'ee Av. J-B Cl\'ement 93430 Villetaneuse, France Abstract: We define and study a certain class of spaces which includes $p$-completed classifying spaces of compact Lie groups, classifying spaces of $p$-compact groups, and $p$-completed classifying spaces of certain locally finite discrete groups. These spaces are determined by fusion and linking systems over ``discrete $p$-toral groups'' --- extensions of $(\Z/p^\infty)^r$ by finite $p$-groups --- in the same way that classifying spaces of $p$-local finite groups as defined in \cite{BLO2} are determined by fusion and linking systems over finite $p$-groups. We call these structures ``$p$-local compact groups''. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Wilkerson/PiOneHopf The fundamental group of a $p$-compact group W. G. Dwyer and C. W. Wilkerson The notion of a $p$-compact group is a homotopy theoretic version of the geometric or analytic notion of compact Lie group, although the homotopy theory differs from the geometry is that there are parallel theories of $p$-compact groups, one for each prime number~$p$. A key feature of the theory of compact Lie groups is the relationship between centers and fundamental groups; these play off against one another, at least in the semisimple case, in that the center of the simply connected form is the fundamental group of the adjoint form. There are explicit ways to compute the center or fundamental group of a compact Lie group in terms of the normalizer of the maximal torus. For some time there has in fact been a corresponding formula for the center of $p$-compact groups, but in general the fundamental group has eluded analysis. The purpose of the present paper is to remedy this deficit. For any space $Y$, we let $\HZp_i(Y)$ denotes $\lim{}_n\HH_i(Y;\Z/p^n)$. Suppose that $X$ is a connected $p$-compact group, with maximal torus $T$ and torus normalizer $\NT$. It is known that the map $\pi_1(T)\to\pi_1(X)$ is surjective or equivalently that the map $\HZp_2(\BB T)\to\HZp_2(\BB X)$ is surjective. We prove the following statement. Main Theorem: If $X$ is a connected $p$-compact group, then the kernel of the map $\HZp_2 \BB T\to \HZp_2\BB\NT$ is the same as the kernel of the map $\HZp_2 \BB T\to \HZp_2\BB X$. Equivalently, the image of the map $\HZp_2\BB T\to \HZp_2(\BB\NT)$ is (naturally) isomorphic to $\pi_1X$. There is a proof of the corresponding statement for compact Lie groups which relies on the Feshbach double coset formula Our proof of the MainTheorem uses a transfer calculation that in practice amounts to a weak homological reflection of the double coset formula; we can get away with this because we have a splitting of $\HZp_2(\BB\NT)$. It is possible to derive from the MainTheorem a more explicit formula for $\pi_1X$; this formula is known for $p$~odd as a consequence of the classification theorem for $p$ odd. Our demonstration does not use the classification theorem. Let $W$ denote the Weyl group of $X$. If $p$ is odd, then $\pi_1X$ is naturally isomorphic to the module of coinvariants $\HH_0(W;\HZp_2(\BB T))$ . If $p=2$, then up to factors which do not contribute to $\pi_1X$, the normalizer of the torus in $X$ is derived by $\Ftwo$-completion from the normalizer $\NT_G$ of a maximal torus $T_G$ in a connected compact Lie group~$G$ . The image of the map $\HH_2(\BB T_G;\Z)\to\HH_2(\BB\NT_G;\Z)$ is isomorphic to $\pi_1G$ , and so by the MainTheorem the tensor product of this image with $\Ztwo$ is $\pi_1X$. This image can be computed from the marked reflection lattice $(\pi_1T_G, \{b_\sigma,\beta_\sigma\})$ corresponding to the root system of $G$ or, after tensoring with $\Ztwo$, from the marked complete reflection lattice $(\pi_1T,\{b_\sigma,\beta_\sigma\})$ associated to $X$ The upshot is that $\pi_1X$ is the quotient of $\pi_1T=\pi_2\BB T=\HZtwo_2\BB T$ by the $\Ztwo$--submodule generated by the elements $\{b_\sigma\}$. Another way to describe this calculation is the following. For each reflection $s_\alpha$ in the Weyl group~$W$, let $u_\alpha$ be a generator over $\Zp$ of the rank~1 submodule of $\pi_1T$ given by the image of $(1-s_\alpha)$. If $p$ is odd let $v_\alpha=u_\alpha$; if $p=2$, let $v_\alpha=u_\alpha$ or $u_\alpha/2$, according to whether the marking of $s_\alpha$ is trivial or non-trivial. Then $\pi_1X$ is the quotient of $\pi_1T$ by the $\Zp$-span of the elements~$v_\alpha$. See the upcoming even classification by Andersen and Grodal for more details. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Gillespie/quasi-coherent Title: A Quillen Approach to Derived Categories and Tensor Products Author: James Gillespie AMS Classification numbers: 55U35, 18G15, 18E30 4000 University Drive Penn State McKeesport McKeesport, PA 15132 Abstract: We put a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves over a quasi-compact and semi-separated scheme X. The approach generalizes and simplifies methods used by the author to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space. Indeed, much of the paper is dedicated to showing that in any Grothendieck category G, a nice enough class of objects, which we call a Kaplansky class, induces a model structure on the category Ch(G) of chain complexes. We also find simple conditions to put on the Kaplansky class which will guarantee that our model structure in monoidal. We see that the common model structures used in practice are all induced by such Kaplansky classes. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Naumann/qisoneu Niko Naumann Quasi-isogenies and Morava stabilizer groups For every prime $p$ and integer $n\ge 3$ we explicitly construct an abelian variety $A/\F_{p^n}$ of dimension $n$ such that for a suitable prime $l$ the group of quasi-isogenies of $A/\F_{p^n}$ of $l$-power degree is canonically a dense subgroup of the $n$-th Morava stabilizer group at $p$. We also give a variant of this result taking into account a polarization. This is motivated by a perceivable generalization of topological modular forms to more general topological automorphic forms. For this, we prove some results about approximation of local units in maximal orders which is of independent interest. For example, it gives a precise solution to the problem of extending automorphisms of the $p$-divisible group of a simple abelian variety over a finite field to quasi-isogenies of the abelian variety of degree divisible by as few primes as possible. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Ulrich-Wilkerson/uw06rev1 Field degrees and multiplicities for non-integral extensions Bernd Ulrich Clarence W. Wilkerson Department of Mathematics, Purdue University, West Lafayette, IN 47907 Department of Mathematics, Purdue University, West Lafayette, IN 47907 ulrich@math.purdue.edu cwilkers@purdue.edu Let $k$ be a field and $S = k[t_1,\hdots,t_d]$ a polynomial ring with variables $t_i$ of degree one. Consider a $k$-subalgebra $R$ generated by $m$ homogeneous elements $\{x_1,\hdots,x_m\}$. In general, if $x$ is a homogeneous element in a graded object, we denote its degree by $|x|$. {\bf Problem.} {\it Let $[S:R]$ denote the degree of the underlying fraction field extension. If $S$ is algebraic over $R$, calculate $[S:R]$ from the $\{|x_i|\}$ }. First, one has a form of Bezout's Theorem: \begin{thm}\label{BezoutsThm} If $S$ is integral over $R$, the following hold: \begin{enumerate} \item $[S:R]$ divides $\prod{|x_i|}$. \item If $m=d$, then $[S:R] = \prod{|x_i|}$. \end{enumerate} \end{thm} In this paper, we consider the case that $m = d$ and obtain a converse to part (b) above: \begin{thm}\label{MainTheorem} If $S$ is algebraic over $R$, $m=d$, and $[S:R] = \prod{|x_i|}$, then $S$ is integral over $R$ $($equivalently, $S$ is finitely generated as an $R$-module$)$. \end{thm} We also note that if $S$ is not integral over $R$, then $[S:R]$ need not even divide $\prod{|x_i|}$. Our proofs rely on reduction to the case of standard graded $k$-algebras. An interesting application of Theorem 1.2 is in the study of rings of invariants of finite groups acting on a polynomial ring: \begin{thm}\label{Invariants} Let $V$ be a $d$-dimensional vector space over the field $k$, $V^\#$ its $k$-dual, and $S = S[V^\#] = k[t_1,\hdots,t_d]$ the algebra of polynomial functions on $V$. Let $W \subset GL(V)$ be a finite group. There is an induced action on $S$. Then $S^W = R$ is a polynomial algebra over $k$ if and only if there exist homogeneous elements $\{x_1, \hdots,x_d\}$ of $R$ such that \begin{enumerate} \item $S$ is algebraic over $k[x_1,\hdots,x_d]$, and \item $|W| = \prod{|x_i|}$. \end{enumerate} \end{thm} 7. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/module_alg Title: Cohomology and deformation of module-algebras Author: Donald Yau Email: dyau@math.ohio-state.edu Abstract: An algebraic deformation theory of module-algebras over a bialgebra is constructed. The cases of module-coalgebras, comodule-algebras, and comodule-coalgebras are also considered. --------------- ----------------------------- I took a month off; sorry about the delay. There are 9 new papers this time, from Bergner, Chebolu-Christensen-Minac, DavisDaniel, Dugger-Isaksen, Fausk (2), GrayB, Hovey-Lockridge-Puninski, and Wuethrich. Mark Hovey New papers appearing on hopf between 8/4/06 and 10/6/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bergner/ReedyFib Title: A characterization of fibrant Segal categories Author: Julia E. Bergner AMS Classification: 55U35, 18G30 Author's address: Kansas State University 138 Cardwell Hall Manhattan, KS 66506 Abstract: In this note we prove that Reedy fibrant Segal categories are fibrant objects in the model category structure Secat_c. Combining this result with a previous one, we thus have that the fibrant objects are precisely the Reedy fibrant Segal categories. We also show that the analogous result holds for Segal categories which are fibrant in the projective model structure on simplicial spaces, considered as objects in the model structure Secat_f. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu-Christensen-Minac/ghost [Note: I had trouble with the dvi file of this paper. I expect to have it up by 10/7/06--Mark] 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: We study ghosts in the stable module category of a finite group. That is, we study maps between modular representations of finite groups which are invisible in Tate cohomology. We establish various sets of conditions which guarantee the existence of a non-trivial ghost out of a given representation. We then investigate the generating hypothesis which is the statement that there are no non-trivial ghosts between finite-dimensional representations. This is done by focusing on three quintessential examples: the cyclic $p$-groups (finite representation type), the Klein four group (domestic representation type), and the quaternion groups (tame representation type). In the examples where the generating hypothesis fails, we obtain bounds on the ghost number: the smallest integer $l$ such that the composition of any $l$ ghosts between finite-dimensional representations is trivial. In particular, we obtain bounds on the ghost numbers for all $2$-groups which have a cyclic subgroup of index $2$. Projective classes in the stable module category play a key role in getting these bounds. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/subhg3 Title: The homotopy orbit spectrum for profinite groups Author: Daniel G. Davis Abstract: Let G be a profinite group. We define an S[[G]]-module to be a G-spectrum X that satisfies certain conditions, and, given an S[[G]]-module X, we define the homotopy orbit spectrum X_{hG}. When G is countably based and X satisfies a certain finiteness condition, we construct a homotopy orbit spectral sequence whose E_2-term is the continuous homology of G with coefficients in the graded profinite Z[[G]]-module pi_*X. Let G_n be the extended Morava stabilizer group and let E_n be the Lubin-Tate spectrum. As an application of our theory, we show that the function spectrum F(E_n,L_{K(n)}(S^0)) is an S[[G_n]]-module with an associated homotopy orbit spectral sequence. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger-Isaksen/etdq Title: Etale homotopy and sums-of-squares formulas Authors: Daniel Dugger, Daniel C. Isaksen AMS classification number: 55P60, 15A63 Abstract: This paper uses a relative of BP-cohomology to prove a theorem in characteristic p algebra. Specifically, we obtain some new necessary conditions for the existence of sums-of-squares formulas over fields of characteristic p > 2. These conditions were previously known in characteristic zero by results of Davis. Our proof uses a generalized etale cohomology theory called etale BP2. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Fausk/ArtinBrauer Generalized Artin and Brauer induction for compact Lie groups Halvard Fausk Abstract: Let $G$ be a compact Lie group. We present two induction theorems for certain generalized $G$-equivariant cohomology theories. The theory applies to $G$-equivariant $K$-theory $K_G$, and to the Borel cohomology associated to any complex oriented cohomology theory. The coefficient ring of $K_G$ is the representation ring $R(G)$ of $G$. When $G$ is a finite group the induction theorems for $K_G$ coincide with the classical Artin and Brauer induction theorems for $R(G)$. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Fausk/Gspectra-Fausk Title: Equivariant homotopy theory for pro--spectra Author: Halvard Fausk Abstract. We extend the theory of equivariant orthogonal spectra from finite groups to profinite groups, and more generally from compact Lie groups to compact Hausdorff groups. The $G-$homotopy theory is ``pieced together'' from the $G/U-$homotopy theories for suitable quotient groups $G/U$ of $G$; a motivation is the way continuous group cohomology of a profinite group is built out of the cohomology of its finite quotient groups. In this category Postnikov towers are studied from a general perspective. We introduce pro$-G-$spectra and construct various model structures on them. A key property of the model structures is that pro-spectra are weakly equivalent to their Postnikov towers. We give a careful discussion of two version of a model structure with ``underlying weak equivalences''. One of the versions only make sense for pro$-$spectra. In the end we use the theory to study homotopy fixed points of pro$-G-$spectra. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/GrayB/fiber Filtering the fiber of the pinch map Brayton Gray This paper develops the similarity between the loops on an odd dimensional sphere and the fiber F of the pinch map from an odd dimensional mod p^r Moore space to the sphere, for p odd. In particular, a Hopf invariant map is defined and there is an EHP sequence up to a factor which is the loops on a bouquet of higher dimensiona Moore spaces. As a consequence we have two technical results about the mysterious connecting map from the double loops on the sphere to the loops on F. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey-Lockridge-Puninski/derived-gen-hyp Title: The generating hypothesis in the derived category of a ring. Authors: Mark Hovey, Keir Lockridge, and Gena Puninski Abstract: We show that a strong form (the fully faithful version) of the generating hypothesis, introduced by Freyd in algebraic topology, holds in the derived category of a ring R if and only if R is von Neumann regular. This extends results of the second author. We also characterize rings for which the original form (the faithful version) of the generating hypothesis holds in the derived category of R. These must be close to von Neumann regular in a precise sense, and, given any of a number of finiteness hypotheses, must be von Neumann regular. However, we construct an example of such a ring that is not von Neumann regular, and therefore does not satisfy the strong form of the generating hypothesis. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Wuethrich/thickenings_rev Title: Infinitesimal thickenings of Morava K-theories (revised version) Author: Samuel Wuethrich AMS classification number: 55P42, 55P43; 55U20, 55N22 Abstract: This is a revised version. A few points have been clarified and some typos have been corrected. 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. ----------------- ------------------------ There are 4 new papers this time, from Chebolu-Christensen-Minac, DavisDaniel, Stacey-Whitehouse, and Yagita. Mark Hovey New papers appearing on hopf between 10/6/06 and 11/5/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu-Christensen-Minac/GH-StMod TITLE: Groups which do not admit ghosts 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 ghost in the stable module category of a group G is a map between representations of G that is invisible to Tate cohomology. We show that the only non-trivial finite p-groups whose stable module categories have no non-trivial ghosts are the cyclic groups of order 2 and 3. We compare this to the situation in the derived category of a commutative ring. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisDaniel/siterplusg Title: The site R^+_G for a profinite group G Author: Daniel G. Davis AMS classification number: 55P42, 55U35, 18B25 Abstract: Let G be a non-finite profinite group and let G-Sets_{df} be the canonical site of finite discrete G-sets. Then the category R^+_G, defined by Devinatz and Hopkins, is the category obtained by considering G-Sets_{df} together with the profinite G-space G itself, with morphisms being continuous G-equivariant maps. We show that R^+_G is a site when equipped with the pretopology of epimorphic covers. Also, we explain why the associated topology on R^+_G is not subcanonical, and hence, not canonical. We note that, since R^+_G is a site, there is automatically a model category structure on the category of presheaves of spectra on the site. Finally, we point out that such presheaves of spectra are a nice way of organizing the data that is obtained by taking the homotopy fixed points of a continuous G-spectrum with respect to the open subgroups of G. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Stacey-Whitehouse/deloopv2 Title: Stable and Unstable Operations in mod p Cohomology Theories Authors: Andrew Stacey and Sarah Whitehouse AMS classification number: 55S25, 55P47 Other useful information: math.AT/0605471 Abstract: We consider operations between two multiplicative, complex orientable cohomology theories. Under suitable hypotheses, we construct a map from unstable to stable operations, left-inverse to the usual map from stable to unstable operations. In the main example, where the target theory is one of the Morava K-theories, this provides a simple and explicit description of a splitting arising from the Bousfield-Kuhn functor. This is an updated version of an earlier submission. The proof of proposition 3.2 has been corrected; other minor improvements have been made. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Yagita/abp Algebraic BP-theory and norm varieties Nobuaki Yagita Department of Mathematics, Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan Primary 14C15, 57T25; Secondary 55R35, 57T05 Let X be a smooth variety over a field k of characteristic zero. For a fixed prime p, the algebraic BP-theory ABP(X) is the algebraic version of the topological BP-theory. Given a nonzero symbol a in K_{n+1}^M (k)/p, the norm variety V_a is a variety such that a=0 in K_{n+1}^M (k(V_a))/p and V_a(C)=v_n. In this paper, we mainly study ABP(V_a) for p an odd prime. -------------There are 8 new papers this time, from Andersen-Grodal (the completion of the classification theorem for p-compact groups!), Benson-Chebolu-Christensen-Minac, BrownR-Sivera, Bousfield, Kadzisa-Mimura, Kuhn, Morel, and Snaith. Mark Hovey New papers appearing on hopf between 11/5/06 and 12/7/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Andersen-Grodal/2classification Title: The classification of 2-compact groups Authors: Kasper K. S. Andersen and Jesper Grodal Abstract: We prove that any connected 2-compact group is classified by its 2-adic root datum, and in particular the exotic 2-compact group DI(4), constructed by Dwyer-Wilkerson, is the only simple 2-compact group not arising as the 2-completion of a compact connected Lie group. Combined with our earlier work with Moeller and Viruel for p odd, this establishes the full classification of p-compact groups, stating that, up to isomorphism, there is a one-to-one correspondence between connected p-compact groups and root data over the p-adic integers. As a consequence we prove the maximal torus conjecture, giving a one-to-one correspondence between compact Lie groups and finite loop spaces admitting a maximal torus. Our proof is a general induction on the dimension of the group, which works for all primes. It refines the Andersen-Grodal-Moeller-Viruel methods to incorporate the theory of root data over the p-adic integers, as developed by Dwyer-Wilkerson and the authors, and we show that certain occurring obstructions vanish, by relating them to obstruction groups calculated by Jackowski-McClure-Oliver in the early 1990s. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Benson-Chebolu-Christensen-Minac/GH-pgroup-new Title: Freyd's generating hypothesis for the stable module category of a $p$-group Authors: David J. Benson, Sunil K. Chebolu, J. Daniel Christensen, and Jan Minac. Abstract: Freyd's generating hypothesis, interpreted in the stable module category of a finite $p$-group $G$, is the statement that a map between finite-dimensional $kG$-modules factors through a projective if the induced map on Tate cohomology is trivial. We show that Freyd's generating hypothesis holds for a non-trivial $p$-group $G$ if and only if $G$ is either $\mathbb{Z}/2$ or $\mathbb{Z}/3$. We also give various conditions which are equivalent to the generating hypothesis. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Sivera/normalisation Title: Normalisation for the fundamental crossed complex of a simplicial set Author(s): Ronald Brown, Rafael Sivera R. Brown University of Wales, Bangor, Dean St., Bangor, Gwynedd LL57 1UT, U.K. R. Sivera, Departamento de Geometria y Topologia, Universitat de Valencia, 46100 Burjassot, Valencia, Spain Abstract: Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with some historical background, and give a survey of the required basic facts on crossed complexes, such as the monoidal closed structure. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Bousfield/Klocal On the 2-adic K-localizations of H-spaces A.K. Bousfield Department of Mathematics University of Illinois at Chicago Chicago, IL 60607 We determine the 2-adic K-localizations for a large class of H-spaces and related spaces. As in the odd-primary case, these localizations are expressed as fibers of maps between specified infinite loop spaces, allowing us to approach the 2-primary v1-periodic homotopy groups of our spaces. The present v1-periodic results have been applied very successfully to simply-connected compact Lie groups by Davis, using knowledge of the complex, real, and quaternionic representations of the groups. We also functorially determine the united 2-adic K-cohomology algebras (including the 2-adic KO-cohomology algebras) for all simply-connected compact Lie groups in terms of their representation theories, and we show the existence of spaces realizing a wide class of united 2-adic K-cohomology algebras with specified operations. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Kadzisa-Mimura/mbflsc1 Authors: Hiroyuki Kadzisa, Mamoru Mimura Title: Morse-Bott functions and the Lusternik-Schnirelmann category, I The Lusternik-Schnirelmann category of a space is a homotopy invariant. Cone-decompositions are used to give an upper bound for Lusternik-Schnirelmann categories of topological spaces. The purpose of this paper is to show how to construct cone-decompositions of manifolds by using functions of class C^1 and their gradient flows, and to apply the result to some homogeneous spaces to determine their Lusternik-Schnirelmann categories. In particular, the Morse-Bott functions on the Stiefel manifolds considered by Frankel are effectively used for constructing all the cone-decompositions in this paper. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/primitives2 Title: Primitives and central detection numbers in group cohomology Author: Nicholas J. Kuhn Address: Department of Mathematics, University of Virginia, Charlottesville, VA 22903 abstract: Henn, Lannes, and Schwartz have introduced two invariants, d_0(G) and d_1(G), of the mod p cohomology of a finite group G such that H^*(G) is detected and determined by H^d(C_G(V)) for d no bigger than d_0(G) and d_1(G), with V < G p-elementary abelian. We study how to calculate these invariants. We define a number e(G) that measures the image of the restriction of H^*(G) to its maximal central p-elementary abelian subgroup. Using Benson--Carlson duality, we show that when $G$ has a p-central Sylow subgroup P, d_0(G) = d_0(P) = e(P), and a similar exact formula holds for d_1(G). In general, we show that d_0(G) is bounded above by the maximum of the e(C_G(V))'s, if Benson's Regularity Conjecture holds. In particular, this holds for all groups such that the p--rank of G minus the depth of H^*(G) is at most 2. When we look at examples with p=2, we learn that d_0(G) is at most 7 for all groups with 2--Sylow subgroup of order up to 64, unless the Sylow subgroup is isomorphic to that of either Sz(8) (and d_0(G) = 9) or SU(3,4) (and d_0(G)=14). Enroute we recover and strengthen theorems of Adem and Karagueuzian on essential cohomology, and Green on depth essential cohomology, and prove theorems about the structure of cohomology primitives associated to central extensions. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Morel/A1homotopy A^1-algebraic topology over a field Fabien Morel Mathematisches Institut der Universität München Theresienstr. 39 D-80333 München Text of Abstract: In this work we prove some basic results in the context of A1-homotopy theory of smooth schemes over a field k : the analogue of the Brouwer degree, the Hurewicz theorem, the theory of A1-coverings and its relationship to the fundamental A1-homotopy sheaf, and some fundamental computations involving unramified Milnor-Witt K-theory like the fundamental A1-homotopy sheaves of P^n and SL_n . 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Snaith/UTTArf Title: Upper triangular technology and the Arf-Kervaire invariant Author: Victor Snaith address: Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, England Abstract. This paper introduces the upper triangular technology (UTT) into classical homotopy theory. This is a new and easy to use method to calculate the effect of the left unit map in 2-adic connective K-theory; the map which is the basis for operations in bu-theory. By way of application, UTT is used to give a new, very simple proof of a conjecture of Barratt- Jones-Mahowald, which rephrases K-theoretically the existence of framed manifolds of Arf-Kervaire invariant one. ----------------------------