Subject: new Hopf listings From: Mark Hovey Date: 19 Apr 2007 10:10:09 -0400 6 new papers this month, from Benson, Chebolu-Christensen-Minac, DavisD-Mahowald, Muro-Schwede-Strickland, Oliver-Ventura, and Panin-Pimenov-Roendigs. Mark Hovey New papers appearing on hopf between 3/19/07 and 4/19/07 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Benson/loops An algebraic model for chains on $\Omega BG\phat$ Dave Benson Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE Abstract: We provide an interpretation of the homology of the loop space on the $p$-completion of the classifying space of a finite group in terms of representation theory, and demonstrate how to compute it. We then give the following reformulation. If $f$ is an idempotent in $kG$ such that $f.kG$ is the projective cover of the trivial module $k$, and $e=1-f$, then we exhibit isomorphisms for $n\ge 2$: H_n(\Omega BG\phat;k) \cong \Tor_{n-1}^{e.kG.e}(kG.e,e.kG) H^n(\Omega BG\phat;k) \cong \Ext^{n-1}_{e.kG.e}(e.kG,e.kG). Further algebraic structure is examined, such as products and coproducts, restriction and Steenrod operations. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chebolu-Christensen-Minac/ghostnumber TITLE: Ghosts in modular representation theory AUTHORS: Sunil K. Chebolu, J. Daniel Christensen, and Jan Minac Department of Mathematics University of Western Ontario London, ON N6A 5B7, Canada AMS Subject classsification: Primary 20C20, 20J06; Secondary 55P42 ABSTRACT: A \emph{ghost} over a finite group $G$ is a map between modular representations of $G$ which is invisible in Tate cohomology. Motivated by the failure of the \emph{generating hypothesis}---the statement that ghosts between finite-dimensional $G$-representations factor through a projective---we define the \emph{compact ghost number} of $kG$ to be the smallest integer $l$ such that the composition of any $l$ ghosts between finite-dimensional $G$-representations factors through a projective. In this paper we study ghosts and the compact ghost numbers of $p$-groups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation $k$, holds for all $p$-groups. We do this by proving that a map between finite-dimensional $G$-representations is a ghost if and only if it is a \emph{dual ghost}. We then compute the compact ghost numbers of all cyclic $p$-groups and all abelian $2$-groups with $C_2$ as a summand. We obtain bounds on the compact ghost numbers for abelian $p$-groups and for all $2$-groups which have a cyclic subgroup of index $2$. Using these bounds we determine the finite abelian groups which have compact ghost number at most $2$. %Finally, using universal ghosts, we establish various sets of conditions which %guarantee the existence of a non-trivial ghost out of a $G$-representation. Our methods involve techniques from group theory, representation theory, triangulated category theory, and constructions motivated from homotopy theory. COMMENTS: This version replaces an earlier one with file name ghost.tex. This is a substantial improvement with many new results and major reorganisation of the paper. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/DavisD-Mahowald/overlook Nonimmersions of RP^n implied by tmf, revisited Donald M. Davis and Mark Mahowald In a 2002 paper, the authors and Bruner used the new spectrum tmf to obtain some new nonimmersions of real projective spaces. In this note, we complete/correct two oversights in that paper. The first is to note that in that paper a general nonimmersion result was stated which yielded new nonimmersions for RP^n with n as small as 48, and yet it was stated there that the first new result occurred when n=1536. Here we give a simple proof of those overlooked results. Secondly, we fill in a gap in the proof of the 2002 paper. There it was claimed that an axial map f must satisfy f^*(X)=X_1+X_2. We realized recently that this is not clear. However, here we show that it is true up multiplication by a unit in the appropriate ring, and so we retrieve all the nonimmersion results claimed in the original paper. Finally, we present a complete determination of tmf^{8*}(RP^\infty\times RP^\infty) and tmf^*(CP^\infty\times CP^\infty) in positive dimensions. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Muro-Schwede-Strickland/tcwm17 Author(s): Fernando Muro, Stefan Schwede, Neil Strickland Abstract: We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Oliver-Ventura/ov2 Saturated fusion systems over $2$-groups Bob Oliver & Joana Ventura AMS classification: Primary 20D20. Secondary 20D45, 20D08 Abstract: We develop methods for listing, for a given 2-group $S$, all nonconstrained centerfree saturated fusion systems over $S$. These are the saturated fusion systems which could, potentially, include minimal examples of exotic fusion systems: fusion systems not arising from any finite group. To test our methods, we carry out this program over four concrete examples: two of order $27$ and two of order $2^{10}$. Our long term goal is to make a wider, more systematic search for exotic fusion systems over 2-groups of small order. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Panin-Pimenov-Roendigs/BGL-post Author: Ivan Panin Author2: Konstantin Pimenov Author3: Oliver Roendigs Title: On Voevodsky's algebraic K-theory spectrum BGL Under a certain normalization assumption we prove that the Voevodsky's spectrum BGL which represents algebraic $K$-theory is unique over the integers. Following an idea of Voevodsky, we equip the spectrum BGL with the structure of a commutative ring spectrum in the motivic stable homotopy category. Furthermore, we prove that under a certain normalization assumption this ring structure is unique over the integers We pull this structure back to get a distinguished monoidal structure on BGL for an arbitrary Noetherian base scheme. ---------------------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 should submit 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 at math.purdue.edu telling him what you have uploaded. 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). I am solely responsible for these messages---don't send complaints about them to Clarence. Thanks to Clarence for creating and maintaining the archive.