
Homotopy Theory Program
Workshop on Homotopy, Geometry and Physics
The Fields Institute
April 1922,1996
Abstracts
Fred Cohen,
University of Rochester 
Configuration Spaces and Mapping Class groups
This lecture describes some work on the overlap between configuration
spaces and the mapping class groups of orientable Riemann surfaces.
Some of this work has appeared while some is in progress with
H.W. Henn.
Natural subbundles of the tautological (Hopf) bundle and flag
varieties were constructed along with natural actions of the unitary
group. In some cases, the resulting bundles are K(G,1)'s where
G is closely related to the mapping class group of an orientable
surface. A specific case is given by the space of n particles
on CP(1) where each particle is equipped with a parameter in the
circle and where the entire space is taken modulo a natural U(2)action.
A specific case is n = 6 where G is the mapping class group for
genus 2 surfaces.
These spaces were assembled into a single space which is an analogue
of the DoldThom construction. The homology (with any field coefficients)
of this construction has homology which is given by the homology
of certain mapping class groups and with coefficients given by
various choices of representations. This "picture" is
a straightforward analogue of the cyclic homology for certain
algebras via two natural fibrations. This analogue is used in
the next paragraph.
Some explicit and elementary calculations for the homology of
mapping class groups are then given for the certain (easy) cases.
Some examples are listed which touch upon (1) automorphic forms
and (2) rings of invariants obtained from tensoring a representation
of the symmetric group on nletters of rank (n1)! which is sometimes
named Lie(n) (and which is a special case of [SLM, v 533, Thm
12.3, page 302]) with a polynomial ring supporting a permutation
representation.
In joint work with Henn, there is further partial information
given for the 2torsion in the cohomology of mapping class groups
associated to the surface of least genus for which a fixed elementary
abelian 2group of rank q is maximal. One such example is genus
three with 2rank 3. This twotorsion does not arise from the
cohomology of Sp(6,F) for any finite field F.
Looping these sorts of constructions and taking the Lie algebra
of primitives in homology gives a Lie algebra obtained from the
infinitesimal braid relations and which appears in work of Falk,
Randell, Kohno, Drinfel'd, et al and which is related to the KniznikZamolodchikov
equations. An explication of these facts will be given as (1)
this audience will probably be able to give an "explanation"
and (2) I would like to know whether there is a "useful"
explanation.


Charles Boyer,
University of New Mexico

Stability Theorems for Spaces of Rational Curves
This talk is based on joint work with Jacques Hurtubise and
Jim Milgram. We outline the proofs of topological theorems for
spaces of rational curves in certain smooth compact varieties.
These varieties have a dense open subvariety on which a complex
solvable group acts freely. They appear to be the largest class
of smooth varieties for which the poles and principal parts description
used in our previous work applies.


Pawel Gajer,
Texas A&M 
Geometry of higher line bundles, Deligne cohomology, and
algebraic cycles
The talk will be devoted to differential geometric and holomorphic
structures on higher line bundles and their relationship with
Deligne cohomology and groups of algebraic cycles.


Jim Milgram,
Stanford

Holomorphic maps from a Riemann surface to complex projective
space
While much is known about the stable topology of spaces of holomorphic
maps of Riemann surfaces of positive genus to projective space,
the complete structure remains to be elucidated. When the Riemann
surface is elliptic or hyperelliptic, we can however give essentially
complete results.


Ezra Getzler,
Northwestern University 
The homology of the moduli spaces M_{1,n} and their compactifications
We apply methods from mixed Hodge theory and modular operad theory
(a higher genus analogue of the theory of operads) to study the
homology groups of moduli spaces of genus 1 curves. This work
is motivated by applications to the theory of quantum cohomology.


Takashi Kimura,
Boston University

Moduli Spaces, Graph Complexes, and Their Representations
The moduli spaces of (decorated) punctured Riemann surfaces and
their compactifications have natural composition maps between
them which arise from the operations of sewing or attaching surfaces
together. The geometric structure of these moduli spaces induces
homotopy theoretic algebraic structures on the representation
space of these moduli spaces through complexes of graphs. Such
representations arise naturally in the context of quantum cohomology
as well as in mathematical physics.


Alexander A. Voronov,
University of Pennsylvania

Homotopy Gerstenhaber algebras in topological field theory
We prove that the BRST complex of a topological conformal field
theory is a homotopy Gerstenhaber algebra, as conjectured by Lian
and Zuckerman in 1992. We illustrate the usefulness of our main
tools, operads and "string vertices" by obtaining some
new results on double loop spaces.''


Jack Morava,
Johns Hopkins

Topological gravity and quantum cohomology
There is a variant of Segal's category of Riemann surfaces, in
which morphisms are stable complex algebraic curves [i.e. possibly
with double point singularities], with some smooth points marked;
composition is defined by glueing at marked points. The spaces
of morphisms in this category are therefore built from the compactified
moduli spaces $\hat M_{g,n}$, where $g$ is the genus and $n$ is
the number of marked points. A generalized topological field theory
[taking values in the category of modulespectra over a ringspectrum
$\bf R$] is a family $$\tau_{g,n} : \hat M_{g,n} \rightarrow {\bf
E} \wedge_{\bf R} \dots \wedge_{\bf R} {\bf E} = {\bf E}^{\wedge
n}$$ of maps, which respect composition of morphisms. More precisely,
$\bf E$ is an $\bf R$module spectrum, $\wedge_{\bf R}$ is the
Robinson smash product, and $\bf E$ is endowed with a suitably
nondegenerate bilinear form $${\bf E} \wedge_{\bf R} {\bf E} \rightarrow
{\bf R}.$$ This data endows $\bf E$ with the structure of an $\bf
R$algebra, such that $\tau_{g,1}$ is a morphism of monoids with
respect to the knickers product on the moduli space of curves;
it therefore seems to define a reasonable context for quantum
generalized cohomology.
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 ${\bf R} = {\bf MU} \otimes \Lambda, {\bf E} = F(V,{\bf
R})$; the bilinear pairing is defined by Poincare 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 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 discussed at the
Adams Symposium; the coupling constant of the associated topological
field theory is Manin's exponential $$\sum_{n \geq 0} \hat M_{0,n+3}\frac
{z^{n}}{n!} .$$


Lisa Jeffrey,
McGill

Quantization commutes with reduction
Suppose M is a compact symplectic manifold equipped with the
Hamiltonian action of a compact Lie group G. If M is Kahler, the
quantization of M is usually defined as the space of holomorphic
sections of a line bundle over M whose first Chern class is specified
by the symplectic form: it is a finite dimensional vector space
with a linear action of G.
We describe a new proof (joint with F. Kirwan) of the conjecture
of Guillemin and Sternberg (1982) that the Ginvariant subspace
of the quantization of M has the same dimension as the quantization
of the symplectic quotient or MarsdenWeinstein reduced space
of M: the symplectic quotient is a manifold whenever G acts freely
on the zero level set of the moment map,and it inherits a symplectic
structure. Our proof is valid whenever G is abelian, and under
suitable hypotheses also for nonabelian G.


Stefan Stolz,
Notre Dame

Manifolds of positive scalar curvature  a survey
This talk presents a survey about what is known concerning the
question which compact manifolds admit metrics of positive scalar
curvature. The central conjecture in the subject is the GromovLawsonRosenberg
conjecture which claims that a spin manifold $M$ of dimension
$n\ge 5$ admits a positive scalar curvature metric if and only
if an index obstruction $\alpha(M)\in KO_n(C^*\pi)$ vanishes.
Here $\pi$ is the fundamental group of $M$, and $KO_n(C^*\pi)$
is the $K$@theory of the $C^*$@algebra of $\pi$ (a completion
of the real group ring). Stable homotopy theory is the essential
ingredient for the proof of this conjecture for those groups $\pi$
which have periodic cohomology. We will outline the proof of a
very recent result saying that if the ``BaumConnes map"
is injective for a group $\pi$, then a stable version of the GromovLawsonRosenberg
Conjecture holds for spin manifolds with fundamental group $\pi$.
The BaumConnes map is known to be injective e.g. for discrete
subgroups of Lie groups. More generally, the BaumConnes Conjecture
claims that this map is an isomorphism for {\it all} discrete
groups $\pi$ (the injectivity part of that statement is one form
of the NovikovConjecture).


Ruth Lawrence,
Michigan

A Holomorphic version of the WittenReshetikhinTuraev
invariant
The WRT invariant, $Z_K(M)$, of 3manifolds $M$ has many different
interpretations, but until recently all have involved the assumption
that $K$ be an integer. The talk will present recent work which
enables $Z_K(M)$ to be considered as a holomorphic function of
$K$ for a large class of manifolds $M$ and will discuss its relation
with previous approaches.


Paulo LimaFilho,
Texas A&M 
EulerChow series and projective bundle formulas
We develop techniquees to compute EulerChow series associated
to a projective variety X. These are series obtained from homological
data on the Chow varieties of X, and provide an interesting collection
of invariants, and refining known objects, such as Hilbert series
etc. Various examples are computed, together with projective bundle
formulas for splitting bundles; algebraic suspension formulas
and various cases of Grassmannians and flag varieties.


Michael Kapranov,
Northwestern University 
Rational curves in flag varieties, EisensteinLanglands
series and affine quantum groups
The space of rational maps of given multidegree from the projective
line to a flag variety is ``rational", so its Poincare polynomial
can be found by counting the number of its points over finite
fields $F_q$. For given q the generating functions of such numbers
are particular cases of Eisenstein Langlands series and satisfy
some functional equations. We will show that one can also consider
these numbers as structure constants of some algebra, a version
of Hall algebra. The functional equations for Eisenstein series
can then be interpreted as commutation relations in this algebra
and are identical to the relations defining the affine quantum
group $U_q(\widehat{sl_2})$ in its socalled Òloop realization"
of Drinfeld.


Jim Stasheff,
University of North Carolina 
Compactifications of configuration and moduli spaces
A variety of influences, many from mathematical physics, have
inspired renewed interest in configuration and moduli spaces and
new compactifications. The talk will provide applications and
concentrate on two approaches: real `nonprojectivized' compactification
as a manifold with corners by `blowups' and `operadic completion'.
Illustrations in terms of configurations on the interval and the
circle provide applications to homotopy theory, knot theory, conformal
field theroy and closed string field theory.


