Subject: new Hopf listings Date: 25 Nov 2003 13:32:14 -0500 From: Mark Hovey Reply-To: mhovey@wesleyan.edu To: dmd1@lehigh.edu 12 new papers this time, from Aouina-Klein, Chalupnik (3), Fausk-Oliver, Grandis (2), Knudson-Walker, Notbohm-Ray, Sauvageot, Troesch, and ZhouXueguang. Mark Hovey New papers appearing on hopf between 10/24/03 and 11/25/03 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Aouina-Klein/config_stable Title: On the homotopy invariance of configuration spaces Author(s): Mokhtar Aouina and John R. Klein Author's e-mail address: aouina@math.wayne.edu, klein@math.wayne.edu AMS classification number: Primary 55R80; Secondary 57Q35, 55R70. Abstract: For a closed PL manifold M, we consider the configuration space F(M,k) of ordered k-tuples of distinct points in M. We show that a suitable iterated suspension of F(M,k) is a homotopy invariant of M. The number of suspensions we require depends on three parameters: the number of points k, the dimension of M and the connectivity of M. Our proof uses a mixture of embedding theory and fiberwise algebraic topology. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/cohsdr Title of Paper: Schur_De-Rham complex and its cohomology Author: Marcin Chalupnik Email: mchal@mimuw.edu.pl Abstract: We associate to a Young diagram a complex of strict polynomial functors which we call the Schur-De-Rham complex. Its cohomology turns out to reflect deep combinatorial properties of a diagram. We show that if a ground field is of characteristic p, the Schur-De-Rham complex is acyclic when the p-core of a diagram is nontrivial. We also compute its cohomology for a diagram with a trivial p-core and p-quotient consisting of a single diagram. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/extpol Title of Paper: Extensions of strict polynomial functors Author: Marcin Chalupnik Email: mchal@mimuw.edu.pl Abstract: We compute Ext-groups between various strict polynomial functors important in representation theory (eg. between twisted Weyl and Schur functors). Our method utilizes: computation of the Ext-groups between twisted divided and symmetric powers due to Franjou-Friedlander-Scorichenko-Suslin, resolutions of functors by divided and symmetric powers, interplay between functors and representations of the symmetric group. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Chalupnik/extws Title of Paper: Extensions of Weyl and Schur functors Author: Marcin Chalupnik Abstract: We use here the Schur-De-Rham complex to extend calculations of the Ext-groups between twisted Weyl and Schur functors initiated in the paper ``Extensions of strict polynomial functors''. The main result is a full calculation of those groups in the case of a pair of diagrams which can be obtained from diagrams of the same weights by the operation F described in ``Schur-De-Rham complex and its cohomology''. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Fausk-Oliver/piperfect Title: Continuity of p-perfection for compact Lie groups Authors: Halvard Fausk and Bob Oliver Author's e-mail address: fausk@math.uio.no and bob@math.univ-paris13.fr AMS classification number: 55P91 Abstract: Let G be a compact Lie group, and let pi be any prime or set of primes. We construct a ``pi-perfection map'': a continuous function from the space of conjugacy classes of all closed subgroups of G to the space of conjugacy classes of pi-perfect subgroups with finite index in their normalizer. We use this to show that the idempotent elements of the Burnside ring of G localized at pi are in bijective correspondence with the open and closed subsets of the space of conjugacy classes of pi-perfect subgroups of G with finite index in their normalizer. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Grandis/Grandis.Bsy2 (Note: this paper is only available in pdf format) Normed combinatorial homology and noncommutative tori Marco Grandis Keywords: Cubical sets, noncommutative C*-algebras, combinatorial homology, normed abelian groups. Dipartimento di Matematica Universita` di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it http://www.dima.unige.it/~grandis/ Notes: Dip. Mat. Univ. Genova, Preprint 484 (2003), 14 p. Abstract. Cubical sets have a directed homology, studied in a previous paper and consisting of preordered abelian groups, with a positive cone generated by the structural cubes. By this additional information, cubical sets can provide a sort of "noncommutative topology", agreeing with some results of noncommutative geometry but lacking the metric aspects of C*-algebras. Here, we make such similarity stricter by introducing normed cubical sets and their normed directed homology, formed of normed preordered abelian groups. The normed cubical sets associated with "irrational" rotations have thus the same classification up to isomorphism as the well-known irrational rotation C*-algebras. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Grandis/Grandis.Dht1 (Note: this paper is only available in pdf format) Directed homotopy theory, I. The fundamental category Marco Grandis Key words: homotopy theory, homotopical algebra, directed homotopy, fundamental category. Dipartimento di Matematica Universita di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it http://www.dima.unige.it/~grandis/ Notes: to appear in: Cahiers Topologie Geometrie Differentielle Categoriques Preprint: Dip. Mat. Univ. Genova, Preprint 443 (2001), 26 p. Revised version: 5 Nov 2001. Abstract. Directed Algebraic Topology is beginning to emerge from various applications. The basic structure we shall use for the foundations of such a theory, a d-space, is a topological space equipped with a family of directed paths, closed under some operations. This allows for directed homotopies, generally non reversible, represented by a cylinder and cocylinder functors. The existence of 'pastings' (colimits) yields a geometric realisation of cubical sets as d-spaces, together with homotopy constructs which will be developed in a sequel. Here, the fundamental category of a d-space is introduced and a 'Seifert-van Kampen' theorem proved; its homotopy invariance rests on directed homotopy of categories. In the process, new shapes appear, for d-spaces but also for small categories, their elementary algebraic model. Applications of such tools are briefly considered or suggested, for objects which model a directed image, or a portion of space-time, or a concurrent process. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/Knudson-Walker/hom11-19 Title: Homology of linear groups via cycles in BG x X Author1: Kevin P. Knudson Author2: Mark E. Walker email1: knudson@math.msstate.edu email2: mwalker@math.unl.edu Abstract: Let G be an algebraic group and let X be a smooth integral scheme over a field k. In this paper we construct homology-type groups H_i(X,G) by considering cycles in the simplicial scheme BG x X (an idea suggested by Andrei Suslin). We discuss the basic properties of these groups and construct a spectral sequence, beginning with the groups H_i(\Delta^j,G), which converges to the etale cohomology of the simplicial group BG. These groups are therefore connected with the study of Friedlander's generalized isomorphism conjecture. We also compute some examples, focusing in particular on the case X=Spec(k). In the case where k is the real numbers, there is a connection between the groups H_i and the Z/2-equivariant cohomology of the classifying space of the discrete group G(R). 9. http://hopf.math.purdue.edu/cgi-bin/generate?/Notbohm-Ray/djthrational On Davis-Januszkiewicz Homotopy Types I; Formality and Rationalisation by Dietrich Notbohm} and Nigel Ray For an arbitrary simplicial complex $K$, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of $K$. Subsequently, Buchstaber and Panov gave an alternative construction, which they showed to be homotopy equivalent to Davis and Januszkiewicz's examples. It is therefore natural to investigate the extent to which the homotopy type of a space $X$ is determined by having such a cohomology ring. We begin this study here, in the context of model category theory. In particular, we extend work of Franz by showing that the singular cochain algebra of $X$ is formal as a differential graded noncommutative algebra. We then specialise to the rationals, by proving the corresponding property for Sullivan's {\it commutative\/} cochain algebra; this confirms that the rationalisation of $X$ is unique. In a sequel, we will consider the uniqueness of $X$ at each prime separately, and apply Sullivan's arithmetic square to produce global results in special families of cases. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/Sauvageot/simpl-hopf-model A simplicial model for the Hopf map Orin R. Sauvageot Ecole Polytechnique Federale de Lausanne orin.sauvageot@epfl.ch We give an explicit simplicial model for the Hopf map S^3 -> S^2. For this purpose, we construct a model of S^3 as a principal twisted cartesian product K x_{eta} S^2, where K is a simplicial model for S^1 acting by left multiplication on itself, S^2 is given the simplest simplicial model and the twisting map is eta:(S^2)_n -> (K)_{n-1}. We construct a Kan complex for the simplicial model K of S^1. The simplicial model for the Hopf map is then the projection K x_{eta} S^2 -> S^2. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/Troesch/troesch_Resolution_of_symmetric_powers Title: A propos d'une question de Friedlander et Suslin I -- Une r'esolution injective des puissances sym'etriques twist'ees (in French) Author: Alain Troesch Address of Author: Institut de Mathematiques de Jussieu, Case 82 4 place Jussieu, F-75252 PARIS CEDEX 05 e-mail address: troesch@math.jussieu.fr Abstract. Some years ago, Friedlander and Suslin constructed an explicit injective resolution of twisted symmetric powers in the category of strict polynomial functors over a ground field of characteristic 2. The factors in this resolutions are given by direct sums of tensor products of (non twisted) symmetric powers. The case of a symmetric power twisted only once is a well-known result: it is some kind of Koszul complex. In characteritic p>2, nothing similar was known up to now, even for a single twist. In this paper, we construct such injective resolutions. The resolutions we construct are in fact "p-resolutions", that is, the differential does not vanish when composed twice, but only when composed p times. This result should unable us to constuct an injective resolution of any twisted functor if we know an injective resolution of the corresponding non twisted functor. This will be the subject of another paper. 12. http://hopf.math.purdue.edu/cgi-bin/generate?/ZhouXueguang/zhou2 title of the paper: The answer to an email of Mr. Douglas C. Ravenel author: Zhou Xueguang Address of author:Department of Mathematics, Nankai University, Tianjin 300071, People's Republic of China Email address of author: zhengqb@eyou.com Abstract: In this paper, we answer the question why V(n) exists for all non-negative integers n. ---------------------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.