Subject: new Hopf listings From: Mark Hovey Date: 16 Jul 1999 05:59:12 -0400 Ah, summertime! When everyone gets a chance to finish those projects. 11 new papers this time, including--brace yourself for this one--a new version of Hopkins-Kuhn-Ravenel!!! Mark Hovey New papers uploaded to hopf between 6/16/99 and 7/16/99. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Gorbounov-Malikov-Schechtman/c hiral Title: Gerbes of chiral differential operators Authors: Vassily Gorbounov, Fyodor Malikov, Vadim Schechtman Already submitted to xxxLANL math.AG/9906117 V.G.: Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA;\ vgorb\@ms.uky.edu F.M.: Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA;\ fmalikov\@mathj.usc.edu V.S.: Department of Mathematics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, UK;\ vs\@maths.gla.ac.uk In this note we compute the cohomological obstruction to the existence of certain sheaves of vertex algebras on smooth varieties. These sheaves have been introduced and studied in the previous work by Malikov, Schechtman and Vaintrob, and are canonically defined for an arbitrary $X$. One can try to define a purely even counterpart of $\Omega^{ch}_X$, a sheaf of graded vertex algebras $\CO^{ch}_X$, called a {\it chiral structure sheaf}. The obstraction to its existence turns out to admit a very simple expression in terms of characteristic classes of $X$, namely it is expressed in terms of the second component of Chern character of the tangent bundle of $X$. From a different viewpoint, one can regard the above result as a geometric interpretation of the second component of the Chern character. In particular, it provides a geometric criterion for a Calabi-Yau manifold to be a $BU\langle 6\rangle$-manifold: those are precisely the manifolds which admit the above mentioned sheaf $\CO^{ch}_X$. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hopkins-Kuhn-Ravenel/hkr Generalized group characters and complex oriented cohomology theories Michael J. Hopkins Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 mjh@math.mit.edu Nicholas J. Kuhn Department of Mathematics, University of Virginia, Charlottesville, VA 22903 njk4x@virginia.edu Douglas C. Ravenel Department of Mathematics, University of Rochester, Rochester, NY 14627 drav@math.rochester.edu AMS classification numbers: Primary 55N22; Secondary 20C99, 55N91, 55R35 Though it seems a shame to mess with an undergraound cult classic, this July 1999 preprint is intended to replace earlier versions dating from 1989 and 1992. It is also intended to get those of you who regularly bug us about this off our case. Cheers, Nick Let BG be the classifying space of a finite group G. Given a multiplicative cohomology theory E^*, the assignment G ---> E^*(BG) is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories E^*, using the theory of complex representations of finite groups as a model for what one would like to know. An analogue of Artin's Theorem is proved for all complex oriented theories: the abelian subgroups of G serve as a detecting family for E^*(BG), modulo torsion dividing the order of G. When E^* is a complete local ring, with residue field of characteristic p and associated formal group of height n, we construct a character ring of class functions that computes E^*(BG) tensored with the rationals. The domain of the characters is G(n,p), the set of n--tuples of elements in G each of which has order a power of p. A formula for induction is also found. The ideas we use are related to the Lubin Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, E_n^--theory, etc. The nth Morava K--theory Euler characteristic for BG is computed to be the number of G--orbits in G(n,p). For various groups G, including all symmetric groups, we prove that K(n)^*(BG) is concentrated in even degrees. Our results about E^*(BG) extend to theorems about E^*(EG\times_G X), where X is a finite G--CW complex. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/IsaksenD/prospace Title: A Model Structure on the Category of Pro-Simplicial Sets Author: Daniel C. Isaksen Address: Department of Mathematics University of Chicago Chicago, IL 60637, USA Email: dci@math.uchicago.edu Abstract: We study the category pro-SS of pro-simplicial sets, which arises in etale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on pro-SS so that it is possible to do homotopy theory in this category. This model structure is related to the strict structure of Edwards and Hastings. In order to understand the notion of homotopy groups for pro-spaces we use local systems on pro-spaces. We also give several other descriptions of weak equivlences, including a cohomological characterization. An appendix contains dual constructions for ind-spaces. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Asymptotics Title: Cobordism of symplectic manifolds and asymptotic expansions Author: Jack Morava Department of Mathematics The Johns Hopkins University Baltimore 21218 Maryland USA jack@math.jhu.edu AMS classification numbers: 55N22, 58Z05, 81S10 Abstract: The cobordism ring defined by manifolds with symplectic structure, in the sense of V.L. Ginzburg [which is NOT the cobordism ring of the Thom spectrum MSp] is shown to be isomorphic to the cobordism ring defined by almost-complex manifolds together with a complex line bundle, and as well to a cobordism ring defined by prequantized manifolds in the sense of Kostant and others. The author uses this as a hook upon which to hang some far-fetched speculations about formal group laws in topology and asymptotic expansions in physics. This paper is to appear in the Mathematical Publications of the Steklov Institue, in a volume dedicated to S.P. Novikov. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Invols Title: Cobordism of involutions revisited, revisited Author: Jack Morava Department of Mathematics The Johns Hopkins University Baltimore 21218 Maryland USA jack@math.jhu.edu AMS classification numbers: 55-03, 55N22, 55N91 Abstract: This is the writeup of a talk at the 1998 AMS Winter meeting, on Mike Boardman's early work on the Conner-Floyd five-halves conjecture; it will appear in Contemporary Mathematics 239. The main point is that Boardman's technical innovations [in the case of unoriented geometric bordism of involutions] foreshadow recent work by Greenlees and Kriz, on equivariant homotopy-theoretic bordism. Attention is also drawn to related old work of Quillen on the use of the language of residues in algebraic topology. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Sympquestions Title: Questions about cobordism of symplectic and toric manifolds Author: Jack Morava Department of Mathematics The Johns Hopkins University Baltimore 21218 Maryland USA jack@math.jhu.edu AMS classification numbers: 55N22, 14M25, 58Z05 Abstract: This note contains no results. It is a kind of footnote to an earlier paper, about V.L. Ginzburg's cobordism ring of manifolds with symplectic structure. Toric manifolds have such structure, and the purpose of this note is to raise some natural questions about the symplectic cobordism classes of such manifolds. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Topgrav Title: Topological gravity in dimensions two and four Author: Jack Morava Department of Mathematics The Johns Hopkins University Baltimore 21218 Maryland USA jack@math.jhu.edu AMS classification numbers: 55P, 58D, 83C Abstract: This is a writeup of a talk at the Utrecht conference on Operads in June 1999: the main observation is that the category with d-manifolds as objects, and (d+1)-dimensional cobordisms as morphisms, is naturally a two-category, with diffeomorphisms as the two-morphisms. The corresponding topological category obtained by replacing the morphism categories with their classifying spaces has deep connections with Riemannian geometry; its monoidal representations are the physicists' theories of topological gravity. Five examples are sketched, four corresponding to d=1 and one to d=3, and the paper concludes with a remark about adjoint structures on such categories. A mistake in some earlier papers on 2D gravity [posted in this Archive] is noted. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Rudyak-Tralle/MasseyThom On Thom spaces, Massey products and non-formal symplectic manifolds Yuli Rudyak and Aleksy Tralle July 6, 1999 We suggest a simple general method of constructing of non-formal manifolds. In particular, we construct a large family of non-formal symplectic manifolds. Here we detect non-formality via non-triviality of rational Massey products. In fact, we analyze the behaviour of Massey products of closed manifolds under the blow-up construction. In this context Thom spaces play the role of a technical tool which allows us to construct non-trivial Massey products in an elegant way. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Salvatore/conflab Title: Configuration spaces with summable labels Author: Paolo Salvatore AMS Classification numbers: 55R35; 55S15; 57N65 xxx preprint math.AT/9907073 Address: University of Bonn Beringstrasse 1 53115 Bonn Germany e-mail: salvator@math.uni-bonn.de Let M be an n-manifold, and let A be a space with a partial sum behaving as an n-fold loop sum. We define the space C(M;A) of configurations in M with summable labels in A via operad theory. Some examples are symmetric products, labelled configuration spaces, and spaces of rational curves. We show that C(I^n,dI^n;A) is an n-fold classifying space of C(I^n;A), and for n=1 it is homeomorphic to the classifying space by Stasheff. If M is compact, parallelizable, and A is path connected, then C(M;A) is homotopic to the mapping space Map(M,C(I^n,\de I^n;A)). 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Sinha/moz2 Real Equivariant Bordism and Stable Transversality Obstructions for $\ints/2$ by Dev Sinha Mathematics Department Box 1917 Brown University Providence, RI 02912 E-mail: dps@math.brown.edu In this paper we compute homotopical equivariant bordism for the group ${\bf Z/2}$, namely $MO^{\bf Z/2}$, geometric equivariant bordism $\Omega^{\bf Z/2}_*$, and their quotient as modules over geometric bordism. This quotient is a module of stable transversality obstructions. In doing these computations, we use the techniques of \cite{Si1}. Because we are working in the real setting only with $\ints/2$, these techniques simplify greatly. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Sinha/mugcomp Computations of Complex Equivariant Bordism Rings by Dev Sinha Mathematics Department Box 1917 Brown University Providence, RI 02912 E-mail: dps@math.brown.edu This paper is a significantly revised version of a previous submission to the Hopf archive. In this paper we compute homotopical bordism rings $MU^G_*$ for abelian compact Lie groups G, giving explicit generators and relations. The key constructions are operations on equivariant bordism which should play an important role in equivariant stable homotopy theory more generally. The main technique used is localization of the theory by inverting Euler classes. Applications to homotopy theory include analysis of the completion map from $MU^G_*$ to $MU^*(BG)$. Applications to geometry include classification up to cobordism of $S^1$ actions on stably complex four-manifolds with precisely three fixed points, answering a question of Bott. ---------------------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://www.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 WWW client (Mosaic, Netscape) to the URL listed. The general Hopf archive URL is http://hopf.math.purdue.edu There are links to conference announcements, Purdue seminars, and other math related things on this page as well. The largest archive of math preprints is at http://xxx.lanl.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 xxx, send e-mail to math@xxx.lanl.gov 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, cd to /pub/incoming. Transfer the dvi file using binary, by first typing binary then put You should also transfer an abstract as well. 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