Abstracts of Talks at the Cornell Topology Festival, May 7-9, 1999




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.



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 is the
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.




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.




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 retract.
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, etc.




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.




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 3-manifold groups.
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.



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.



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 exotic spheres.
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.