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