Abstracts of talks at the Cornell Topology Festival, May 1-3, 1998

 

DARYL COOPER: "Some surface groups survive surgery"

 

Suppose M is a 3-manifold and N is M with an open solid torus neigborhood of a simple closed curve deleted. Then M is said to be obtained by Dehn-filling N. It has long been known that every closed orientable 3-manifold, M, can be obtained from the complement of finitely many disjoint solid tori in the 3-sphere by suitable Dehn-fillings. Thurston has shown that most 3-manifolds are hyperbolic. Earlier, Haken and Waldhausen developed an extensive theory of 3-manifolds which contain an essential embedded surface.

 

We show that all but finitely many Dehn fillings of a one-cusped hyperbolic 3-manifold contain a surface group. The method involves quasi-Fuchsian subgroups and analysing certain convex sets.

 

The pre-print is available from http://www.math.ucsb.edu/~cooper/index.html

 

JAMES STASHEFF: "From operads to string field theory"

 

String theory regards particles as (tiny) one-dimensional objects in apace-time -- paths or loops. String FIELD theory deals with functions defined on the space of all such strings. String field theory is multi-layered, often presented as involving topology, geometry, algebra and analysis, especially analysis in the sense of Riemann surfaces. The bottom layer is the topology of string configurations which in turn gives rise to `convolution' algebras of fields. The talk will provide interpretation of these algebraic structures from the point of view of operads. This focuses particularly on the subtleties of and fits naturally into the existing framework in algebraic topology developed for studying spaces of the homotopy type of (iterated) loop spaces. The topological operads involved are moduli spaces for Riemann surfaces with marked points and decorations or compactifications as well as the more traditional configuration spaces.

 

JON McCAMMOND "General versions of small cancellation theory"

 

Various generalizations of small cancellation theory have arisen over the years. These include word hyperbolic groups, automatic groups, and groups of non-positive curvature. In a forthcoming article, I have given my own generalization of small cancellation theory which overlaps significantly with the above theories but is not contained in any one of them. Although the original motivation for its development was to provide a geometric approach to the study of the Burnside groups of large exponent, the theory is also applicable to many other types of groups which are well-known in geometric group theory.

 

In my talk I will focus on the development of the general small cancellation techniques. The key generalization is an expansion of the usual notion of a relator from a word in the free group on the generators to a special type of high-dimensional simplicial complex with a labeled 1-skeleton. Lots of examples and pictures will be drawn.

 

MICHAEL HUTCHINGS: "Reidemeister torsion in generalized Morse Theory (with an application to Seiberg-Witten theory)"

 

Let V be the vector field dual to a closed 1-form on a closed Riemannian manifold X with Euler characteristic zero and positive first Betti number. We define an invariant which counts closed orbits and flow lines between critical points of V. We show that this invariant depends only on the cohomology class of the 1-form and equals a form of topological Reidemeister torsion. When dim(X)=3, our invariant is related to the Seiberg-Witten invariants. This is joint work with Yi-Jen Lee.

 

ALAN REID: "Thue equations and Dehn surgery"

 

Let F(X,Y) be a polynomial with integer coefficients which is irreducible and homogeneous. A Thue equation is an equation of the form F(X,Y) = m where m is an integer. This talk will concern work in progress on connections between solutions to Thue equations F(X,Y) = ±1 in integers and problems arising from Dehn surgeries on knots.

 

MICHAEL HANDEL: "The mapping torus of a free group automorphism is coherent"

 

A group is coherent if all of its finitely generated subgroups are finitely presented. Free groups are obviously coherent. The classification of surfaces and the fact that every covering space of a surface is itself a surface, imply that surface groups are coherent. Fundamental groups of three dimensional manifolds make up the most interesting known family of coherent groups. Scott and Shalen independently established the coherence of this family in the early 1970's. There are many parallels between isotopy classes on a surface and free group outer automorphisms. A surface isotopy class determines a three manifold via the mapping torus construction. There is a group theoretic analog that is called either a mapping torus or an ascending HNN-construction. It is natural to ask if there are parallels between three manifold groups and groups obtained as the mapping torus of a free group automorphism. In this talk we prove that these latter groups are coherent. The main technique involves the folding operation of Stallings.

 

WILLIAM THURSTON: "Three-manifolds that slither around the circle"

 

A manifold M slithers around a manifold N when the universal cover of M fibers over N so that deck transformations are bundle automorphisms. Three-manifolds that slither around the circle are like a hybrid between 3-manifolds that fiber over the circle and certain kinds of Seifert-fibered 3-manifolds. There are examples of non-Haken hyperbolic manifolds that slither around the circle. It seems conceivable that every hyperbolic 3-manifold slithers around the circle, and it seems reasonable that every hyperbolic 3-manifold has a finite sheeted cover that slithers around the circle.

The paper is available from: http://front.math.ucdavis.edu/math.GT/9712268.