Abstracts of Talks at the Cornell Topology Festival, May 4-6, 2001

MIGUEL ABREU

THE TOPOLOGY OF SYMPLECTOMORPHISM GROUPS

In sharp contrast with Riemannian geometry, symplectic geometry has no local invariants (Darboux's theorem) and this gives rise to an infinite-dimensional group of transformations preserving the symplectic structure (symplectomorphisms). Despite this local flexibility, there is a certain rigidity that controls global symplectic phenomena, and symplectic topology studies this global rigidity.

 

In this talk I will discuss how rigidity manifests itself in the global topological structure of some groups of symplectomorphisms. In particular, I will give a complete description of the rational cohomology ring of the symplectomorphism groups of rational ruled surfaces (due to Gromov, Abreu and Abreu-McDuff). This will show that, although in general these groups do not have the homotopy type of a finite-dimensional Lie group (flexibility), their topology reflects and is determined by the various different subgroups of (Kaehler) isometries they have (rigidity). The main idea behind the proof of these results goes back to Gromov's 1985 seminal paper, introducing pseudo-holomorphic methods in symplectic topology.

DANIEL ALLCOCK

REFLECTION GROUPS ON THE OCTAVE HYPERBOLIC PLANE

The octave (or Cayley) hyperbolic plane is the exceptional hyperbolic space, and there are some groups acting on it with finite covolume that are generated by reflections. These are the natural generalization of Coxeter groups to this setting. We will introduce the geometry of the plane and discuss the construction of the groups.

DANNY CALEGARI

PROMOTING ESSENTIAL LAMINATIONS

We show that every essential lamination of an atoroidal 3-manifold either contains a genuine sublamination or admits a transverse genuine lamination. As a corollary, by results of Gabai and Kazez this implies that a 3-manifold with an essential lamination is toroidal or has word-hyperbolic fundamental group (i.e. it satisfies the "weak geometrization conjecture") and its mapping class group is finite.

 

Our results fit into Thurston's program to geometrize 3-manifolds with taut foliations, by adapting the proof for surface bundles over circles.

ROBIN FORMAN

THE DIFFERENTIAL TOPOLOGY OF COMBINATORIAL SPACES

We will show how combinatorial analogues of some ingredients of differential topology, such as Vector Fields and their corresponding Flows, can play an important role in the investigation of combinatorial spaces (i.e. simplicial complexes, or more general cell complexes). We will also provide some hints as to how these ideas can be applied to problems in topology, combinatorics and computer science.

JOHN ROGNES

TWO-PRIMARY ALGEBRAIC K-THEORY OF POINTED SPACES

Waldhausen's algebraic K-theory spectrum A(*) of the category of finite pointed CW-complexes is closely related to the smooth concordance spaces of highly-connected manifolds, such as discs and spheres. These are in turn related to the diffeomorphism spaces of such manifolds, e.g. by work of Hatcher.

 

The rational homotopy type of A(*) agrees with that of the algebraic K-theory spectrum K(Z) of the integers, which was determined by Borel, and used by Farrell and Hsiang over 20 years ago to determine the rational homotopy type of the diffeomorphism spaces of discs and spheres. For prime numbers p the p-primary homotopy type of A(*) has been harder to pin down.

 

In the talk I will explain how to assemble the identification of (a) the 2-primary algebraic K-theory of the integers (Voevodsky, Rognes and Weibel, et al), (b) the topological cyclic homology of a point (Bokstedt, Hsiang, Madsen), and (c) the 2-primary topological cyclic homology of the integers (Rognes), using a comparison result due to Dundas, to compute the mod 2 spectrum cohomology of A(*) as a module over the Steenrod algebra. This determines the 2-primary homotopy type of A(*), and thus of the smooth concordance spaces of discs and spheres in a stable range.

DEV SINHA

THE TOPOLOGY OF SPACES OF KNOTS

We describe models which are homotopy equivalent to spaces of knots (that is, spaces whose points are embeddings of one-dimensional manifolds topologized as subspaces of the corresponding mapping spaces). In knot theory, one is mainly concerned with understanding the components of these spaces. We look at the topology more generally. The first model we describe is an inverse limit of certain mapping spaces. The second model is cosimplicial. Both are related to the "evaluation map" and both use compactifications of configuration spaces inspired by Fulton and MacPherson, which we spend some time developing. At the end of the day, we give spectral sequences which converge to the cohomology groups and homotopy groups of embedding spaces when the ambient manifold has dimension greater than three.

PETER TEICHNER

L-THEORY OF KNOTS

We study isotopy classes of classical knots, i.e. knotted circles in 3-space. This set has a commutative addition via connected sum without inverses. There are several ways to introduce inverses, and hence to get interesting abelian groups of knots. The most well known are the knot concordance group and the Goussarov-Vassiliev "finite type" quotients. We will describe recent developments in both these theories and explain a geometric connection between them.

GANG TIAN

SYMPLECTIC SURFACES IN RATIONAL COMPLEX SURFACES

In this talk, we discuss the isotopy of symplectic surfaces in a symplectic 4-manifold. We will discuss related problems and possible applications to symplectic topology.