--------------------------------- Happy New Year! This is the beginning of the 10th year I have been doing this. 10 new papers this time, from Bubenik, ChornyB (2), Gillespie, Hovey, Jardine, Lueck, Mitchell, Pengelley-Williams, and Vavpatic-Viruel. Mark Hovey New papers appearing on hopf between 11/25/03 and 1/05/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bubenik/fsi Title: Free and semi-inert cell attachments Author: Peter Bubenik Author's e-mail address: peter.bubenik---epfl.ch AMS classification number: 55P35 (Primary) 16E45 (Secondary) arXive submission number: math.AT/0312387 Abstract: Let $Y$ be the space obtained by attaching a finite-type wedge of cells to a simply-connected, finite-type CW-complex. We introduce the free and semi-inert conditions on the attaching map which broadly generalize the previously studied inert condition. Under these conditions we determine $H_*(\Omega Y;R)$ as an $R$-module and as an $R$-algebra respectively. Under a further condition we show that $H_*(\Omega Y;R)$ is generated by Hurewicz images. As an example we study an infinite family of spaces constructed using only semi-inert cell attachments. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/ChornyB/Localization1 Title: Localization with respect to a class of maps I - Equivariant localization of diagrams of spaces Author: Boris Chorny Author's e-mail address: bchorny2---uwo.ca Abstract: Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions). In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model categories and, more important, will allow for a localization with respect to a class of maps (satisfying some restrictive conditions). We illustrate our technique by applying it to the equivariant model category of diagrams of spaces. This model category is not cofibrantly generated. We give conditions on a class of maps which ensure the existence of the localization functor; these conditions are satisfied by any set of maps and by the classes of maps which induce ordinary localizations on the generalized fixed-points sets. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/ChornyB/Localization2 Title: Localization with respect to a class of maps II - Equivariant cellularization and its application Author: Boris Chorny Author's e-mail address: bchorny2---uwo.ca Abstract: We present an example of a homotopical localization functor which is not a localization with respect to any set of maps. Our example arises from equivariant homotopy theory. The technique of equivariant cellularization is developed and applied to the proof of the main result. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Gillespie/sheafproblem Title: The flat model structure on Ch(O) Author: James Gillespie Email: jrg21---psu.edu Abstract: Let Ch(O) be the category of chain complexes of O-modules on a topological space T (where O is a sheaf of rings on T ). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on Ch(O). As a corollary, we have a general framework for doing homological algebra in the category O-MOD of O-modules. I.e., we have a natural way to define the functors Ext and Tor in O-MOD. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/morava-E-operations Operations and co-operations in Morava $E$-theory Mark Hovey Wesleyan University mhovey---wesleyan.edu In this paper, we revisit the calculations of the operations and co-operations in Morava E-theory. Recall that the co-operations are the continuous functions from a profinite group G that is a version of the Morava stabilizer group to E_*. The operations are the completed twisted group ring E_*[[G]]. These results have appeared in the literature before. The advantage of this paper is that it is self-contained, works out all the details that are usually skipped over, and uses a new approach, not directly dependent on Morava's Annals paper on comodules, that the author finds fairly simple and elegant. Most of all, though, the author wrote this paper because he was unable to understand the proofs in the literature. He hopes it will be useful for people in the same unhappy situation. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/cat5 Categorical homotopy theory J.F. Jardine This paper is an exposition and extension of the ideas and methods of Cisinksi, set at the level of A-presheaves on a small Grothendieck site, where A is an arbitrary test category in the sense of Grothendieck. The model structures for the category of simplicial presheaves and all of its localizations can be modelled by A-presheaves in the sense that there is a corresponding model structure for A-presheaves with an equivalent homotopy category. The theory specializes, for example, to the homotopy theories of cubical sets, cubical presheaves, and gives a cubical model for motivic homotopy theory. The applications of Cisinski's ideas are explained in some detail for cubical sets. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Lueck/lueck_classifyingspaces1203 Title: Survey on Classifying Spaces for Families of Subgroups Author: Wolfgang Lueck AMS Classification numbers: 55R35, 57S99, 20F65, 18G99 Address: Mathematisches Institut der Westfaelischen Wilhelms Universitaet Einsteinstr. 62 48149 Muenster Germany Abstract: We define for a topological group G and a family of subgroups F two versions for the classifying space for the family F, the G-CW-version E_F(G) and the numerable G-space version J_F(G). They agree if G is discrete, or if G is a Lie group and each element in F compact, or if G is totally disconnected and F is the family of compact subgroups or of compact open subgroups. We discuss special geometric models for these spaces for the family of compact open groups in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also discuss the relevance of these spaces for the Baum-Connes Conjecture about the topological K-theory of the reduced group C^*-algebra, for the Farrell-Jones Conjecture about the algebraic K- and L-theory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Mitchell/sw Author: Stephen A. Mitchell Title: Stiefel-Whitney classes, united K-theory and real embeddings of number rings e-mail: mitchell---math.washington.edu We study the relations among the Stiefel-Whitney classes associated to the real embeddings of a number ring. Our results depend on a computation of the real and self-conjugate K-theory of the algebraic K-theory spectrum of the number ring. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Pengelley-Williams/subsmalg Global Structure of the mod 2 Symmetric Algebra over the Steenrod algebra. David J. Pengelley (davidp---nmsu.edu) Frank Williams (frank---nmsu.edu) The algebra S of symmetric invariants over the field with two elements is an unstable algebra over the Steenrod algebra A and is isomorphic to the mod two cohomology of BO, the classifying space for vector bundles. We provide a minimal presentation for S in the category of unstable A-algebras, i.e., a minimal set of generators and a minimal set of relations. From this we produce minimal presentations for various unstable A-algebras associated with the cohomology of related spaces, such as the BO(2^n - 1) that classify finite dimensional vector bundles, and the connected covers of BO. The presentations then show that certain of these unstable A-algebras coalesce to produce the mod 2 Dickson algebras, and we speculate about possible related topological realizability. Our methods also produce a related simple A-module presentation of the cohomology of infinite-dimensional real projective space, with a filtration having well-known filtered quotients. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Vavpetic-Viruel/PU On the mod p cohomology of BPU(p) Ales Vavpetic Fakulteta za matematiko in fiziko Univerza v Ljubljani Jadranska 19 SI-1111 Ljubljana Slovenia ales.vavpetic---fmf.uni-lj.si Antonio Viruel Dpto de Algebra, Geometria y Topologia Universidad de Malaga Apdo correos 59 E29080 Malaga Spain viruel---agt.cie.uma.es AMS Classification numbers: 55R35, 55P15 ABSTRACT: We study the mod p cohomology of the classifying space of the projective unitary group PU(p). We first proof that old conjectures due to J.F. Adams, and Kono and Yagita about the structure of the mod p cohomology of classifying space of connected compact Lie groups held in the case of PU(p). Finally, we proof that the classifying space of the projective unitary group PU(p) is determined by its mod p cohomology as an unstable algebra over the Steenrod algebra for p>3, completing previous works of Dwyer, Miller, Wilkerson at prime 2 and Broto, Viruel at prime 3. ---------------- 10 new papers this time, from BrownR-Kamps-Porter, Chalupnik, ChornyB, Clarke-Crossley-Whitehouse, GrayB, Hikida, Hornbostel, IsaksenD, Lupton-SmithSB, and Porter. Mark Hovey New papers appearing on hopf between 1/5/04 and 2/1/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR-Kamps-Porter/vkt7 TITLE: A van Kampen theorem for the homotopy double groupoid of a Hausdorff space AUTHORS: R. Brown, K.H. Kamps, T.Porter EMAILS: r.brown---bangor.ac.uk, heiner.kamps---fernuni-hagen.de, t.porter---bangor.ac.uk ADDRESSES: Mathematics Department, Dean St., Bangor, Gwynedd LL57 1UT, UK. Fachbereich Mathematik, FernUniversit\"at in Hagen, D-58084 Hagen, Germany Mathematics Department, Dean St., Bangor, Gwynedd LL57 1UT, UK. ABSTRACT: We show that the homotopy double groupoid of a Hausdorff space defined by the authors in a previous paper satisfies a version of the van Kampen theorem, and so is a suitable tool for non abelian, 2-dimensional, local-to-global problems. The methods are analogous to those developed by Brown and Higgins for similar theorems for other higher homotopy groupoids. There is a detailed discussion of commutative cubes in a double category with connections, and a proof of the key result that any composition of commutative cubes is commutative. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/koeks Title of Paper: Koszul duality and extensions of exponential functors Author: Marcin Chalupnik Email: mchal---mimuw.edu.pl Abstract: We study Koszul duality in the category of strict polynomial functors. We compute Koszul duals for various functors and apply these results to the problem of calculating Ext--groups between exponential functors. The main application is a full description of the Ext--groups between twisted exterior and divided powers and between twisted symmetric and divided powers. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/ChornyB/prospaces Title: A generalization of Quillen's small object argument Author(s): Boris Chorny Author's e-mail address: bchorny2---uwo.ca Abstract: We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen's small object argument). The necessity of such a generalization arose with the appearance of several important examples of model categories which were proven to be non-cofibrantly generated. Our current approach allows for the construction of functorial factorizations and localizations in the equivariant model category on diagrams of spaces and in two different model structures on the category of pro-spaces. The examples above suggest a natural extension of the framework of cofibrantly generated model categories. We introduce the concept of a class-cofibrantly generated model category, which is a model category generated by classes of cofibrations and trivial cofibrations satisfying some reasonable assumptions. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Clarke-Crossley-Whitehouse/ccwops Title: Algebras of operations in K-theory Authors: Francis Clarke, Martin Crossley, Sarah Whitehouse Authors' e-mail addresses: F.Clarke---Swansea.ac.uk, M.D.Crossley---Swansea.ac.uk, S.Whitehouse---sheffield.ac.uk Abstract: We describe explicitly the algebras of degree zero operations in connective and periodic p-local complex K-theory. Operations are written uniquely in terms of certain infinite linear combinations of Adams operations, and we give formulas for the product and coproduct structure maps. It is shown that these rings of operations are not Noetherian. Versions of the results are provided for the Adams summand and for real K-theory. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/GrayB/decomposition On Decompositions in Homotopy Theory Brayton Gray 55P35, 5P30, 55P45 Dept. of Mathematics, Statistics and Computer Science (M/C 249) University of Illinois at Chicago 851 South Morgan Street Chicago, IL 60607-7045 brayton---uic.edu We first describe Krull-Schmidt theorems decomposing $H$ spaces and simply connected co-$H$ spaces into atomic factors in the category of pointed nilpotent $p$-complete spaces of finite type. We use this to construct a 1-1 correspondence between homotopy types of atomic $H$ spaces and homotopy types of atomic co-$H$ spaces, and construct a split fibration which connects them and illuminates the decomposition. Various properties of these constructions are analyzed. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Hikida/Acycle Title: Some acyclic relations in the lambda algebra Author: Mizuho Hikida Author's e-mail address: hikida---bus.hiroshima-pu.ac.jp Author's mailing address: Hiroshima Prefectural University, Shobara-shi, 727-0017, Japan AMS Classification numbers: 55Q40 Abstract: We consider the relations omega gamma = 0 in Lambda, and show that if omega alpha = 0 then alpha = gamma beta for some beta. These relations give the acyclic chain complex Lambda -gamma-> Lambda -omega-> Lambda . We consider various cases, e.g. omega = lambda_n and gamma = lambda_{2n+1}. Especially, we consider the case omega = w_n = d lambda_n for n=2^{e+r} + 2^{e}-1, where gamma = (h_{e+r})^r. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Hornbostel/Motchrom2 Author : Jens Hornbostel Author's e-mail address : jens.hornbostel---mathematik.uni-regensburg.de Author's mailing address: Universitaet Regensburg, NWF I - Mathematik, D- 93047 Regensburg, Germany Abstract: We construct a motivic version of the chromatic filtration and the chromatic spectral sequence. This should be used to study the stable ${\bf A}^1$-homotopy groups of the motivic sphere spectrum. We also study different localization techniques both for classical and motivic spectra. This is an improved version of a preprint posted in June 2003. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/IsaksenD/flasque Title: Flasque model structures for simplicial presheaves Author: Daniel C. Isaksen Author's e-mail address: isaksen---math.wayne.edu Abstract: By now it is well known that there are two useful (objectwise or local) families of model structures on presheaves: the injective and projective. In fact, there is at least one more: the flasque. For some purposes, both the projective and the injective structure run into technical and annoying (but surmountable) difficulties for different reasons. The flasque model structure, which possesses a combination of the convenient properties of both structures, sometimes avoids these difficulties. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Lupton-SmithSB/GseqII Title: Rationalized Evaluation Subgroups of a Map II: Quillen Models and Adjoint Maps Authors: Gregory Lupton and Samuel Bruce Smith Authors' e-mail addresses: G.Lupton---csuohio.edu and smith---sju.edu Authors' mailing addresses: Department of Mathematics, Cleveland State University, 2121 Euclid Ave., Cleveland OH 44115 and Department of Mathematics, Saint Joseph's University, Philadelphia, PA 19131 AMS classification number: 55P62, 55Q52 Other useful information: 33 pages; http://arXiv.org/abs/math.AT/0401178 Abstract: Let w: Map(X,Y;f) -> Y denote a general evaluation fibration. Working in the setting of rational homotopy theory via differential graded Lie algebras, we identify the long exact sequence induced on rational homotopy groups by w in terms of (generalized) derivation spaces and adjoint maps. As a consequence, we obtain a unified description of the rational homotopy theory of function spaces, at the level of rational homotopy groups, in terms of derivations of Quillen models and adjoints. In particular, as a natural extension of a result of Tanre, we identify the rationalization of the evaluation subgroups of a map f: X -> Y in this setting. As applications, we consider a generalization of a question of Gottlieb, within the context of rational homotopy theory. We also identify the rationalization of the G-sequence of f and make explicit computations of the homology of this sequence. In a separate result of independent interest, we give an explicit Quillen minimal model of a product AxX, in the case in which A is a rational co-H-space. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Porter/s-catsv2 Title: S-categories, S-groupoids, Segal categories and quasicategories Author: Timothy Porter Author's e-mail address: t.porter---bangor.ac.uk Author's mailing address: Mathematics Department, School of Informatics, University of Wales Bangor, Bangor, Gwynedd, LL57 1UT, United Kingdom. Included ps or eps files: 5 epsi files AMS classification number: 55U35 Other useful information: arXive submission number: math.AT/0401274 Abstract: These notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguna, the Canary Islands, in September, 2003. They aim (i) to revisit some oldish material on abstract homotopy and simplicially enriched categories, that seems to be being used in today's resurgence of interest in the area and to try to view it in a new light, or perhaps from new directions;(ii) to introduce Segal categories and various other tools used by the Nice-Toulouse group of abstract homotopy theorists and link them into some of the older ideas;(iii) to introduce Joyal's quasicategories, and show how that theory links in with some old ideas of Boardman and Vogt, Dwyer and Kan, and Cordier and Porter; and finally to ask lots of questions of myself and of the reader. ----------------15 new papers this month, from Aguilar-Prieto (2), Arkowitz-Brown, Arkowitz-Stanley-Strom, Arkowitz-Strom, Ausoni, DJGreen, Elmendorf-Mandell, Hovey, Jardine-Luo, Marzantowicz-Prieto, McClure-SmithJH (2), and SchwartzL (2). Mark Hovey New papers appearing on hopf between 2/1/04 and 3/1/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Aguilar-Prieto/transrami-1 Transfers for ramified coverings in homology and cohomology Primary 55R12, 57M12; Secondary 55Q05, 55R35, 57M10 Transfer, ramified covering maps, classifying spaces Marcelo A. Aguilar and Carlos Prieto Abstract Making use of a modified version, due to McCord, of the Dold-Thom construction of ordinary homology, we give a simple topological definition of a transfer for ramified covering maps in homology with arbitrary coefficients. The transfer is induced by a suitable map between topological groups. We also define a cohomology transfer which is dual to the homology transfer. This duality allows us to show that our homology transfer coincides with the one given by L. Smith. With our definition of the homology transfer we can give simpler proofs of the properties of the known transfer and of some new ones. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Aguilar-Prieto/transrami-2 A classification of cohomology transfers for ramified coverings Primary 55R12, 57M12; Secondary 55Q05, 55R35, 57M10 Transfer, covering maps, ramified covering maps, classifying spaces Marcelo A. Aguilar and Carlos Prieto cprieto---math.unam.mx, marcelo---math.unam.mx Abstract We construct a cohomology transfer for $n$-fold ramified covering maps. Then, we define a very general concept of transfer for ramified covering maps and prove a classification theorem for these transfers. This generalizes Roush's classification of transfers for $n$-fold ordinary covering maps. We characterize those representable cofunctors which admit a family of transfers for ramified covering maps that have two naturality properties, as well as normalization and stability. This is analogous to Roush's characterization theorem for the case of ordinary covering maps. Finally, we classify these families of transfers and construct some examples. In particular, we extend the determinant function in $\GL(k,\C)$ to a transfer. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Brown/Lef The Lefschetz-Hopf Theorem and Axioms for the Lefschetz Number Martin Arkowitz and Robert F. Brown martin.arkowitz---dartmouth.edu rfb---math.ucla.edu 55M20 The reduced Lefschetz number, that is, the Lefschetz number minus 1, is proved to be the unique integer-valued function L on selfmaps of compact polyhedra which is constant on homotopy classes such that (1) L(fg) = L(gf), for f:X --->Y and g:Y --->X; (2) if (f_1, f_2, f_3) is a map of a cofiber sequence into itself, then L(f_2) = L(f_1) + L(f_3); (3) L(f) = - (degree(p_1 f e_1) + ... + degree(p_k f e_k)), where f is a map of a wedge of k circles, e_r is the inclusion of a circle into the rth summand and p_r is the projection onto the rth summand. If f:X --->X is a selfmap of a polyhedron and I(f) is the fixed point index of f on all of X, then we show that I minus 1 satisfies the above axioms. This gives a new proof of the Normalization Theorem: If f:X --->X is a selfmap of a polyhedron, then I(f) equals the Lefschetz number of f. This result is equivalent to the Lefschetz-Hopf Theorem: If f: X --->X is a selfmap of a finite simplicial complex with a finite number of fixed points, each lying in a maximal simplex, then the Lefschetz number of f is the sum of the indices of all the fixed points of f. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Stanley-Strom/Cl&Cat The Cone Length and Category of Maps: Pushouts, Products and Fibrations Martin Arkowitz, Donald Stanley and Jeffrey Strom martin.arkowitz---dartmouth.edu stanley---math.uregina.ca Jeffrey.Strom---wmich.edu 55M30; 55P99, 55R05 For any collection of spaces A, we investigate two non-negative integer homotopy invariants of maps: l_A(f), the A-cone length of f, and L_A(f), the A-category of f. When A is the collection of all spaces, these are the cone length and category of f, respectively, both of which have been studied previously. The following results have been obtained: (1) For a map of one homotopy pushout diagram into another, we derive an upper bound for I_A and L_A of the induced map of homotopy pushouts in terms of I_A and L_A of the other maps. This has many applications including an inequality for I_A and L_A of the maps in a mapping of one mapping cone sequence into another. (2) We establish an upper bound for I_A and L_A of the product of two maps in terms of I_A and L_A of the given maps and the A-cone length of their domains. (3) We study our invariants in a pullback square and obtain as a consequence an upper bound for the A-cone length and A-category of the total space of a fibration in terms of the A-cone length and A-category of the base and fiber. We conclude with several remarks, examples and open questions. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Arkowitz-Strom/Secat The Sectional Category of a Map Martin Arkowitz and Jeffrey Strom martin.arkowitz---dartmouth.edu Jeffrey.Strom---wmich.edu 55M30; 55P99 We study a generalization of the Svarc genus of a fiber map. For an arbitrary collection E of spaces and a map f:X--->Y, we define a numerical invariant, the E-sectional category of f, in terms of open covers of Y. We obtain several basic properties of E-sectional category, including those dealing with homotopy domination and homotopy pushouts. We then give three simple properties which characterize the E-sectional category. In the final section we obtain inequalities for the E-sectional category of a composition and inequalities relating the E-sectional category to the Fadell-Husseini category of a map and the Clapp-Puppe category of a map. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Ausoni/thhku-ausoni Author: Christian Ausoni Title: Topological Hochschild Homology of connective complex K-theory Email: ausoni---math.uni-bonn.de Abstract: Let ku be the connective complex K-theory spectrum, completed at an odd prime p. We present a computation of the mod (p,v_1) homotopy algebra of the topological Hochschild homology spectrum of ku. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/DJGreen/essCM Title: The essential ideal is a Cohen-Macaulay module Author: David J. Green Institution: University of Wuppertal, Germany MSC 2000: Primary 20J06; Secondary 13C14 arXiv: math.GR/0402434 Abstract: Let G be a finite p-group which does not contain a rank two elementary abelian p-group as a direct factor. Then the ideal of essential classes in the mod-p cohomology ring of G is a Cohen-Macaulay module whose Krull dimension is the p-rank of the centre of G. This basically answers in the affirmative a question posed by J. F. Carlson. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Elmendorf-Mandell/RMA Title: Rings, modules, and algebras in infinite loop space theory Authors: Anthony D. Elmendorf and Michael A. Mandell Email: aelmendo---math.purdue.edu Email2: mandell---math.uchicago.edu Abstract: 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, 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 Smith's category of symmetric spectra, with an intermediate stop at a new category that may be of interest in its own right, whose objects we call symmetric functors. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Hovey/morava-E-SS Some spectral sequences in Morava E-theory by Mark Hovey mhovey---wesleyan.edu The Morava E-theory of X is the homotopy of the K(n)-localization of E smash X, where E is the completed and extended version of E(n) on which the Morava stabilizer group acts. Because K(n)-localization is not smashing, Morava E-theory is not a homology theory; it is exact, but does not preserve coproducts. Nevertheless, it is the most important theory to use in understanding the K(n)-local stable homotopy category; for example, X is small in the K(n)-local stable homotopy category if and only if the Morava E-theory of X is degreewise finite. In the paper at hand, we show how the usual spectral sequences used with homology theories work for Morava E-theory. The most interesting such spectral sequence is a spectral sequence that converges to the Morava E-theory of an infinite coproduct. The E_2-term involves the derived functors of direct sum in the category of "L-complete" E_*-modules. There are (n-1) such derived functors (n if we try to compute filtered homotopy colimits). Thus, Morava E-theory is "n derived functors away from being a homology theory". In particular, when n=1, we see that p-completed K-theory actually commutes with coproducts, in the category of Ext-p-complete abelian groups. It follows that K(1)-local homotopy also commutes with coproducts as a functor to Ext-p-complete abelian groups. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine-Luo/cocycles6 Title: Higher order principal bundles Authors: J.F. Jardine and Z. Luo AMS Classification numbers: 14F05, 55R65, 14A20 E-mail: jardine---uwo.ca E-mail: zluo---uwo.ca Abstract: We define torsors for sheaves of simplicial groups and sheaves of groupoids enriched in simplicial sets, and give classification results for these torsors in terms of the homotopy theory of simplicial sheaves. The proofs of the classification results use a new, general approach to cocycles taking values in simplicial sheaves. We prove a homotopy classification result for gerbes locally isomorphic to a fixed sheaf of groups. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Marzantowicz-Prieto/decompAMS Computation of the equivariant $1$-stem by a decomposition of equivariant stable homotopy classes W. Marzantowicz and C. Prieto marzan---main.amu.edu.pl, cprieto---math.unam.mx Primary 54H25; Secondary 55M20, 55M25, 55N91 Equivariant stable homotopy groups, equivariant stems, equivariant fixed point index and fixed point transfer Abstract For any compact Lie group $G$, we give a decomposition of the group $\{X,Y\}_G^k$ of (unpointed) stable $G$-homotopy classes as a direct sum of subgroups of fixed orbit types. This is done by interpreting the $G$-homotopy classes in terms of the generalized fixed point transfer and making use of conormal maps. Finally, we give a full computation of the first equivariant (stable) stem for $G$, $\pi\ho{G\,\rm{st}}_1=\{*,*\}_G\ho{-1}$. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/McClure-SmithJH/McClure-Smith_survey Operads and cosimplicial objects: an introduction. James E. McClure and Jeffrey H. Smith AMS Classification Numbers 18D50; 55P48 Submitted to arXiv: math.QA/0402117 mcclure---math.purdue.edu jhs---math.purdue.edu This paper is an introduction to a series of papers in which we have given combinatorial models for certain important operads (including A-infinity and E-infinity operads, the little n-cubes operads, and the framed little disks operad) and combinatorial conditions for them to act on a given space or chain complex. The paper does not assume any prior knowledge of operads---Sections 2, 6 and 9, which can be read independently, are an introduction to the theory of operads. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/McClure-SmithJH/mcclure-smith3 Cosimplicial Objects and little $n$-cubes. I. James E. McClure and Jeffrey H. Smith AMS Classification Numbers 18D50; 55P48 Submitted to arXiv: math.QA/0211368 mcclure---math.purdue.edu jhs---math.purdue.edu This is a revised version of a paper previously posted on Hopf. The main theorem says that if a cosimplicial space has a certain kind of combinatorial structure (called a $\Xi^n$-structure) then its total space has an action of an operad $\cal D_n$ which is weakly equivalent to the little $n$-cubes operad. There are three new sections in the revised version: Section 10 shows that $\Xi^2$-structures are essentially the same thing as operads with multiplication, Section 11 shows that the operad $\cal D_n$ acts on $n$-fold loop spaces, and Section 15 shows that the main results are still valid for the homotopy-invariant version of Tot. 14. http://hopf.math.purdue.edu/cgi-bin/generate?/SchwartzL/Erd L'alg`ebre de Steenrod, modules injectifs, et foncteurs polynomiaux Lionel Schwartz These notes come from talks made in Nantes in December 2001 at a session " Etat de la Recherche " of the french mathematic society. They are an introduction to the algebraic aspects of the theory of unstable modules over the Steenrod algebra and to the relations of the related category to functor categories. The Steenrod algebra is introduced using the additive group scheme. Reduced injective modules are described follwing the point of view of Campbell and Selick. Most of the material is classical, however there are new (at least in an accessible form) remarks concerning the odd prime case, as well as some new proofs of classical results, in particular the structure of Miller's algebra. The Adem relations are discussed following Bullett and MacDonald. 15. http://hopf.math.purdue.edu/cgi-bin/generate?/SchwartzL/Grot Sur l'anneau de Grothendieck de la cat'egorie des modules instables Lionel Schwartz 19 octobre 2003 R'esum'e Dans cet article on calcule l'anneau de Grothendieck de la cat'egorie des modules instables de type fini et de la cat'egorie obtenue par quotient par la sous-cat'egorie des modules instables nilpotents. Les r'esultats principaux montrent que la s'erie de Poincar'e, ou un substitut ad'equat d'eterminent ces groupes. On peut de plus caract'eriser les s'eries repr'esentant un module instable. Ce type de sujet a d'ej`a 'et'e abord'e par N. Kuhn dans [K2] d'un point de vue de th'eorie des repr'esentations, on retrouve ses r'esultats au long du d'eveloppement. ----------------------------------------- 4 new papers this month, from Badzioch-Chung-Voronov, Broto-Castellana-Grodal-Levi-Oliver, Christensen-Isaksen, and KrauseH. Mark Hovey New papers appearing on hopf between 3/1/04 and 4/5/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Badzioch-Chung-Voronov/bcv Title: Yet another delooping machine Authors: Bernard Badzioch, Kuerak Chung, and Alexander A. Voronov Author's e-mail address: voronov---math.umn.edu Authors' mailing address: School of Mathematics, University of Minnesota, Minneapolis, MN 55455 Included ps or eps files: mor.eps AMS classification number: 55P48 (Primary); 18C10 (Secondary) ArXiv submission number: math.AT/0403098 Abstract: We suggest a new delooping machine, which is based on recognizing an n-fold loop space by a collection of operations acting on it, like the traditional delooping machines of Stasheff, May, Boardman-Vogt, Segal, and Bousfield. Unlike in the traditional delooping machines, which carefully select a nice space of such operations, we consider all natural operations on n-fold loop spaces, resulting in the algebraic theory Map (V_. S^n, V_. S^n). The advantage of this new approach is that the delooping machine is universal in a certain sense, the proof of the recognition principle is more conceptual, works the same way for all values of n, and does not need the test space to be connected. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Castellana-Grodal-Levi-Oliver/bcglo1 Subgroup families controlling $p$-local finite groups by C. Broto, N. Castellana, J. Grodal, R. Levi, 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 examine which subgroups control this structure. More precisely, we prove that the question of whether an abstract fusion system $F$ over a finite $p$-group $S$ is saturated can be determined by just looking at smaller classes of subgroups of $S$. We also prove that the homotopy type of the classifying space of a given $p$-local finite group is independent of the family of subgroups used to define it, in the sense that it remains unchanged when that family ranges from the set of $F$-centric $F$-radical subgroups (at a minimum) to the set of $F$-quasicentric subgroups (at a maximum). Finally, we look at constrained fusion systems, analogous to $p$-constrained finite groups, and prove that they in fact all arise from groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Christensen-Isaksen/duality Duality and Pro-spectra J. Daniel Christensen and Daniel C. Isaksen jdc---uwo.ca isaksen---math.wayne.edu Keywords: Spectrum, pro-spectrum, Spanier-Whitehead duality, closed model category, colocalization Arxiv: math.AT/0403451 MSC-class: 55P42 (Primary); 55P25, 18G55, 55U35, 55Q55 (Secondary) Abstract: Cofiltered diagrams of spectra, also called pro-spectra, have arisen in diverse areas, and to date have been treated in an ad hoc manner. The purpose of this paper is to systematically develop a homotopy theory of pro-spectra and to study its relation to the usual homotopy theory of spectra, as a foundation for future applications. The surprising result we find is that our homotopy theory of pro-spectra is Quillen equivalent to the opposite of the homotopy theory of spectra. This provides a convenient duality theory for all spectra, extending the classical notion of Spanier-Whitehead duality which works well only for finite spectra. Roughly speaking, the new duality functor takes a spectrum to the cofiltered diagram of the Spanier-Whitehead duals of its finite subcomplexes. In the other direction, the duality functor takes a cofiltered diagram of spectra to the filtered colimit of the Spanier-Whitehead duals of the spectra in the diagram. We prove the equivalence of homotopy theories by showing that both are equivalent to the category of ind-spectra (filtered diagrams of spectra). To construct our new homotopy theories, we prove a general existence theorem for colocalization model structures generalizing known results for cofibrantly generated model categories. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/KrauseH/stable Title: The stable derived category of a noetherian scheme Author: Henning Krause E-mail: hkrause---math.upb.de Abstract: For a noetherian scheme, we introduce its unbounded stable derived category. This leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. Some applications are included, for instance an analogue of maximal Cohen-Macaulay approximations, a construction of Tate cohomology, and an extension of the classical Grothendieck duality. -------------------------4 new papers this month, from Bousfield, Castellana-Crespo-Scherer, IsaksenD, and Kitchloo-Morava. Mark Hovey New papers appearing on hopf between 4/5/04 and 5/4/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bousfield/vper On the 2-primary v1-periodic homotopy groups of spaces A.K. Bousfield bous---uic.edu AMS Classification Numbers: 55Q51(Primary),55N15,55P60,55S25,57T20 We develop foundations of a general approach for calculating p-primary v1-periodic homotopy groups of spaces using their p-adic KO-cohomologies and K-cohomolgies with particular attention to the case p = 2. As a main application, we derive a method for calculating v1-periodic homotopy groups of simply-connected compact Lie groups using their complex, real, and quaternionic representation theories. This method has been applied very effectively by D.M. Davis in recent work. We rely heavily on the p-primary v1-stabilization functor Phi from spaces to spectra. Roughly speaking, we obtain the p-primary v1-periodic homotopy of a space X from the p-adic KO-cohomology of Phi X, which we obtain from the p-adic KO-cohomology and K-cohomology of X by a v1-stabilization process under suitable conditions. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Castellana-Crespo-Scherer/DeconstructH Title: Deconstructing Hopf spaces Authors: Natalia Castellana, Juan A. Crespo, Jerome Scherer email: natalia---mat.uab.es, JuanAlfonso.Crespo---uab.es, jscherer---mat.uab.es AMS classification number: 55P45; 55S10; 55P60; 55P47; 55S45 Abstract: We characterize Hopf spaces with finitely generated cohomology as algebra over the Steenrod algebra. We ``deconstruct" the original space into an H-space Y with finite mod p cohomology and a finite number of p-torsion Eilenberg-Mac Lane spaces. One reconstructs X from Y by taking extensions by principal H-fibrations. We give a precise description of homotopy commutative H-spaces in this setting and give a criterion to recognize connected covers of H-spaces with finite mod p cohomology. The key observation is that the module of indecomposables lies in some stage of the Krull filtration of the category of unstable modules over the Steenrod algebra. We compare this algebraic condition with a topological one, namely that some iterated loop space of X is BZ/p-local. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/IsaksenD/completion Author: Daniel C. Isaksen Author's e-mail address: isaksen---math.wayne.edu Author's mailing address: Department of Mathematics \\ Wayne State University \\ Detroit, MI 48202 Included ps or eps files: None AMS classification number: 55P60, 55N10 (Primary); 18G55, 55U35 (Secondary) Abstract: For every ring R, we present a pair of model structures on the category of pro-spaces. In the first, the weak equivalences are detected by cohomology with coefficients in R. In the second, the weak equivalences are detected by cohomology with coefficients in all R-modules (or equivalently by pro-homology with coefficients in R). In the second model structure, fibrant replacement is essentially just the Bousfield-Kan R-tower. When R = Z/p, the first homotopy category is equivalent to a homotopy theory defined by Morel but has some convenient categorical advantages. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Morava/Thomprospectra2 THOM PROSPECTRA FOR LOOP GROUP REPRESENTATIONS NITU KITCHLOO, JACK MORAVA We construct an S^1-equivariant prospectrum that models the Atiyah dual of a free loop space of a manifold. By applying a suitably completed S^1-equivariant K-theory to the Atiyah dual, we show how to recover the Witten genus of a manifold. The main technical tool is a Tits building for the loop group. We use the Tits building to construct a dualizing spectrum for the loop group and relate it to work of Freed, Hopkins and Teleman. ------------------2 new papers this month, from Biss-Farb and Flores. Mark Hovey New papers appearing on hopf between 5/4/04 and 6/2/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Biss-Farb/kg Title: K_g is not finitely generated Authors: Daniel Biss and Benson Farb Author's email addresses: daniel---math.uchicago.edu, farb---math.uchicago.edu Included files: curve1.eps and curve3.eps Abstract: We prove that for any genus g>1, the subgroup K_g of the mapping class group of a closed genus g surface generated by Dehn twists about separating curves is not finitely generated. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Flores/draft2 Nullification functors and the homotopy type of the classifying space for proper bundles Ram'on J. Flores Departamento de Matem'aticas, Universidad Aut'onoma de Barcelona, E-08193 Bellaterra, Spain E-mail address: ramonj---mat.uab.es Abstract. Let G be a discrete group. In this note we build a bridge between the homotopy theory of BG and the theory of proper G-actions, by showing that under mild restrictions, the classifying space for proper G-bundles has the homotopy type of the W-nullification of BG for some space W. This allows us to use properties of the localization functors to obtain spaces that are homotopy equivalent to this "proper" classifying space for a wide range of groups, and on the other hand, we take profit of the existence of well-known geometrical and finite-dimensional models of it for some infinite groups to deduce homotopical information about the p-primary part of their classifying spaces. -----------------10 new papers this month, from Bergner, Broto-Moller, Bruner-Rognes, Galvez-Whitehouse, Intermont-Strom, Jardine, Rezk, and YauD (3 papers). Mark Hovey New papers appearing on hopf between 6/2/04 and 7/2/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bergner/SimplicialCategoryMC Title: A model category structure on the category of simplicial categories Author: Julia E. Bergner Author's e-mail address: jbergner---nd.edu AMS Classification: 18G55, 18D20 arXiv submission number: math.AT/0406507 Author's address: Department of Mathematics University of Notre Dame Notre Dame, IN 46556 Abstract: In this paper we put a cofibrantly generated model category structure on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Broto-Moller/Chev Title: Finite Chevalley versions of p-compact groups Authors: Carles Broto, Jesper M. Moller Author's e-mail address: broto---mat.uab.es moller---math.ku.dk Address: Carles Broto Departament de Matematiques, Universitat Autonoma de Barcelona, 08193 Bellaterra, Spain Address: Jesper M. Moller Matematisk Institut Universitetsparken 5 DK-2100 Copenhagen Denmark AMS class: 55R35, 55P15, 55P10 Abstract: We describe the spaces of homotopy fixed points of unstable Adams operations acting on p-compact groups and also of unstable Adams operations twisted with a finite order automorphism of the p-compact group. We obtain new exotic p-local finite groups. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Bruner-Rognes/bruner Title: Leibniz Formulas for Cyclic Homotopy Fixed Point Spectra Authors: Robert R. Bruner and John Rognes MSC-class: 19D55, 55P43, 55P91, 55S12, 55T05. ArXiv ID: math.AT/0406081 Addresses: Robert R. Bruner Department of Mathematics Wayne State University Detroit, Michigan 48067 USA rrb---math.wayne.edu John Rognes Department of Mathematics University of Oslo Box 1053, Blindern NO-0316 Oslo Norway rognes---math.uio.no Abstract: We analyze the homotopy fixed point spectrum of a circle-equivariant commutative S-algebra R in homological terms. There is a homological homotopy fixed point spectral sequence that converges conditionally to the continuous homology of the homotopy fixed point spectrum. We show that there are Dyer-Lashof operations Q^i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the E^{2r}-term of the spectral sequence there are 2r other classes in the E^{2r}-term (obtained mostly by Dyer-Lashof operations on x) that are infinite cycles, i.e., survive to the E^infty-term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R = THH(B) of many S-algebras, including B = MU, BP, ku, ko and tmf. Similar results apply for all finite subgroups of the circle, and for the Tate- and homotopy orbit spectra. This work is part of a homological approach to calculating topological cyclic homology and algebraic K-theory of commutative S-algebras. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Galvez-Whitehouse/centres Title: Infinite Sums of Adams Operations and Cobordism Authors: Imma Galvez, Sarah Whitehouse E-mail: i.galvezicarrillo---londonmet.ac.uk, s.whitehouse---sheffield.ac.uk Addresses: Computing, Communications Technology and Mathematics, London Metropolitan University, Holloway Road, London N7 8DB, UK. Pure Mathematics, University of Sheffield, Sheffield S3 7RH, UK. Included ps or eps files: centrediag1.ps, centrediag2.ps AMS classification number: Primary: 55S25; Secondary: 55N22, 19L41. Abstract: In recent work by Clarke, Crossley and the second author, various algebras of stable degree zero operations in p-local K-theory were described explicitly. The elements are certain infinite sums of Adams operations. Here we show how to make sense of the same expressions for p-local cobordism and for BP, thus identifying the "Adams subalgebra" of the algebras of operations. We prove that the Adams subalgebra is the centre of the ring of degree zero operations. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Intermont-Strom/GoodSp Complexity and Good Spaces M. Intermont (Kalamazoo College) and J. Strom (Western Michigan University) intermon---kzoo.edu jeffrey.strom---wmich.edu This paper is an exploration of two ideas in the study of closed classes: the A-complexity of a space X and the notion of good spaces (spaces A for which C(A) = \overline{C(A)}). A variety of formulae for the computation of complexity are given, along with some calculations. Good spaces are characterized in terms of the functors CW_A and P_A. The main result is a countable upper bound for the complexity with respect to the suspension of A when A is a good space. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/stack-coh6 Title: Fibred sites and stack cohomology Author: J.F. Jardine AMS Classification numbers: 55P42, 18F20, 14A20 J.F. Jardine Department of Mathematics University of Western Ontario London, ON N6A 5B7 Canada E-mail: jardine---uwo.ca The usual notion of a site fibred over a stack is expanded to a definition of a site C/A fibred over a presheaf of categories A. Presheaves of simplicial sets on the site fibred over a presheaf of categories A are contravariant enriched diagrams defined on A, taking values in simplicial sets. The standard model structure for presheaves of simplicial sets induces a coarse equivariant structure for enriched contravariant A-diagrams. If the presheaf of categories is a presheaf of groupoids G, then the associated homotopy theory is Quillen equivalent to the homotopy theory of simplicial presheaves over BG, and so the homotopy theory for the fibred site C/G is an invariant of the homotopy type of G. Similar homotopy invariance results obtain for presheaves of spectra and presheaves of symmetric spectra on C/G. In particular, stack cohomology can be calculated on the fibred site for a representing presheaf of groupoids. 7. 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 Authors e-mail address: rezk---math.uiuc.edu 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. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/cohom Cohomology of $\lambda$-rings Donald Yau (University of Illinois at Urbana-Champaign), dyau---math.uiuc.edu A cohomology theory for $\lambda$-rings is developed. This is then applied to study deformations of $\lambda$-rings. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/lambda-rev2 On $\lambda$-ring structures over Z[[x]] Donald Yau (University of Illinois at Urbana-Champaign), dyau---math.uiuc.edu It is shown that the $\lambda$-ring structure over the power series ring Z[[x]] given by the $K$-theory of $CP^\infty$ is uniquely determined by the following condition: \psi^p(x) = px mod{x^2} for each prime $p$, where $\psi^p$ is the Adams operation. Applications to algebraic topology and formal group laws are given. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/YauD/truncated Extensions of filtered $\lambda$-ring structures over the dual number ring Donald Yau (University of Illinois at Urbana-Champaign), dyau---math.uiuc.edu We study problems related to the existence and uniqueness of filtered $\lambda$-ring structures over the truncated polynomial ring Z[x]/(x^3) that extend a given filtered $\lambda$-ring structure over Z[x]/(x^2). ---------------5 new papers this month, from Basterra-Mandell, BrownR, Diaz-Ruiz-Viruel, Dugger-Isaksen, and Kuhn. Mark Hovey New papers appearing on hopf between 7/2/04 and 8/7/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Basterra-Mandell/Basterra-Mandell-stability Title: Homology and Cohomology of E-infinity Ring Spectra Authors: Maria Basterra Department of Mathematics, University of New Hampshire, Durham, NH basterra---math.unh.edu Michael A. Mandell Department of Mathematics, University of Chicago, Chicago, IL mandell---math.uchicago.edu AMS Subject class: Primary 55P43; Secondary 55P48, 55U35 Abstract: We show that every homology or cohomology theory on a category of E-infinity ring spectra is Topological Andr'e-Quillen homology or cohomology with appropriate coefficients. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/BrownR/IMA-talk Author: Ronald Brown Author's e-mail address: r.brown---bangor.ac.uk Author's web page: http://www.bangor.ac.uk/~mas010 Author's mailing address: Professor Emeritus R. Brown, Department of Mathematics, University of Wales, Bangor Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom AMS classification number: 18D05,18D15,18G50,55P15,55U40,57M07 arXive submission number: math.AT/0407275 Abstract: We sketch the background to a book with the title `Nonabelian algebraic topology' being written under support of a Leverhulme Emeritus Fellowship (2002-2004) by the speaker and Rafael Sivera (Valencia). The aim is to give in one place a full account of work by R. Brown and P.J. Higgins and others since the 1970s which defined and applied crossed complexes and cubical higher homotopy groupoids to local-to-global problems and homotopy classification of maps. This yields a distinctive account of that part of algebraic topology which lies between homology theory and homotopy theory, in which the fundamental group and its actions plays an essential role, and which allows for nonabelian calculations in dimension 2. This is an extended account of a short presentation with this title given at the Minneapolis IMA Workshop on `$n$-categories: foundations and applications', June 7-18, 2004, organised by John Baez and Peter May. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Diaz-Ruiz-Viruel/drv Title: All p-local finite groups of rank two for odd prime p Authors: Antonio Diaz, Albert Ruiz, Antonio Viruel Author's e-mail address: adiaz---agt.cie.uma.es, Albert.Ruiz---uab.es, viruel---agt.cie.uma.es ArXive submission number: math.AT/0407324 Abstract: In this paper we give a classification of the rank two p-local finite groups for odd p. This study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group and the proper F-radical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p-local finite group, getting some new examples of these for p = 3. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger-Isaksen/ksumDI Algebraic K-theory and sums-of-squares formulas Daniel Dugger and Daniel C. Isaksen Email: ddugger---math.uoregon.edu and isaksen---math.wayne.edu Addresses: Daniel Dugger Department of Mathematics University of Oregon Eugene, OR 97403 Daniel C. Isaksen Department of Mathematics Wayne State University Detroit, MI 48202 Abstract: We prove a result about the non-existence of certain sums-of-squares formulas over a field. This generalizes an old theorem which used topological K-theory to produce obstruction conditions when the field is the real numbers. Our result applies to arbitrary fields not of characteristic 2, making use of algebraic K-theory in place of topological K-theory. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/homologyiso Title: Mapping spaces and homology isomorphisms Author: Nicholas J. Kuhn AMS classification numbers: 55P35, 55N20, 55P42 arXiv no.: math.AT/0407146 address: University of Virginia, Charlottesville, VA USA email: njk4x---virginia.edu Abstract: Let Map(K,X) denote the space of pointed continuous maps from a finite cell complex K to a space X. Let E_* be a generalized homology theory. We use Goodwillie calculus methods to prove that under suitable conditions on K and X, Map(K, X) will send an E_*--isomorphism in either variable to a map that is monic in E_* homology. Interesting examples arise by letting E_* be K--theory, K be a sphere, and the map in the X variable be an exotic unstable Adams map between Moore spaces. ---------------4 new papers this month, from Devinatz, Dugger, IsaksenD, and Sinha. Mark Hovey New papers appearing on hopf between 8/7/04 and 9/2/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Devinatz/recog Title: Recognizing Hopf algebroids defined by a group action Author: Ethan Devinatz e-mail: devinatz---math.washington.edu Abstract: Let A be a complete noetherian regular local ring, and suppose that S is a profinite group acting continuously on A via ring homomorphisms. Let T be the algebra of continuous functions from S to A. Then (A,T) has a canonical structure of a complete Hopf algebroid, determined by the action of S on A. We give necessary and sufficient conditions for a general Hopf algebroid to be of this form. Applications to Morava theory are also discussed. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Dugger/milnor Title: Notes on the Milnor conjectures Author: Daniel Dugger email: ddugger---math.uoregon.edu Abstract: These are some expository notes on the two Milnor conjectures and their proofs (due to Voevodsky, Orlov-Vishik-Voevodsky, and Morel). 3. http://hopf.math.purdue.edu/cgi-bin/generate?/IsaksenD/gencohlgy Title: Generalized cohomology of pro-spectra Author: Daniel C. Isaksen E-mail: isaksen---math.wayne.edu AMS classification: 55T25, 55P42, 55U35, 55N20, 18G55 (Primary), 19L99 (Secondary) Abstract: We present a closed model structure for the category of pro-spectra in which the weak equivalences are detected by stable homotopy pro-groups. With some bounded-below assumptions, weak equivalences are also detected by cohomology as in the classical Whitehead theorem for spectra. We establish an Atiyah-Hirzebruch spectral sequence in this context, which makes possible the computation of topological K-theory (and other generalized cohomology theories) of pro-spectra. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Sinha/opknot Title: Operads and knot spaces Author: Dev Sinha E-mail: dps---math.uoregon.edu Abstract: Let F_m be the space of knotted intervals in I^m equipped with a trivialization through immersions. We show that the totalization of the Kontsevich operad provides a model for the embedding calculus tower for F_m. Combined with results of Goodwillie-Klein-Weiss and Volic, this resolves Kontsevich's conjecture of existence of such a model which captures the homotopy type of F_m when m>3 and which classifies finite-type framed knot invariants when m=3. We carefully develop the Kontsevich operad, which is closely related to the Fulton-MacPherson operad and weakly equivalent to the little cubes operad. In doing so we show that the standard simplicial model for the two-sphere carries an operad structure in the opposite category of pointed sets. We apply the well-developed machinery of McClure and Smith on operads with multiplication to deduce that our model has a little two-cubes action. (Note: if you want the dvi file to contain the figures, you need to download the directory Figures as well. The pdf file already has the figures built in.) ----------------As you have heard from Clarence, the Hopf Archive is now a virtual server on the Purdue Math Department's server. This means there is no more ftp access to Hopf, only web access. Also, because of this changeover, my October message was a bit delayed. 3 new papers this month, from Blanc, Devinatz, and Kuhn. Mark Hovey New papers appearing on hopf between 9/2/04 and 10/15/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc/mod03 Moduli spaces of homotopy theory by David Blanc Dept. of Mathematics, Univ. of Haifa, 31905 Haifa, Israel E-mail address: blanc---math.haifa.ac.il The moduli spaces refered to are topological spaces whose path components parametrize homotopy types. Such objects have been studied in two separate contexts: rational homotopy types, in the work of several authors in the late 1970's; and general homotopy types, in the work of Dwyer-Kan and their collaborators. We here explain the two approaches, and show how they may be related to each other. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Devinatz/homotopydev Title: Homotopy groups of homotopy fixed point spectra associated to E_n Author: Ethan Devinatz e-mail: devinatz---math.washington.edu Abstract: We compute the mod(p) homotopy groups of the continuous homotopy H_2 fixed points of E_2 for p>2, where E_n is the Landweber exact spectrum whose coefficient ring is the ring of functions on the Lubin-Tate moduli space of lifts of height n formal group laws, and H_n is the semi-direct product of the group of diagonal matrices in the nth Morava stabilizer group with an appropriate Galois group. We examine some consequences of this related to Brown-Comenetz duality and to finiteness properties of homotopy groups of K(n)_*-local spectra. We also indicate a plan for generalizing this computation to n>2. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Kuhn/Kinosaki Title: Goodwillie towers and chromatic homotopy: an overview Author: Nicholas J. Kuhn Email:njk4x at virginia.edu Address: University of Virginia, Charlottesville, VA 22904 arXive no: math.AT/0410342 Abstract: This paper is based on talks I gave in Nagoya and Kinosaki in August of 2003. I survey, from my own perspective, Goodwillie's work on towers associated to continuous functors between topological model categories, and then include a discussion of applications to periodic homotopy as in my work and the work of Arone--Mahowald. --------------- 6 new papers this month, from Boardman-Wilson, Goerss-Henn-Mahowald-Rezk, Lewis-Mandell, McClure, Turiel, and Wodarz. Mark Hovey New papers appearing on hopf between 10/15/04 and 11/04/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Boardman-Wilson/BWonPn Title: k(n)-torsion-free H-spaces and P(n)-cohomology Authors: J. Michael Boardman, W. Stephen Wilson E-mail: boardman---math.jhu.edu, wsw---math.jhu.edu Address: Dept. of Mathematics, Johns Hopkins University, 3400 N. Charles St., Baltimore MD 21218-2686 AMS Classifications: Primary 55N22, 55P45 Abstract: In his thesis, the second author split the H-space that represents Brown-Peterson cohomology BP^k(-) into indecomposable factors, which have torsion-free homotopy and homology. Here, we do the same for the related spectrum P(n), by constructing idempotent operations in P(n)-cohomology P(n)^k(-) in the style of Boardman-Johnson-Wilson; this relies heavily on the Ravenel-Wilson determination of the relevant Hopf ring. The resulting (i-1)-connected H-spaces Y_i have free connective Morava K-homology k(n)_*(Y_i), and may be built from the spaces in the Omega-spectrum for k(n) using only v_n-torsion invariants. We also extend Quillen's theorem on complex cobordism to show that for any space X, the P(n)_*-module P(n)^*(X) is generated by elements of P(n)^i(X) for i>=0. This result is essential for the work of Ravenel-Wilson-Yagita, which in many cases allows one to compute BP-cohomology from Morava K-theory. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Goerss-Henn-Mahowald-Rezk/ghmr Title: A resolution of the K(2)-local sphere at the prime 3 Authors: Paul Goerss, Hans-Werner Henn, Mark Mahowald and Charles Rezk Northwestern University, Universit\'e Louis Pasteur et CNRS, Northwestern University, University of Illinois Urbana, IL 61801 (This is an updated version) ABSTRACT We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum L_{K(2)}S^0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E_2^{hF} where F is a finite subgroup of the Morava stabilizer group and E_2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n=2 at p=3 represents the edge of our current knowledge: n=1 is classical and at n=2, the prime 3 is the largest prime where the Morava stabilizer group has a p-torsion subgroup, so that the homotopy theory is not entirely algebraic. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Lewis-Mandell/Lewis-Mandell-UCT Equivariant Universal Coefficient and Kunneth Spectral Sequences L. Gaunce Lewis, Jr. Department of Mathematics Syracuse University Syracuse, NY 13244-1150 lglewis---syr.edu Michael A. Mandell DPMMS, University of Cambridge Wilberforce Road Cambridge CB3 0WB UK M.A.Mandell---dpmms.cam.ac.uk AMS Classification: Primary 55N91; Secondary 55P43,55U20,55U25} Abstract We construct hyper-homology spectral sequences of Z-graded and ROG-graded Mackey functors for Ext and Tor over G-equivariant S-algebras (A-infty ring spectra) for finite groups G. These specialize to universal coefficient and Kunneth spectral sequences. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/McClure/intersection On the chain-level intersection pairing for PL manifolds. J.E. McClure mcclure---math.purdue.edu AMS classification numbers: 57Q65; 18D50 Posted on arXiv: math.QA/0410450 Abstract: Let M be a compact oriented PL manifold and let C_*M be its PL chain complex. The domain of the chain-level intersection pairing is a subcomplex G of C_*M\otimes C_*M. We prove that G is a ``full'' subcomplex, that is, the inclusion of G in C_*M \otimes C_*M is a quasi-isomorphism. An analogous result is true for the domain of the iterated intersection pairing. Using this, we show that the intersection pairing gives C_*M a structure of partially defined commutative DGA, which in particular implies that C_*M is canonically quasi-isomorphic to an E_\infty chain algebra. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Turiel/poly Polynomial Maps and Even Dimensional Spheres Javier Turiel turiel---agt.cie.uma.es Abstract: We construct, for every even dimensional sphere $S^n$, $n >1$, and every odd integer $k$, a homogeneous polynomial map $f: S^{n}\to S^{n}$ of Brouwer degree $k$ and algebraic degree $2|k|-1$. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Wodarz/ExactHomotopyFunctors Title: Exactness of Homotopy Functors of Spaces Author: Nathan Wodarz AMS Classification: 55P65, 55T25 Address: Grand Valley State University, Allendale, MI E-mail: wodarzn---gvsu.edu Abstract: We will provide an analysis of the generalized Atiyah--Hirzebruch spectral sequence (GAHSS), which was introduced by Hakim-Hashemi and Kahn. To do so, we introduce a new class of functors, called $n$--exact functors, which are analogous to Goodwillie's $n$--excisive functors. In the study of these functors, we introduce a new spectral sequence, the homological Barratt--Goerss spectral sequence (HBGSS), which has properties similar to those of the classical Barratt--Goerss Spectral Sequence on homotopy. We close by giving an identification of the $E^2$ term of the GAHSS in the case of 2--exact functors on Moore spaces. ---------------- -------------------------------------- Sorry for the delay this month. The semester is finally over! 6 new papers this month, from Angeltveit, Bokstedt-Ottosen (2), Castellana-Crespo-Scherer, Kitchloo-Wilson, and May. Mark Hovey New papers appearing on hopf between 11/04/04 and 12/14/04 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Angeltveit/Ainfinity Title: $A_\infty$ obstruction theory and the strict associativity of $E/I$ Author: Vigleik Angeltveit E-mail address: vigleik---math.mit.edu Address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Abstract: We prove that for a ring spectrum $K$ with a perfect universal coefficient formula, the obstructions to extending the multiplication to an $A_\infty$ multiplication lie in $Ext^{*,*}_{K_*K^{op}}(K_*,K_*)$. As a corollary, we show that if $E$ is even and $I=(x_1,x_2,\ldots)$ is a regular sequence in $E_*$, then any product on $E/I$ can be extended to an $A_\infty$ multiplication. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Bokstedt-Ottosen/hiem Ttile: An alternative approach to homotopy operations Authors: Marcel Bokstedt and Iver Ottosen Email: marcel---imf.au.dk, ottosen---imf.au.dk Address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 530, DK-8000 Aarhus C, Denmark Abstract: We give a particular choice of the higher Eilenberg-MacLane maps of a simplicial ring by a recursive formula. This choice leads to a simple description of the homotopy operations for simplicial Z/2-algebras. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Bokstedt-Ottosen/kkp Title: A splitting result for the free loop space of spheres and projective spaces Authors: Marcel Bokstedt and Iver Ottosen Email: marcel---imf.au.dk, ottosen---imf.au.dk Address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 530, DK-8000 Aarhus C, Denmark MSC: 55P35, 18G50, 55S10 Abstract: Let X be a 1-connected compact space such that the algebra H*(X;Z/2) is generated by one single element. We compute the cohomology of the free loop space H*(LX;Z/2) including the Steenrod algebra action. When X is a projective space CP^n, HP^n, the Cayley projective plane CaP^2 or a sphere S^m we obtain a splitting result for integral and mod two cohomology of the suspension spectrum of LX_+. The splitting is in terms of the suspension spectrum of X_+ and the Thom spaces of the q-fold Whitney sums of the tangent bundle over X for non negative integers q. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Castellana-Crespo-Scherer/CWPostH Title: Postnikov pieces and BZ/p-homotopy theory Authors: Natalia Castellana, Juan A. Crespo, Jerome Scherer email: natalia---mat.uab.es, JuanAlfonso.Crespo---uab.es, jscherer---mat.uab.es AMS classification number: 55R35; 55P60, 55P20, 20F18 ArXiv submission number: math.AT/0409399 Abstract: We present a constructive method to compute the cellularization with respect to K(Z/p, m) for any integer m > 0 of a large class of H-spaces, namely all those which have a finite number of non-trivial K(Z/p, m)-homotopy groups (the pointed mapping space map( K(Z/p, m), X) is a Postnikov piece). We prove in particular that the K(Z/p, m)-cellularization of an H-space having a finite number of K(Z/p, m)-homotopy groups is a p-torsion Postnikov piece. Along the way we characterize the BZ/p^r-cellular classifying spaces of nilpotent groups. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Kitchloo-Wilson/kitchloo-wilson-ER2 Title: On the Hopf ring for ${ER(n)}$ Authors: Nitu Kitchloo Department of Mathematics University of California, San Diego (UCSD) La Jolla, CA 92093-0112 nitu---math.ucsd.edu W. Stephen Wilson Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 wsw---math.jhu.edu Kriz and Hu construct a real Johnson-Wilson spectrum, $ER(n)$, which is $2^{n+2}(2^n-1)$ periodic. $ER(1)$ is just $KO_{(2)}$. We do two things in this paper. First, we compute the homology of the $2^{n+2}k}$ spaces in the Omega spectrum for $ER(n)$. There are $2^n-1$ of them and their double is the Hopf ring for $E(n)$. As a byproduct of this we get the homology of the zeroth spaces for the Omega spectrum for real complex cobordism and real Brown-Peterson cohomology. The second result is to compute the homology Hopf ring for all 48 spaces in the Omega spectrum for $ER(2)$. This turns out to be generated by very few elements. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/May/Split A note on the splitting principle J.P. May may---math.uchicago.edu 55R40, 55N99 We offer a new* perspective on the splitting principle. We give an easy proof that applies to all classical types of vector bundles and in fact to $G$-bundles for any compact connected Lie group $G$. The perspective gives precise calculational information and directly ties the splitting principle to the specification of characteristic classes in terms of classifying spaces. * Note to the list: if this is not new, please let me know --- it shouldn't be, but it was to those experts I tried it out on. --------------