June 20, 2018

Homotopy Theory Program
Workshop on Homotopy, Geometry and Physics

The Fields Institute
April 19-22,1996

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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 sub-bundles 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 Dold-Thom 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 n-letters of rank (n-1)! 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 2-torsion in the cohomology of mapping class groups associated to the surface of least genus for which a fixed elementary abelian 2-group of rank q is maximal. One such example is genus three with 2-rank 3. This two-torsion 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 Kniznik-Zamolodchikov 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,

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 module-spectra over a ring-spectrum $\bf R$] 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,

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 G-invariant subspace of the quantization of M has the same dimension as the quantization of the symplectic quotient or Marsden-Weinstein 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 Gromov-Lawson-Rosenberg 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 ``Baum-Connes map" is injective for a group $\pi$, then a stable version of the Gromov-Lawson-Rosenberg Conjecture holds for spin manifolds with fundamental group $\pi$. The Baum-Connes map is known to be injective e.g. for discrete subgroups of Lie groups. More generally, the Baum-Connes 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 Novikov-Conjecture).

Ruth Lawrence,

A Holomorphic version of the Witten-Reshetikhin-Turaev invariant

The WRT invariant, $Z_K(M)$, of 3-manifolds $M$ 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 Lima-Filho,
Texas A&M

Euler-Chow series and projective bundle formulas

We develop techniquees to compute Euler-Chow 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, Eisenstein-Langlands 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 so-called 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 `non-projectivized' compactification as a manifold with corners by `blow-ups' 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.