Subject: new Hopf listings Date: 01 Mar 2004 08:37:41 -0500 From: Mark Hovey Reply-To: mhovey@wesleyan.edu To: dmd1@lehigh.edu 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. ---------------------Instructions----------------------------- To subscribe or unsubscribe to this list, send a message to Don Davis at dmd1@lehigh.edu with your e-mail address and name. Please make sure he is using the correct e-mail address for you. To see past issues of this mailing list, point your WWW browser to http://math.wesleyan.edu/~mhovey/archive/ If this doesn't work or is missing a few issues, try http://www.lehigh.edu/~dmd1/algtop.html which also has the other messages sent to Don's list. To get the papers listed above, point your Web browser to the URL listed. The general Hopf archive URL is http://hopf.math.purdue.edu There is a web form for submitting papers to Hopf on this site as well. You can also use ftp, explained below. The largest archive of math preprints is at http://arxiv.gov There is an algebraic topology section in this archive. The most useful way to browse it or submit papers to it is via the front end developed by Greg Kuperberg: http://front.math.ucdavis.edu To get the announcements of new papers in the algebraic topology section at the arXiv, send e-mail to math@arxiv.org with subject line "subscribe" (without quotes), and with the body of the message "add AT" (without quotes). You can also access Hopf through ftp. Ftp to hopf.math.purdue.edu, and login as ftp. Then cd to pub. Files are organized by author name, so papers by me are in pub/Hovey. If you want to download a file using ftp, you must type binary before you type get . To put a paper of yours on the archive, go to http://hopf.math.purdue.edu and use the web form. You can also use anonymous ftp as above. First cd to /pub/incoming. Transfer the dvi file using binary, by first typing binary then put You should also transfer an abstract as well. Clarence has explicit instructions for the form of this abstract: see http://hopf.math.purdue.edu/new-html/submissions.html In particular, your abstract is meant to be read by humans, so should be as readable as possible. I reserve the right to edit unreadable abstracts. You should then e-mail Clarence at wilker@math.purdue.edu telling him what you have uploaded. I am solely responsible for these messages---don't send complaints about them to Clarence. Thanks to Clarence for creating and maintaining the archive.