Abstracts of Talks at the Cornell Topology Festival,
May 7-10, 2004
TAMENESS OF HYPERBOLIC 3-MANIFOLDS
- We show that a hyperbolic 3-manifold with finitely generated
fundamental group is homeomorphic to the interior of a compact manifold
with boundary. This answers a question of Marden (conjectured by Canary).
Many cases were already known, including geometrically finite manifolds
(Marden), indecomposable fundamental group (Bonahon), and limits of tame
hyperbolic manifolds (Brock-Souto, extending work of Canary-Minsky, Evans,
Ohshika). Applications will be discussed as well to conjectures and
problems of Ahlfors, Bers, Canary, Hass, Simon, and Thurston.
WORKSHOP ON 3-MANIFOLD THEORY:
Geometrization conjecture and covers of 3-manifolds
- I'll give an outline of the geometrization conjecture for
3-dimensional closed manifolds. Then I'll explain how this gives a
classification of the universal covers of compact manifolds, implying
the sphere theorem and the Poincare conjecture.
ENDING LAMINATIONS AND THE WEIL-PETERSSON VISUAL SPHERE
- Recent advances in the combinatorial topology of surfaces have led
to new classification results for infinite volume hyperbolic 3-manifolds
via their 'ending laminations'. These developments have ramifications
toward another goal, namely, to develop an analogous classification
for geodesic rays in the Weil-Petersson metric on Teichmüller space.
I will discuss recent results in the CAT(0) geometry of the Weil-Petersson
completion and analogies that arise between its geodesics and the geometry of
hyperbolic 3-manifolds. In particular I will address cases in which a
kind of 'ending lamination conjecture' for Weil-Petersson rays can be shown.
Portions of this talk represent joint work with Yair Minsky and Howard Masur.
DOES A RANDOM 3-MANIFOLD FIBER
OVER THE CIRCLE
- I'll discuss the question of when a tunnel number one 3-manifold
fibers over the circle. In particular, I will discuss a criterion of
Brown which answers this question from a presentation of the
fundamental group. I will describe how techniques of Agol, Hass and
W. Thurston can be adapted to calculate this very efficiently by using
that the relator comes from an embedded curve on the boundary of a
genus 2 handlebody. I will then describe some experiments which
strongly suggest the answer to the question: Does a random
tunnel-number one 3-manifold fiber over the circle? (Joint work with
INVARIANTS OF EMBEDDINGS VIA CONTACT GEOMETRY
- I will describe a method to define, hopefully new, invariants of any
embedded submanifold of Euclidean space. To define this invariant we
will need to take an excursion into the realm of contact geometry and
a recent generalization of Floer homology called contact
homology. More specifically, after recalling various notions from
contact geometry, I will show how to associate a Lagrangian
submanifold to any embedded submanifold of Euclidean space. The
invariant of the embedding will be the contact homology of this
Lagrangian. Though the definition of this invariant is somewhat
complicated it is possible to compute it for knots in Euclidean
3-space. Lenny Ng has combinatorially studied this invariant for such
knots and has shown that it does not seem to be determined by
previously known invariants but non the less has some connections with
the classical Alexander polynomial of a knot. I will concentrate on
the more geometric aspects of the invariant and ongoing work of Tobias
Ekholm, Michael Sullivan and myself aimed at a better understanding of
the invariant (in particular, showing that it is well defined in some
GROUPS OF BRANCH TYPE AND
FINITELY PRESENTED GROUPS
- We dicsuss finite presentability of branch groups and (nice)
embeddings of branch groups into finitely presented groups. Some
interesting notions will be discussed, such as amenability, growth,
and residual finiteness, along with their relation to finitely
presented groups. We will pay particular attention to some problems
of geometric nature regarding branch groups.
PROPERTY P FOR KNOTS
- According to Bing, a knot K has "Property P" if no surgery on K
can yield a counterexample to the Poincare conjecture. Bing asked
whether every knot has Property P. We give an affirmative answer to
this question, drawing on techniques and recent results from gauge
theory and symplectic topology.
CATEGORY WEIGHT AND THE ARNOLD
CONJECTURE ON FIXED POINTS OF
- The well-known Arnold conjecture claims that, for every closed
symplectic manifold M, the number of fixed points of any Hamiltonian
symplectomorphism of M is at least the minimal number of critical
points of a smooth function on M. We prove that this conjecture
holds provided π2(M)=0 (more precisely, if the first Chern class
and the symplectic form vanish on π2(M)). This improves the
results of Floer and Hofer.
SPHERICAL CR GEOMETRY AND DEHN SURGERY?
- A complete spherical CR 3-manifold is the manifold at infinity
for a discrete group acting isometrically on the complex hyperbolic
plane. I will describe how Dehn filling arises in the context of
these manifolds: Under fairly general conditions, when such a
discrete group is perturbed, it remains discrete and the manifold at
infinity undergoes a Dehn filling. This phenomenon is, of course,
parallel to Thurston's well-known hyperbolic Dehn surgery theorem. As
an application of the spherical CR Dehn surgery theorem I will show
that a positive density subset of the Dehn fillings of the Whitehead
link complement result in closed 3-manifolds which bound complex
STRING BACKGROUND IN ALGEBRAIC TOPOLOGY
- Using transversality and classical algebraic topology in
manifolds, one can construct a string background or string algebra.
Then more sophisticated theories like Gromov-Witten invariants of
symplectic manifolds and the symplectic field theory for contact
manifolds can be formulated in terms of solutions to equations written
in the string background. Hopefully there are similar backgrounds for
other appearances of quantum invariants in topology. A background is
to a given theory as say, the deRham algebra of forms on a manifold is
to the cohomology algebra. Namely, higher order products or
invariants can be constructed from the background.
HEEGAARD DIAGRAMS AND HOLOMORPHIC DISKS
- The aim of this talk is to give a quick introduction to
Heegaard Floer homology and discuss some applications to knots.
(This is a joint work with Peter Ozsvath.)
- Abstract not yet available.
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Last modified: Thu May 06 17:56:35 Eastern Daylight Time 2004