---------------------------------- 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 $2^7$ 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. -----------------