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.
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More useful is the front end developed by Greg Kuperberg:
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You can also use ftp to hopf.math.purdue.edu, and login as ftp. Then cd
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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
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I am solely responsible for these messages---don't send complaints
about them to Clarence. Thanks to Clarence for creating and maintaining
the archive.
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