Topology Seminar, Fall 2004

Topology Seminar

Fall 2004

Tuesdays 1:30-3:00 pm

Goldsmith 226

This semester the topic of the Topology seminar will be The mapping class group.
The mapping class group of a closed orientable surface Σ is the group of isotopy classes of orientation-preserving self-diffeomorphisms (or homeomorphisms) of Σ. The study of this group has several applications in mathematics.

In algebraic geometry and complex analysis it is related to the space of holomorphic structures on a complex curve (the Teichmuller space).
In 3-manifold topology one associates to any element of the mapping class group a closed orientable 3-manifold by gluing together two handlebodies (Heegaard decomposition). It is known exactly when two such elements give rise to the same manifold.
Some references:

Joan Birman. Braids, links and mapping class groups

Nikolai Ivanov, Mapping class groups, available here.
We will be following the notes from a course given by Andrew Casson on the work of Nielsen-Thurston.
Excerpt from the description of the book at Amazon.
This book, which grew out of Steven Bleiler's lecture notes from a course given by Andrew Casson at the University of Texas, is designed to serve as an introduction to the applications of hyperbolic geometry to low dimensional topology. In particular it provides a concise exposition of the work of Neilsen and Thurston on the automorphisms of surfaces.
Schedule

Date Speaker Title

September 21 Ophir Every diffeomorphism is a composition of Dehn twists

September 28 Ruth Charney Automorphisms of surfaces, an introduction

Abstract: I will mention some general theorems about the structure of mapping class groups and then talk about classifying individual automorphism of a surface, a la Thurston and Nielson. This will be an overview with no proofs!

October 5--guest speaker Sarah Rees (U. of Newcastle, UK) Are hyperbolic groups star-free?

Abstract: In the early 1980's, in the early days of the theory of automatic groups (which associates with some groups normal forms which are regular languages, that is, sets of strings recognised by finite state automata, essentially as an expression of certain finiteness properties of these groups) questions started to be asked about what the structure of a regular language associated with a group might reveal about the group itself.
In particular semigroup theorists Rhodes and Margolis conjectured that the regular set of all geodesics of a word hyperbolic group in any presentation must always be a star-free set, that is, expressible in terms of finite sets using only the operations of union, intersection, concatenation and complementation (but without the Kleene closure operation which is needed in addition to find expressions for the full range of regular sets). Since the star-free languages form a natural, low complexity subclass of the regular languages, such a result would be in line with the low complexity of the solution of the word problem for these groups. Recently, Holt (Warwick), Hermiller (Nebraska) and I started to examine Margolis and Rhodes's conjecture.
We found it to be false, indeed we found a presentation of a free group for which the set of geodesics is not star-free. But nontheless we can prove that certain small cancellation conditions on a presentation (which imply word hyperbolicity) do make a group star-free, so Margolis and Rhodes were not far off.
I shall discuss these results and others which relate the structure of sets of geodesics in a group to the properties of the group itself.

October 19 Georgi Review of hyperbolic geometry

November 2 Tim Cochran (Rice U.--visiting MIT) Knots, Gropes and von Neumann Signatures

Abstract: I will sketch the proof of the theorem (joint with Peter Teichner) that, for any natural number n, there exist knots in 3-space that bound embedded Gropes of height (n+2) in the 4-ball but do not bound any embedded Grope of larger height. I will explictly describe the examples and (briefly !) prove the theorem modulo a few "black boxes". Then I will "open up" as many of the black boxes as time permits. For example I will draw nice pictures to show that the knots really do bound embedded gropes of large height. Then maybe I will discuss the Cheeger-Gromov-Atiyah-von Neumann rho-invariant which involves discussing how one can define a real-val;ued signature of a (Hermitian) matrix over the group ring of a group G.

November 9 Danny Hyperbolic structures

November 16 Danny Hyperbolic structures II

November 18 (Thursday), 3-4 PM--guest speaker Allan Edmonds (Indiana U.) 3-manifolds in 4-manifolds

Abstract: This talk will discuss aspects of the conjecture that any rational homology 3-sphere embeds topologically in some connected sum of copies of the complex projective plane. Most do. And all embed in suitable positive definite 4-manifolds. But for certain 3-manifolds, the conjectured embedding, if it exists, must have a non-simply connected complementary domain.

November 23 Heejung Hyperbolic structures III

November 30 Jerry Geodesic laminations

December 7 Jerry Geodesic laminations II

Spring 2004 schedule

Date	Speaker	Title
September 21	Ophir	Every diffeomorphism is a composition of Dehn twists
September 28	Ruth Charney	Automorphisms of surfaces, an introduction
Abstract: I will mention some general theorems about the structure of mapping class groups and then talk about classifying individual automorphism of a surface, a la Thurston and Nielson. This will be an overview with no proofs!
October 5--guest speaker	Sarah Rees (U. of Newcastle, UK)	Are hyperbolic groups star-free?
Abstract: In the early 1980's, in the early days of the theory of automatic groups (which associates with some groups normal forms which are regular languages, that is, sets of strings recognised by finite state automata, essentially as an expression of certain finiteness properties of these groups) questions started to be asked about what the structure of a regular language associated with a group might reveal about the group itself. In particular semigroup theorists Rhodes and Margolis conjectured that the regular set of all geodesics of a word hyperbolic group in any presentation must always be a star-free set, that is, expressible in terms of finite sets using only the operations of union, intersection, concatenation and complementation (but without the Kleene closure operation which is needed in addition to find expressions for the full range of regular sets). Since the star-free languages form a natural, low complexity subclass of the regular languages, such a result would be in line with the low complexity of the solution of the word problem for these groups. Recently, Holt (Warwick), Hermiller (Nebraska) and I started to examine Margolis and Rhodes's conjecture. We found it to be false, indeed we found a presentation of a free group for which the set of geodesics is not star-free. But nontheless we can prove that certain small cancellation conditions on a presentation (which imply word hyperbolicity) do make a group star-free, so Margolis and Rhodes were not far off. I shall discuss these results and others which relate the structure of sets of geodesics in a group to the properties of the group itself.
October 19	Georgi	Review of hyperbolic geometry
November 2	Tim Cochran (Rice U.--visiting MIT)	Knots, Gropes and von Neumann Signatures
Abstract: I will sketch the proof of the theorem (joint with Peter Teichner) that, for any natural number n, there exist knots in 3-space that bound embedded Gropes of height (n+2) in the 4-ball but do not bound any embedded Grope of larger height. I will explictly describe the examples and (briefly !) prove the theorem modulo a few "black boxes". Then I will "open up" as many of the black boxes as time permits. For example I will draw nice pictures to show that the knots really do bound embedded gropes of large height. Then maybe I will discuss the Cheeger-Gromov-Atiyah-von Neumann rho-invariant which involves discussing how one can define a real-val;ued signature of a (Hermitian) matrix over the group ring of a group G.
November 9	Danny	Hyperbolic structures
November 16	Danny	Hyperbolic structures II
November 18 (Thursday), 3-4 PM--guest speaker	Allan Edmonds (Indiana U.)	3-manifolds in 4-manifolds
Abstract: This talk will discuss aspects of the conjecture that any rational homology 3-sphere embeds topologically in some connected sum of copies of the complex projective plane. Most do. And all embed in suitable positive definite 4-manifolds. But for certain 3-manifolds, the conjectured embedding, if it exists, must have a non-simply connected complementary domain.
November 23	Heejung	Hyperbolic structures III
November 30	Jerry	Geodesic laminations
December 7	Jerry	Geodesic laminations II