Subject: new Hopf listings From: Mark Hovey Date: 17 Jul 1998 02:53:54 -0400 Both Clarence and I have been out of town, consecutively, with the result that there are 13 papers to announce this time. The two papers by Jack Morava below are "postfinal" versions--each has an appendix not present in the published version! So Jack is taking Tibor Beke's suggestion, for which I confess a sentimental attraction myself, to heart. Jack also wants me to let you know that he did not intend to imply that one should not replace preliminary postings on Hopf with final versions, merely that xxx may not be the place for preliminary versions. Mark Hovey New papers uploaded to hopf between 6/29/98 and 7/17/98: 1. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Hopkins-Strickland/eswgtc Elliptic spectra, the Witten genus, and the theorem of the cube. M. J. Hopkins, M. Ando, and N. P. Strickland MIT mjh@math.mit.edu University of Virginia and Johns Hopkins University ando@math.jhu.edu Trinity College, Cambridge n.strickland@dpmms.cam.ac.uk 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. 2. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Ando-Strickland/pairings Weil pairings and Morava K-theory M. Ando and N. P. Strickland University of Virginia and Johns Hopkins University ando@math.jhu.edu Trinity College, Cambridge n.strickland@dpmms.cam.ac.uk An important component of joint work with M. Hopkins (Elliptic spectra, the Witten genus, and the theorem of the cube) is that the complex-orientable cohomology of BU<6> represents the group of "cubical structures on the trivial torsor over the formal group". We give a proof of this result for Morava K-theories which demonstrates the close relationship of the topological situation to the algebro-geometric situation in which the notion of cubical structure originally arose. 3. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Casacuberta-Rodriguez-Tai/rigi d TITLE: "Localizations of abelian Eilenberg--Mac Lane spaces of finite type" AUTHORS: Carles Casacuberta Universitat Autonoma de Barcelona 08193 Bellaterra, Spain casac@mat.uab.es http://mat.uab.es/casac Jose L. Rodriguez Universitat Autonoma de Barcelona 08193 Bellaterra, Spain jlrodri@mat.uab.es http://mat.uab.es/jlrodri Jin-Yen Tai Department of Mathematics, Dartmouth College, Hanover, NH 03755-3551, Jin-Yen.Tai@Dartmouth.edu ABSTRACT: Using recent techniques of unstable localization, we extend earlier results on homological localizations of Eilenberg--Mac Lane spaces, and show that several deep properties of such localizations can be explained by the preservation of certain algebraic structures under the effect of idempotent functors. We study localizations $L_f K(G,n)$ of Eilenberg--Mac Lane spaces with respect to any map $f$, where $n\ge 1$ and $G$ is abelian. We find that, if $G$ is finitely generated, then the result is a $K(A,n)$, where $A$ can be computed using cohomological data derived from $f$. If $G=\Z$, then $A$ is a commutative ring which is isomorphic to the ring $\End(A)$ of its own additive endomorphisms; such rings, which we call rigid, form a proper class which contains the set of solid rings. From this fact it follows that there is a proper class of distinct homotopical localizations of the circle $S^1$. Among other applications of our results, we show that, if $X$ is a product of abelian Eilenberg--Mac Lane spaces and $f$ is any map, then the homotopy groups $\pi_m(L_f X)$ become modules over the ring $\pi_1(L_f S^1)$. 4. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Devadoss/mosaic Tessellations of Moduli Spaces and the Mosaic Operad Satyan L. Devadoss Primary: 14H10 Secondary: 05B45, 52B11 Department of Mathematics Johns Hopkins University Baltimore, MD 21218 devadoss@math.jhu.edu The following are all EPS files: assoc bcollide blow6 braid6 btp cubes hyperbolic k2tok4 k5codim1 k6codim1 kapdc m04 m05c m05d m05pieces m06c onepoly pairpants pcollide polycomp simpose twist twistpf Abstract: We construct a new (cyclic) operad of \emph{mosaics} defined by polygons with marked diagonals. Its underlying (aspherical) spaces are the sets \overline{\mathcal {M}}^n_0({\mathbb R}) of real points of the moduli space of punctured stable curves of genus zero, which are naturally tiled by Stasheff associahedra. We (combinatorially) describe them as iterated blow-ups and show that their fundamental groups form an operad with similarities to the operad of braid groups. 5. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Gilkey-Leahy-Sadofsky/GLSeigen Title: Riemannian manifolds whose skew-symmetric curvature operator has constant eigenvalues Authors: Peter B. Gilkey, John V. Leahy, Hal Sadofsky AMS classification: 53B20 Address: Department of Mathematics, University of Oregon, Eugene, OR 97403. Email: gilkey@math.uoregon.edu, leahy@math.uoregon.edu, sadofsky@math.uoregon.edu Abstract: A Riemannian metric on a manifold is said to be IP if the eigenvalues of the skew-symmetric curvature operator are pointwise constant, i.e. they depend upon the point of the manifold but not upon the particular $2$ plane in the tangent bundle at that point. We classify the IP metrics for manifolds of dimensions $m=5$, $m=6$, and $m>8$. 6. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hirschowitz-Simpson/descente Title: Descente pour les $n$-champs (Descent for $n$-stacks) Authors: Andr\'e Hirschowitz, Carlos Simpson Authors' addresses: Universit\'e de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice cedex 2, France; Laboratoire Emile Picard, Universit\'e Toulouse 3, 31062 Toulouse cedex, France Authors' email addresses: ah@math.unice.fr; carlos@picard.ups-tlse.fr Subj-class: Algebraic Geometry; Algebraic Topology; Category Theory Abstract: We develop the theory of $n$-stacks (or more generally Segal $n$-stacks which are $\infty$-stacks such that the morphisms are invertible above degree $n$). This is done by systematically using the theory of closed model categories (cmc). Our main results are: a definition of $n$-stacks in terms of limits, which should be perfectly general for stacks of any type of objects; several other characterizations of $n$-stacks in terms of ``effectivity of descent data''; construction of the stack associated to an $n$-prestack; a strictification result saying that any ``weak'' $n$-stack is equivalent to a (strict) $n$-stack; and a descent result saying that the $(n+1)$-prestack of $n$-stacks (on a site) is an $(n+1)$-stack. As for other examples, we start from a ``left Quillen presheaf'' of cmc's and introduce the associated Segal $1$-prestack. For this situation, we prove a general descent result, giving sufficient conditions for this prestack to be a stack. This applies to the case of complexes, saying how complexes of sheaves of $\Oo$-modules can be glued together via quasi-isomorphisms. This was the problem that originally motivated us. 7. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Hovey/stable-model Stabilization of Model Categories by Mark Hovey Wesleyan University hovey@member.ams.org Suppose C is a (nice enough) model category, and G: C --> C is a left Quillen endofunctor of C. Think of C as the category of pointed topological spaces, and G as the suspension. Then we construct a new model category Sp(C,G), an embedding C --> Sp(C,G), and an extension of G to a Quillen EQUIVALENCE of Sp(C,G). Essentially, we have inverted the functor G, up to homotopy. When C is the category of pointed topological spaces and G is the suspension, we recover the Bousfield-Friedlander model category of spectra. The trouble with the Bousfield-Friedlander model category is that it is not symmetric monoidal, and we have the same problem with Sp(C,G). But there is also the same fix. Suppose C is a (nice enough) symmetric monoidal model category, K is a cofibrant object of C, and D is a (nice enough) C-model category. Think of C as pointed simplicial sets, D as a pointed simplicial model category, and K as the simplicial circle. Then we construct a model category Sp^Sigma(D,K), so that Sp^Sigma(C,K) is a symmetric monoidal model category, Sp^Sigma(D,K) is a Sp^Sigma(C,K)-model category, and smashing with K is a Quillen equivalence on Sp^Sigma(D,K). When C is pointed simplicial sets, and K is S^1, we get the symmetric spectra of Hovey-Shipley-Smith. The method used is the Bousfield localization technology of Hirschhorn, so the words "nice enough" mean "left proper cellular", though occasionally we also need to assume to domains of the generating cofibrations are cofibrant. 8. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Jardine/sym Title of paper: Presheaves of symmetric spectra Author: J.F. Jardine AMS Classification numbers: 55P42 18F20 55U35 Address of Author: Mathematics Department University of Western Ontario London, Ontario N6A 5B7 Canada Email: jardine@uwo.ca This paper shows that there is a proper closed simplicial model category on the category of presheaves of symmetric spectra on an arbitrary Grothendieck site, and that the resulting homotopy category is equivalent to the stable category of presheaves of spectra. The argument follows the outline established by Hovey, Shipley and Smith, while many of the techniques of proof originate in the Goerss-Jardine paper "Localization theories for simplicial presheaves". This paper was written in lamstex and requires the lamstex fonts to view or print. A postscript version is available at http://www.math.uwo.ca/~jardine/papers/ 9. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Kuhn/kuhnloops "New relationships among loopspaces, symmetric products, and Eilenberg MacLane s paces" Nicholas J. Kuhn AMS classification number: 55P42 Mathematics Department University of Virginia Charlottesville, VA 22903 njk4x@virginia.edu This is a revised version of the 1996 preprint "New cohomological relationships among loopspaces, symmetric products, and Eilenberg MacLane spaces". The paper studies a bigraded family of finite spectra T(n,j), at p=2, which specialize to the dual Brown-Gitler spectra when n=1. One can take hocolimits of these as either j goes to infinity or n goes to infinity. When one lets j go to infinity, one gets in cohomology A-modules, which are shown to be related to the cohomology of K(V,n)'s in the same way that the Carlsson modules are related to the cohomology of K(V,1)'s. When one lets n go to infinity, one gets a filtration of HZ/2 that cohomologically looks like the mod 2 Whitehead conjecture filtration (a modified symmetric products of spheres filtration). A result new in the revision is that this IS the modified symmetric products of spheres filtration. Also new in the revision is an appendix which relates my constructions to work of Arone-Mahowald, and Arone-Dwyer on the Goodwillie tower of spheres. 10. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Luminy6-final \begin{center}{Abstract for {\bf Quantum generalized cohomology}}\bigskip \end{center} \noindent There is a variant of Segal's category of Riemann surfaces, in which morphisms are stable complex algebraic curves [i.e. double points are allowed], with some smooth points marked; composition is defined by glueing at marked points. The spaces of morphisms in this category are built from the compactified moduli spaces $\overline M_{g,n}$ of Deligne, Mumford, and Knudesen; here $g$ is the genus and $n$ is the number of marked points. A generalized topological field theory taking values in the category of module-spectra over a ring-spectrum $\bf R$ is a family $$\tau_{g,n} : \overline M_{g,n} \rightarrow {\bf M} \wedge_{\bf R} \dots \wedge_{\bf R} {\bf M} = {\bf M}^{\wedge n}$$ of maps, which respect composition of morphisms. More precisely, $\bf M$ is an $\bf R$-module spectrum, $\wedge_{\bf R}$ is the Robinson smash product, and $\bf M$ is endowed with a suitably nondegenerate bilinear form $${\bf M} \wedge_{\bf R} {\bf M} \rightarrow {\bf R}.$$ This data entails the existence of an $\bf R$-algebra structure on $\bf M$, such that $\tau_{g,1}$ is a morphism of monoids if the moduli space of curves is given the pair-of-pants product; it seems to define a natural context for quantum generalized cohomology.\medskip \noindent There is an interesting example of all this, associated to a smooth algebraic variety $V$. It is closely related to the Tate $\bf MU$-cohomology of the universal cover of the free loopspace of $V$, but it can be described more concretely in terms of the rational Novikov ring $\Lambda = {\Bbb Q} [H_{2}(V,{\Bbb Z})]$ of $V$ by setting ${\bf R} = {\bf MU} \otimes \Lambda$; then {\bf E} is the function spectrum $F(V,{\bf R})$ representing the cobordism of $V$ tensored with $\Lambda$, and the bilinear pairing is defined by Poincar\'e duality. In this case $\tau_{g,n}$ represents the cobordism class of the space of stable maps [in the sense of Kontsevich] from a curve of genus $g$, marked with $n$ ordered smooth points together with an indeterminate number of unordered smooth points, to $V$. A variant construction requires the unordered points to lie on a cycle $z$ in $V$; this defines a parameterized family of multiplications satisfying the analogue of the WDVV equation. When $V$ is a point, the resulting theory boils down to the version of topological gravity I advertised at the Adams Symposium; the coupling constant of the associated topological field theory is the cobordism analogue of Manin's exponential $$\sum_{n \geq 0} \overline M_{0,n+3} \frac {z^{n}}{n!} .$$ Although much of the machinery used here comes from fields adjacent to topology, this paper is concerned with the old problem of constructing complex cobordism out of Riemann surfaces by some analogue of the plus-construction. Having hacked through the physics background, I hope to produce a more topological account in the near future. \medskip \noindent This is to appear in Contemporary Math., in the Proceedings of the Hartford/Luminy Conference on the Renaissance of Operads, ed. J.-L. Loday, J. Stasheff, and A. A. Voronov. \end{document} [ Jack tells me that this is the final version. I've labeled the DVI file as Luniny6-final.dvi and reclused the original version, CWW 7/13/98] 11. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Morava/Schur2-final Schur cohomology and a Kontsevich-Witten genus Jack Morava AMS Classification: Primary 14H10, Secondary 55N35, 81R10 Johns Hopkins University: jack@math.jhu.edu ABSTRACT: Two-dimensional topological gravity is a kind of physicist's interpretation of the rational cohomology of the group completion of the monoid of Riemann surfaces under glueing. It has a natural algebra of operations, which look vaguely like the operations in complex cobordism, and Witten has raised the question of their possible homotopy-theoretic interpretation. Over the integers this theory turns out to have an interesting model, which looks a lot like (a double of) the cohomology of Sp/U. There is an associated formal-group-like object, which looks unfamiliar because its coordinate seems to be centered at infinity, corresponding to asymptotic expansions of interest in physics. [This paper is a kind of sequel to 'Generalized quantum cohomology' posted previously on {\bf Hopf}, which has since appeared [in Contemporary Math. 202, Proceedings of the operads renaissance conference, ed. Loday, Stasheff, & Voronov] [ Jack tells me that this is the final version. I've labeled the DVI file as Schur2-final.dvi and reclused the original version, CWW 7/13/98] 12. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Stanley/ls1 Title:``Spaces with Lusternik-Schnirelmann category n and cone length n+1'' Author: Don Stanley AMS-classification number: 55P50 Address: Don Stanley Freie Universitaet Berlin Institut fur Mathematik II Arnimallee 3 14195 Berlin Germany email: stanley@math.fu-berlin.de Abstract: We construct a series of spaces, $X(n)$, for each $n>0$, such that $cat(X(n))=n$ and $cl(X(n))=n+1$. We show that the Hopf invariants determine whether the category of a space goes up when attaching a cell of top dimension. We give a new proof of counterexamples to Ganea's conjecture. Also we introduce some techniques for manipulating cone decompositions. 13. http://hopf.math.purdue.edu/cgi-bin/generate?/pub/Voronov/swiss/voronov-swiss The Swiss-Cheese Operad Alexander A. Voronov AMS Classification: Primary 55P99, 18C99; Secondary 14H10, 17A30, 17A42, 81T40 Department of Mathematics M.I.T., 2-246 77 Massachusetts Ave. Cambridge, MA 02139-4307 Email address: voronov@math.mit.edu Included EPS or PS files: disks.eps and semidisks.eps Abstract. We introduce a new operad, which we call the Swiss-cheese operad. It mixes naturally the little disks and the little intervals operads. The Swiss-cheese operad is related to the configuration spaces of points on the upper half-plane and points on the real line, considered by Kontsevich for the sake of deformation quantization. This relation is similar to the relation between the little disks operad and the configuration spaces of points on the plane. The Swiss-cheese operad may also be regarded as a finite-dimensional model of the moduli space of genus-zero Riemann surfaces appearing in the open-closed string theory studied recently by Zwiebach. We describe algebras over the homology of the Swiss-cheese operad. ---------------------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.cs.wesleyan.edu/Math/Guests/Mark If this doesn't work or is missing a few issues, try http://www.lehigh.edu/~dmd1/public/www-data/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 general xxx archive URL is http://xxx.lanl.gov. More useful is the front end developed by Greg Kuperberg: http://front.math.ucdavis.edu You can also use 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/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. For instructions on uploading papers to xxx, see http://front.math.ucdavis.edu I am solely responsible for these messages---don't send complaints about them to Clarence. Thanks to Clarence for creating and maintaining the archive. ------- End of forwarded message ------- ------- End of forwarded message -------