Abstracts of Talks at the Cornell Topology Festival, May 7-9, 1999
A PROOF OF THE FINITE FILLING CONJECTURE
- Let M be a compact, connected, orientable 3-manifold whose boundary
is a torus. The Dehn fillings of M are the manifolds M(r) obtained
by attaching a solid torus to M by a homeomorphism from its boundary torus
to the boundary of M, taking a meridian circle of the torus to a curve
of slope r in the boundary of M. Here "slope" can be interpreted
to mean homology class in the boundary. The distance between two
slopes is the absolute value of their homological intersection number.
- One of the primary goals of recent research in 3-manifold topology
has been to understand the topology of fillings, in particular when M is
hyperbolic, i.e. the interior of M admits a complete hyperbolic structure
of finite volume. In this case Thurston has proven that the set E(M) of
exceptional slopes, for which M(r) is not hyperbolic, is finite.
One theme of current research has been to determine sharp bounds on the
cardinality of E(M) as well as on the distance between two of its elements.
A strategy that has proven quite effective has been to work modulo Thurston's
hyperbolisation conjecture by considering ETOP(M), the slopes
for which M(r) is either reducible, toroidal, Seifert fibred, or has finite
fundamental group. It is well-known that ETOP(M) is a subset
of E(M) and Thurston conjectured that the two sets are equal. Excellent
bounds on numbers and distances have been found by several authors for
various classes of slopes in ETOP(M). In recent work with Xingru
Zhang, the following theorem has been proved.
- Theorem (Boyer-Zhang) If M is a hyperbolic manifold, then
there are at most five slopes whose associated Dehn fillings have either
a finite or an infinite cyclic fundamental group. Furthermore, the distance
between two slopes yielding such manifolds is no more than three, and there
is at most one pair of slopes which realizes the distance three.
- This result is sharp in that each of the bounds is realized when M
is taken to be the exterior of the figure-8 sister knot.
- The proof of the theorem is based on the graph-theoretic methods of
C. McA. Gordon and J. Luecke and the SL(2,C)-character variety methods
of M. Culler and P. Shalen. Of particular importance are the bounds on
the Culler-Shalen norms of finite filling classes we develop, and the precise
relationships between Culler-Shalen norms and A-polynomials we determine.
CANTOR GROUPS, THEIR CLASSIFYING SPACES, AND THEIR ACTIONS ON ENR's
- A cantor group is a topological group which is homeomorphic to the
cantor set (i.e., is an infinite second-countable profinite group, if you
wish). Basic examples are 1) any countably infinite direct product of nontrivial
finite groups, and 2) the p-adic integers, for your favorite prime p. A
fundamental open problem concerning how cantor groups can act on nice spaces
- Free-Set Z-Set (FSZS) Conjecture: Given any action by a cantor group
on an ENR (= euclidean neighborhood retract), the free set of the action
is a homology Z-set (in the ENR).
- A homology Z-set is one whose removal does not change the homology
of any open subset of the ENR.
- The FSZS Conjecture can be regarded as a sort of Super Hilbert-Smith
Conjecture, the HSC being the case where the ENR is a manifold. This talk
will discuss the (natural) classifying space approach to the FSZS Conjecture,
and my work on the key free-action, finite-dimensional-quotient case. The
main theorem can be paraphrased as follows: Although it is well known that
the classical (principal action) cohomological dimension of the p-adic
integers is 1, the free-action cohomological dimension is infinite.
INTRODUCTION TO SYMPLECTIC FIELD THEORY
- Symplectic field theory is a new theory (currently under a joint construction
by A. Givental, H. Hofer and the author), whose goal is to provide, on
the one hand, new invariants of contact manifolds and Legendrian knots
in them and, on the other hand, to give a way of computing Gromov-Witten
invariants of symplectic manifolds and their Lagrangian submanifolds by
cutting them into elementary pieces.
GROUP ACTIONS ON COARSE POINCARE DUALITY SPACES
- This is a joint work with Bruce Kleiner. We study free discrete simplicial
group actions on coarse n-dimensional Poincare duality spaces, i.e.
simplicial complexes which behave homologically (in the large-scale) like
real n-space. Basic examples of such spaces are: real n-space with a bounded
geometry uniformly acyclic triangulation, universal covers of compact n-dimensional
PL manifolds and universal covers of compact Poincare complexes of formal
dimension n. A k-dimensional duality group is a group G whose cohomology
with coefficients in ZG is nontrivial only in dimension k. A special
case of Scott's compact core theorem asserts that if G is a finitely generated
1-ended group acting freely and discretely on a contractible 3-dimensional
manifold X, then G is the fundamental group of a compact 3-dimensional
manifold Q with incompressible boundary and Q embeds in X/G as a deformation
- We prove a homological analogue of this theorem for n-1 dimensional
duality group actions on coarse n-dimensional Poincare duality spaces X.
In particular, every such group G has to have the structure of an (n-1)-dimensional
relative Poincare duality pair, where the peripheral structure on G comes
from the "peripheral structure" of its action on X.
- This result is new even when X is the universal cover of an aspherical
n-dimensional PL manifold.
- As an application I will give a number of examples of groups which
do not admit actions on coarse n-dimensional Poincare duality spaces, for
instance: if G is the direct product of a Baumslag-Solitar group BS(p,q)
(|p| not equal to |q|) with a k-dimensional duality group (for example
the product of k copies of Z), then G cannot act on coarse (k+3)-dimensional
Poincare duality spaces. If G1 and G2 are 1-relator
torsion-free one-ended groups which are not surface groups, then G1
x G2 cannot act on a coarse 5-dimensional Poincare duality space,
BOUNDARY SLOPES OF KNOTS, AND 3-MANIFOLDS WITH CYCLIC FUNDAMENTAL
- Let K be a nontrivial knot in the 3-sphere. Recent work by Culler-Shalen
and Dunfield gives topological restrictions on the set of all bounded surfaces
in the exterior M of K which are essential (i.e. incompressible and non-boundary-parallel),
and in particular on the set of all isotopy classes of simple closed curves
in the boundary of M that arise as boundary components of such surfaces.
These isotopy classes are customarily parametrized by certain elements
of Q (or infinity) which are called boundary slopes of K.
These results can be thought of as giving characterizations of the trivial
knot in terms of its essential surfaces or its boundary slopes. More generally,
in a closed, orientable, irreducible 3-manifold with cyclic fundamental
group, knots which are round, in the sense that their exteriors
are solid tori, can be characterized among all knots with irreducible exterior
in terms of their essential surfaces or their boundary slopes.
- These characterizations of round knots, which depend for their proofs
on the use of the SL(2,C) character variety, do not extend to knots
in manifolds for which the fundamental group is noncyclic. Indeed, it seems
possible that in a manifold with cyclic fundamental group, round knots
can be characterized by some property, stated in terms of essential surfaces
or boundary slopes, which is "ubiquitous" in the sense that every
closed, irreducible, orientable 3-manifold contains a knot having the property
in question. If one could give such a characterization it would follow
that every closed, orientable 3-manifold with cyclic fundamental group
contains a round knot and is therefore a lens space. The Poincare Conjecture
is a special case of this conjecture. I will discuss this general program
and mention some recent progress.
SUBGROUP SEPARABILITY OF THE FIGURE 8 KNOT
- A subgroup H of a group G is said to be separable provided that H is
the intersection of finite index subgroups of G. A group is called subgroup
separable provided that every finitely generated subgroup is separable.
Well-known examples of groups which are subgroup separable include free
groups (M. Hall) and surface groups (P. Scott).
- Important work building on these earlier results has been done by Brunner-Burns-Solitar,
Gitik, Long, Niblo, Tretkoff and others. These authors obtain many examples
of subgroup separable 3-manifold groups. However, an example of a finitely
generated 3-manifold group which is not subgroup separable was given by
Burns-Karrass-Solitar. Other examples were given more recently by Rubinstein-Wang.
- I have proven that every geometrically finite subgroup of the figure
8 knot group is separable. The same proof works for many other hyperbolic
- One reason why low dimensional topologists are interested in subgroup
separability is because it allows certain immersions to lift to embeddings
in a finite cover. For instance, as an application of the result, we find
that if M is the figure 8 knot complement, then every properly immersed
incompressible surface S in M, of minimal self-intersection, lifts to an
embedding in a finite cover of M.
EIGENVALUE INEQUALITIES AND QUANTUM COHOMOLOGY OF THE GRASSMANNIAN
- Beginning with Weyl, many mathematicians have wondered about the following
question: given two Hermitian matrices, what are the possible eigenvalues
of the sum? I will discuss a paper of Klyachko which uses the Hilbert-Mumford
criterion to give a solution in terms of cohomology of the Grassmannian,
and a generalization of myself and Agnihotri of this result to products
of unitary matrices, where the answer involves quantum cohomology.
CURVATURE AND SYMMETRY OF MILNOR SPHERES
- In joint work with Karsten Grove, we explore the geometry and topology
of cohomogeneity one manifolds, i.e. manifolds with a group action whose
principal orbits are hypersurfaces.
- As a consequence we prove that every vector bundle and every sphere
bundle over the 4-sphere has a complete metric with non-negative curvature.
In particular the 3-sphere bundles over the 4-sphere admit such metrics.
It is well known that 15 of the 27 exotic spheres in dimension 7 can be
written as 3-sphere bundles over the 4-sphere in infinitely many ways,
and hence we obtain infinitely many non-negatively curved metrics on these
- A further consequence will be that there are infinitely many almost
free actions by SO(3) on the 7-sphere which preserve the Hopf fibration
over the 4-sphere and which do not extend to the 8-disc. We also construct
infinitely many such actions on the 15 exotic 7-spheres mentioned above.