Subject: new Hopf listings From: Mark Hovey Date: 05 Jun 2006 08:59:53 -0400 There are 7 new papers this time, from Bartels-Rosenthal, Chermak-Oliver-Shpectorov, Dwyer-Wilkerson, Jardine (2), Stacey-Whitehouse, and Wilkerson. Mark Hovey New papers appearing on hopf between 5/3/06 and 6/5/06 1. http://hopf.math.purdue.edu/cgi-bin/generate?/Bartels-Rosenthal/asymptotic Authors: Arthur Bartels, David Rosenthal arXiv submission number: math.KT/0605088 Abstract: It is proved that the assembly maps in algebraic K- and L-theory with respect to the family of finite subgroups is injective for groups with finite asymptotic dimension that admit a finite model for the classifying space for proper actions. The result also applies to certain groups that admit only a finite dimensional model for this space. In particular, it applies to discrete subgroups of virtually connected Lie groups. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/Chermak-Oliver-Shpectorov/fundsol The simple connectivity of $B\Sol(q)$ by Andrew Chermak, Bob Oliver, and Sergey Shpectorov Andrew Chermak Kansas State University Bob Oliver LAGA, Institut Galil\'ee Sergey Shpectorov University of Birmingham Abstract: A $p$-local finite group is an algebraic structure which includes two categories, a fusion system and a linking system, which mimic the fusion and linking categories of a finite group over one of its Sylow subgroups. The $p$-completion of the geometric realization of the linking system is the classifying space of the finite group. In this paper, we study the geometric realization, \emph{without} completion, of linking systems of certain exotic 2-local finite groups whose existence was predicted by Solomon and Benson, and prove that they are all simply connected. The file "Co3graph.eps" must be included with the dvi file. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Wilkerson/GorensteinCoinvariants POINCAR'E DUALITY AND STEINBERG'S THEOREM ON RINGS OF COINVARIANTS W. G. DWYER AND C. W. WILKERSON In this note we use elementary methods to prove Steinberg's result for fields of characteristic 0 or of characteristic prime to the order of W . This gives a new proof even in the characteristic zero case. 1.1. Theorem. Let k be a field, V an r-dimensional k-vector space, and W a finite subgroup of Aut k(V ). Let S = S[V #] be the symmetric algebra on V # the k-dual of V, and R = S^W the ring of invariants of under the natural action of W on S. Define P* to be the quotient algebra S i\tensor_R k. If the characteristic of k is zero or prime to the order of W and P* satisfies Poincar'e duality, then R is isomorphic to a polynomial algebra on r generators. Steinberg [9] has shown that R is polynomial if k is the field of complex numbers and the quotient algebra P* = S\tensor_R k satisfies Poincar'e duality (1.3). Steinberg's result was extended by Kane [3, 4] to other fields of characteristic zero, and by T.-C. Lin [5] to the case in which k is a finite field of characteristic prime to the order of W . The current proof is independent of previous methods. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/coc-cat3 Title: Cocycle categories Author: J.F. Jardine arXive submission number: math.AT/0605198 Abstract: A cocycle category H(X,Y) is defined for objects X and Y in a model category, and it is shown that the set of morphisms [X,Y] is isomorphic to the set of path components of H(X,Y) provided the ambient model category is right proper and satisfies the extra condition that weak equivalences are closed under finite products. Various applications of this result are displayed, including the homotopy classification of torsors, abelian cohomology groups, group extensions and gerbes. The older classification results have simple new proofs involving canonically defined cocycles. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/Jardine/gerbes6 Title: Homotopy classification of gerbes Author: J.F. Jardine arXive submission number: math.AT/0605200 Abstract: Gerbes are locally connected presheaves of groupoids. They are classified up to local weak equivalence by path components in a 2-cocycle category taking values in all sheaves of groups, their isomorphisms and homotopies. If F is a full presheaf of sheaves of groups, isomorphisms and homotopies, then [*,BF] is isomorphic to equivalence classes of gerbes locally equivalent to groups appearing in F. Giraud's non-abelian cohomology object of equivalence classes of gerbes with band L is isomorphic to morphisms in the homotopy category from the point * to the homotopy fibre over L for a map defined on BF and taking values in the classifying space for the stack completion of the fundamental groupoid of F. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/Stacey-Whitehouse/deloopv2 Title: Stable and Unstable Operations in mod p Cohomology Theories Authors: Andrew Stacey and Sarah Whitehouse Other useful information: math.AT/0605471 Abstract: We consider operations between two multiplicative, complex orientable cohomology theories. Under suitable hypotheses, we construct a map from unstable to stable operations, left-inverse to the usual map from stable to unstable operations. The main example is where the target theory is one of the Morava K-theories in which case our map is closely related to the Bousfield-Kuhn functor. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/Wilkerson/newfred Loop Spaces and Finiteness Clarence W. Wilkerson Purdue University This expository note began as comments on a shorter note of F.R. Cohen \cite{cohen}. Cohen's paper is an elegant application of powerful recent results in unstable homotopy theory to a problem of interest to analysts. {\sl {\bf Theorem :} (F.\,R. Cohen,\cite{cohen}) Let $X$ be a simply connected finite complex which is not contractible and let $\Omega^j_0X$ be the component of the constant map in the $j$-th pointed loop space of $X$. If $j \geq 2$, then the Lusternik-Schnirlman category of $\Omega^j_0X$ is not finite. }\\ This note includes a rederivation of the above theorem using H-space methods of W. Browder from the 60's, \cite{Browder-loop}, \cite{Browder-Torsion}. The aim is to reduce the prerequisites for Cohen's theorem to those available after a second course in algebraic topology. We end with a discussion of recent work of Lannes-Schwartz on various notions of finiteness properties and the behavior under looping. The common theme is extensive use of the action of the Steenrod algebra on the cohomology of a topological space. ---------------------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. 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