Subject: new Hopf listings From: Mark Hovey Date: 14 Sep 2000 11:55:24 -0400 Sorry for the long delay since the last such announcement. One big factor contributing to the delay is e-mail attachments. Clarence has trouble dealing with these, and it also messes up my system. So it would be a big help to us if you could follow the old ftp method, or the newer web browser method, of uploading papers to Hopf. 14 new papers this time, including the abstract of Larry Smith's paper that was announced last time. Mark Hovey New papers appearing on hopf between 7/16/00 and 9/14/00. 0. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/SmithL/koszulii (This paper was announced last time without abstract. Here is the abstract.) Title of Paper: Invariant Theory and the Koszul Complex Representations of Z/p in Characteristic p Applications Author: Larry Smith AMS Code: 13A50 Invariant Theory Address: Mathematisches Institut Bunsenstrasse 3--5 D 37073 Goettingen Federal republic of germany e-mail: larry@sunrise.uni-math.gwdg.de THIS IS a POstScript file. Summary: We study the ring of invariants $\F[V]^{\Z/p}$\/, and its derived functors $H^i(\Z/p\semicolon \F[V])$\/, of the cyclic group $\Z/p$ of prime order $p$ over a field $\F$ of characteristic $p$\/. We verify a formula of Ellingsrud and Skjelbred \cite{norway} for the homological codimension, show the quotient algebra $\F[V]^{\Z/p}/\Im(\Tr^{\Z/p})$ is Cohen-Macaulay, and that the ideal generated by the elements in the image of the transfer homomorphism, $\Im(\Tr^{\Z/p}) \subset \F[V]^{\Z/p}$\/, is primary of height $n-1$ when $V$ is an $n$-dimensional irreducible representation of $\Z/p$\/. Using our cohomological computations and a previous result \cite{vectors} about permutation representations we are able to obtain an upper bound for the degree of homogeneous forms in a minimal algebra generating set for $\F[V]^{\Z/p}$\/. 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Basterra/abwgeec Title: The Witten genus and equivariant elliptic cohomology Authors: Matthew Ando mando@math.uiuc.edu Maria Basterra basterra@math.uiuc.edu Department of Mathematics, The University of Illinois at Urbana-Champaign Abstract: We construct a Thom class in complex equivariant elliptic cohomology extending the equivariant Witten genus. This gives a new proof of the rigidity of the Witten genus, which exhibits a close relationship to recent work on non-equivariant orientations of elliptic spectra. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Hopkins-Strickland/eswgtc -2/ Elliptic spectra, the Witten genus, and the theorem of the cube. (revised version) M. Ando, M. J. Hopkins, and N. P. Strickland University of Illinois at Urbana-Champaign mando@math.uiuc.edu MIT mjh@math.mit.edu University of Sheffield N.P.Strickland@sheffield.ac.uk This is a revised version of an earlier paper (1998) with the same title. We show that every elliptic spectrum receives a natural MU<6>-orientation. For the elliptic spectrum defined by the Tate curve, this orientation specializes to the Witten genus. The naturality of the orientation implies that the modularity of the Witten genus for MU<6>-manifolds. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Broto-Levi-Oliver/blo1 Homotopy equivalences of p-completed classifying spaces of finite groups by Carles Broto, Ran Levi, and Bob Oliver We study homotopy equivalences of p-completions of classifying spaces of finite groups. To each finite group G and each prime p, we associate a finite category with the following properties. Two p-completed classifying spaces BG_p^\wedge and BG'_p^\wedge have the same homotopy type if and only if the associated categories are equivalent. And the topological group Aut(BG_p^\wedge) of self equivalences is determined by the self equivalences of the associated category. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Bruner-Davis-Mahowald/eo2/ Nonimmersions of real projective spaces implied by eo2 Robert R. Bruner Wayne State University, Detroit, MI 48202 rrb@math.wayne.edu Donald M. Davis Lehigh University, Bethlehem, PA 18018 dmd1@lehigh.edu Mark Mahowald Northwestern University, Evanston, IL 60201 mark@math.nwu.edu AMS Classifications: 57R42, 55N20 Abstract: Recently Hopkins and Mahowald constructed a new 2-primary ring spectrum eo2, satisfying H^*(eo2)=A//A2. We use eo2 to obtain new results regarding nonimmersions of real projective spaces in Euclidean space. The method is to say enough about eo2-cohomology of a product of real projective spaces to obtain nonexistence of certain axial maps. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Chacholski-Dwyer-Intermont/com plication The A-complication of a space W. Chacholski, W. G. Dwyer, and M. Intermont Suppose that A is a pointed CW-complex. We look at how difficult it is to construct an A-cellular space B from copies of A by repeatedly taking homotopy colimits; this is determined by an ordinal number called the complication of B. Studying the complication leads to an iterative technique, based on resolutions, for constructing the A-cellular approximation CW_A(X) of an arbitrary space X. Yale University, New Haven, CT 06520 USA University of Notre Dame, Notre Dame IN 46556 USA Kalamazoo College, Kalamazoo MI, 49006 USA MSC2000: 55P60, 55P99 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Fors/AugmHom Title: Augmental Homology Theory and the Künneth Formula for Topological Joins. Author: Göran Fors. AMS Classification numbers: 55N10. Address: Department of Mathematics, University of Stockholm, SE-106 91 Stockholm, Sweden E-mail address: goranf@matematik.su.se We prove topological join versions of the relative Eilenberg-Zilber Theorem and the relative Künneth Formula. We also express the local homology groups for topological joins and products in terms the local homology groups for the factors. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Gorbunov-Malikov-Schechtman/gr oup-all-fedin1 On chiral differential operators over homogeneous spaces Vassily Gorbounov, Fyodor Malikov, Vadim Schechtman 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.: IHES, 35 Route de Chartres, 91440 Bures-sur-Yvette, France;\ vadik\@ihes.fr The notion of an algebra of chiral differential operators (cdo for short) over a smooth algebraic variety X has been studied by the authors previously. We give a classification of cdo over X in the following cases: X=G is an affine algebraic group; X=G/N or G/P where N is a unipotent subgroup and P is a parabolic subgroup and G is simple (the extension to the case of a semisimple G being straightforward). The above sheaves are constructed using the BRST (or quantum Hamiltonian) reduction of the corresponding cdo's on G. The classification of cdo over homogeneous spaces is exactly reflected in the BRST world: namely the square of the corresponding BRST charge is zero at all levels for G/N, only at the critical level for G/B and is never zero for G/P. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ishiguro/G2 "Classifying spaces and a subgroup of the exceptional Lie group G_2" Kenshi Ishiguro Mathematics subject classification: 55R35 Abstract: We consider a problem on the conditions of a compact Lie group that its loop space of the p-completed classifying space be a p-compact group, as well as some related problems. A previously obtained necessary condition is shown to be not sufficient. Our counterexample is given by a quotient group \Gamma_2 of a subgroup of the exceptional Lie group G_2 at p=3. The 3-adic K-theory of B\Gamma_2 and BG_2 are isomorphic , though the loop space of the 3-completion of B\Gamma_2 is not a 3-compact group. 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Klein/quinn Title: The dualizing spectrum of a topological group Author: John R. Klein AMS subjclass: Primary: 55P91, 55N91, 55P42, 57P10. Secondary: 55P25, 20J05,18G15. Address: Dept. Of Mathematics, Wayne State University, Detroit, MI 48202 e-mail: klein@math.wayne.edu Abstract: To a topological group G, we assign a naive G-spectrum D_G, called the "dualizing spectrum" of G. When the classifying space BG is finitely dominated, we show that D_G detects Poincare duality in the sense that BG is a Poincare duality space if and only if D_G is a homotopy finite spectrum. Secondly, we show that the dualizing spectrum behaves multiplicatively on certain topological group extensions. In proving these results we introduce a new tool: a "norm map" which is defined for any G and for any naive G-spectrum E. Applications include: (1) a homotopy theoretic solution to a problem posed by Wall which says that in a fibration sequence of finitely dominated spaces, the total space satisfies Poincare duality if and only if the base and fiber do. (2) An entirely homotopy theoretic construction of the Spivak fibration of a finitely dominated Poincare duality space. (3) A new proof of Browder's theorem that every finite H-space satisfies Poincare duality. (4) We show how to define a variant of Farrell-Tate cohomology for any topological or discrete group G, with coefficients in any naive equivariant cohomology theory E. We prove a vanishing result for this theory. In an appendix, we identify the homotopy type of D_G for certain kinds of groups. The class includes all compact Lie groups, torsion free arithmetic groups and Bieri-Eckmann duality groups. (This paper has already been accepted for publication in Math. Annalen.) 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Mandell/chains Cochain Multiplications Michael A. Mandell mandell@math.uchicago.edu Abstract We describe a refinement of the Eilenberg--Steenrod axioms that provides a necessary and sufficient condition for functors from spaces to algebras or E-infty algebras to be naturally quasi-isomorphic to the singular cochain functor. 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/McAuley/revised-hilbert (This is a revised version of the author's paper proving the Hilbert-Smith conjecture about certain topological groups being forced to be Lie. The abstract has appeared at least twice before here, so I omit it). MH 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Pakianathan-YalcinE/nc Title: On Commuting and Non-Commuting Complexes Authors: Jonathan Pakianathan and Erg\"un Yal\c c\i n 2000 Mathematics Subject Classification. Primary: 20J05; Secondary: 06A09, 05E25. Addresses: Department of Mathematics University of Rochester N.Y., U.S.A. Department of Mathematics Bilkent University Ankara, Turkey Abstract: In this paper we study various simplicial complexes associated to the commutative structure of a finite group $G$. We define $NC(G)$ (resp. $C(G)$) as the complex associated to the poset of pairwise non-commuting (resp. commuting) sets of nontrivial elements in $G$. We observe that $NC(G)$ has only one positive dimensional connected component, which we call $BNC(G)$, and we prove that $BNC(G)$ is simply connected. Our main result is a simplicial decomposition formula for $BNC(G)$ which follows from a result of A. Bj\"orner, M. Wachs and V. Welker on inflated simplicial complexes. As a corollary we obtain that if $G$ has a nontrivial center or if $G$ has odd order, then the homology group $H_{n-1}(BNC(G))$ is nontrivial for every $n$ such that $G$ has a maximal noncommuting set of order $n$. We discuss the duality between $NC(G)$ and $C(G)$, and between their $p$-local versions $NC_p(G)$ and $C_p(G)$. We observe that $C_p(G)$ is homotopy equivalent to the Quillen complexes $A_p(G)$, and obtain some interesting results for $NC_p(G)$ using this duality. Finally, we study the family of groups where the commutative relation is transitive, and show that in this case, $BNC(G)$ is shellable. As a consequence we derive some group theoretical formulas for the orders of maximal non-commuting sets. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/YalcinE/clpg4 Title: Set Covering and Serre's Theorem on the Cohomology Algebra of a $p$-Group Author: Erg\" un Yal\c c\i n 2000 Mathematics Subject Classification. Primary: 20J06; Secondary: 20D15, 20D60, 51E20. Address: Department of Mathematics Bilkent University Ankara, Turkey Email: yalcine@math.mcmaster.ca Abstract: We define a group theoretical invariant, denoted by $s(G)$, as a solution of a certain set covering problem, and show that it is closely related to $chl(G)$, the cohomology length of a $p$-group $G$. By studying $s(G)$, we improve the known upper bounds for the cohomology length of a $p$-group, and determine $chl(G)$ completely for extra-special $2$-groups of real type. ---------------------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 (Netscape< Internet Explorer) to the URL listed. The general Hopf archive URL is http://hopf.math.purdue.edu There are links to 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. Clarence has explicit instructions for the form of this abstract: see http://hopf.math.purdue.edu/pub/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.