Topology Seminar, Spring 2005

Topology Seminar

Spring 2005

Tuesdays 1:30-3:00 pm

Goldsmith 226

This semester the topic of the Topology seminar will continue to 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

January 25 Georgi The structure of geodesic laminations I.

February 1 Georgi The structure of geodesic laminations II.

February 8 Ruth The structure of geodesic laminations III.

February 15 Ophir Surface automorphisms

February 28 John Crisp
Dijon University Non-positive curvature and hyperbolic groups

Abstract: (joint with Noel Brady, Oklahoma)
It is a longstanding open question of Gromov as to whether all word hyperbolic groups act properly cocompactly on some non-positively curved (CAT(0)) space. In this talk I shall explain how to construct an infinite family of examples of two-dimensional torsion free hyperbolic groups which admit no proper action on any proper CAT(0) metric space of dimension 2.

March 8 Stefan Friedl
Rice University The genus of a knot and twisted Alexander polynomials.

Abstract: The degree of the Alexander polynomial gives a lower bound on the genus of a knot. We will show that the degrees of Alexander polynomials are remarkably successful at giving the right genus bounds. We will furthermore generalize this to the Thurston norm of a 3-manifold. This is joint work with Taehee Kim.

March 15 Pierre-Emmanuel Caprace
Université Libre de Bruxelles On triangles in Coxeter complexes.

Abstract: Coxeter groups are those groups having a presentation in which generators have order 2 and all other relations involve only pairs of generators. Every Coxeter group has a natural action "by reflection" on a certain topological space called the Davis complex. This makes it possible to study Coxeter groups from a geometrical viewpoint. For example, the Davis complex Êhas beenÊ used by G. Moussong to give a nice characterization of Gromov hyperbolic Coxeter groups. A related open question is to determine which Coxeter groups are biautomatic. In this talk, I will present some recent developments of a geometric nature which have led to new results in this direction.

March 21, Monday, 1-2 Eiji Ogasa Three spheres in a sphere.

Abstract: In order to research n-dimensional links, we discuss the following.
Take three 4-spheres A, B, C in the 6-sphere. Suppose that each of A, B, C is embedded. Suppose that each pair of (A,B),(B,C),(C.A) intersects transversely and that the intersection of each of (A,B),(B,C),(C.A) is connected. Call each intersection I(A,B),I(B,C),I(C.A).
Then (I(A,B), I(C,A)) in A is a 2-component 2-dimensional surface link in the 4-sphere. Call this link L(A). Similarly (I(B,C), I(A,B)) in B is called L(B). (I(C,A), I(B,C)) in C is called L(C).
We discuss what kind of triple (L(A),L(B),L(C)) we obtain. We also discuss higher dimensional cases.

March 29 Yann Rollin, MIT Construction of Kaehler surfaces with constant scalar curvature .

Abstract: A new construction is presented for Kaehler metrics of constant scalar curvature on non-minimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank-2 parabolically stable holomorphic bundles. This rather general construction is shown also to give new examples of low genus: in particular, it is shown that CP^2 blown up at 10 suitably chosen points, admits a scalar-flat Kaehler metric; this answers a question raised by Claude LeBrun in 1986 in connection with the classification of compact self-dual 4-manifolds.

April 5 Danny

April 12 Kiyoshi Axiomatic higher torsion and Out(F_n)

Abstract: This is actually two talks which I will compress into 90 minutes.
Part I: Axiomatic higher torsion. The purpose of this talk is to explain the relationship between Miller-Morita-Mumford classes, higher Franz-Reidemeister torsion and higher analytic torsion classes. The axiomatic framework clarifies the conjectured relation (now claimed by S. Goette). Also, the axiomatic approach is very easy to understand. Higher torsion invariants can be computed rather easily from the axioms.
Part II: Out(F_n): In joint work with John Klein and Bruce Williams, I showed that graphs have canonical manifold thickenings. Using the Dwyer-Weiss-Williams approach to higher torsion, this allows us to give a short proof of the fact that the map \phi: Out(F_n)\to GL(n,Z) is trivial in rational homology in a stable range. It also implies that higher axiomatic torsion is defined on IOut(F_n):=ker\phi.

April 19 Jimmy (More) on Surface Automorphisms with a View towards Pseudo-Anosov Automorphisms

Fall 2004 schedule

Date	Speaker	Title
January 25	Georgi	The structure of geodesic laminations I.
February 1	Georgi	The structure of geodesic laminations II.
February 8	Ruth	The structure of geodesic laminations III.
February 15	Ophir	Surface automorphisms
February 28	John Crisp Dijon University	Non-positive curvature and hyperbolic groups
Abstract: (joint with Noel Brady, Oklahoma) It is a longstanding open question of Gromov as to whether all word hyperbolic groups act properly cocompactly on some non-positively curved (CAT(0)) space. In this talk I shall explain how to construct an infinite family of examples of two-dimensional torsion free hyperbolic groups which admit no proper action on any proper CAT(0) metric space of dimension 2.
March 8	Stefan Friedl Rice University	The genus of a knot and twisted Alexander polynomials.
Abstract: The degree of the Alexander polynomial gives a lower bound on the genus of a knot. We will show that the degrees of Alexander polynomials are remarkably successful at giving the right genus bounds. We will furthermore generalize this to the Thurston norm of a 3-manifold. This is joint work with Taehee Kim.
March 15	Pierre-Emmanuel Caprace Université Libre de Bruxelles	On triangles in Coxeter complexes.
Abstract: Coxeter groups are those groups having a presentation in which generators have order 2 and all other relations involve only pairs of generators. Every Coxeter group has a natural action "by reflection" on a certain topological space called the Davis complex. This makes it possible to study Coxeter groups from a geometrical viewpoint. For example, the Davis complex Êhas beenÊ used by G. Moussong to give a nice characterization of Gromov hyperbolic Coxeter groups. A related open question is to determine which Coxeter groups are biautomatic. In this talk, I will present some recent developments of a geometric nature which have led to new results in this direction.
March 21, Monday, 1-2	Eiji Ogasa	Three spheres in a sphere.
Abstract: In order to research n-dimensional links, we discuss the following. Take three 4-spheres A, B, C in the 6-sphere. Suppose that each of A, B, C is embedded. Suppose that each pair of (A,B),(B,C),(C.A) intersects transversely and that the intersection of each of (A,B),(B,C),(C.A) is connected. Call each intersection I(A,B),I(B,C),I(C.A). Then (I(A,B), I(C,A)) in A is a 2-component 2-dimensional surface link in the 4-sphere. Call this link L(A). Similarly (I(B,C), I(A,B)) in B is called L(B). (I(C,A), I(B,C)) in C is called L(C). We discuss what kind of triple (L(A),L(B),L(C)) we obtain. We also discuss higher dimensional cases.
March 29	Yann Rollin, MIT	Construction of Kaehler surfaces with constant scalar curvature .
Abstract: A new construction is presented for Kaehler metrics of constant scalar curvature on non-minimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank-2 parabolically stable holomorphic bundles. This rather general construction is shown also to give new examples of low genus: in particular, it is shown that CP^2 blown up at 10 suitably chosen points, admits a scalar-flat Kaehler metric; this answers a question raised by Claude LeBrun in 1986 in connection with the classification of compact self-dual 4-manifolds.
April 5	Danny
April 12	Kiyoshi	Axiomatic higher torsion and Out(F_n)
Abstract: This is actually two talks which I will compress into 90 minutes. Part I: Axiomatic higher torsion. The purpose of this talk is to explain the relationship between Miller-Morita-Mumford classes, higher Franz-Reidemeister torsion and higher analytic torsion classes. The axiomatic framework clarifies the conjectured relation (now claimed by S. Goette). Also, the axiomatic approach is very easy to understand. Higher torsion invariants can be computed rather easily from the axioms. Part II: Out(F_n): In joint work with John Klein and Bruce Williams, I showed that graphs have canonical manifold thickenings. Using the Dwyer-Weiss-Williams approach to higher torsion, this allows us to give a short proof of the fact that the map \phi: Out(F_n)\to GL(n,Z) is trivial in rational homology in a stable range. It also implies that higher axiomatic torsion is defined on IOut(F_n):=ker\phi.
April 19	Jimmy	(More) on Surface Automorphisms with a View towards Pseudo-Anosov Automorphisms