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Geometric Topology
Org: Hans Boden (McMaster), B. Doug Park (Waterloo) and Mainak Poddar (Waterloo)
[PDF]
- TARA BRENDLE, Cornell University, Malott Hall, Ithaca, NY 14853
The Birman-Craggs-Johnson homomorphism and the cohomology
of the Torelli group
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The Torelli group Tg is the subgroup of the mapping class group of
a surface consisting of those diffeomorphism classes acting trivially
on the homology of the surface. The Torelli group is the "mysterious
part" of the mapping class group, and basic questions about its
structure (e.g., finite presentability) are still not known;
in particular its homology is known only in dimension one. In the
1970/80s Birman-Craggs-Johnson constructed a remarkable surjective
homomorphism from Tg to a certain Z/2Z-vector space of Boolean
(square-free) polynomials. The heart of this construction comes from
the Rochlin invariant for homology 3-spheres. This homomorphism,
together with abelian cycles in Tg, can be used to construct
nontrivial elements of H2(Tg, Z/2Z), which had not been detected
rationally. This construction yields a lower bound on the order of
g4 for the rank of this homology group. We will also discuss the
question of lifting these classes to integral classes and the
connection with the "Casson-Morita algebra".
This is joint work in progress with Benson Farb.
- OLIVIER COLLIN, Université du Québec à Montréal
Floer homology of links of complex singularities and
analytical invariants of Milnor fibres
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In this talk, we survey some aspects of Floer homology for homology
spheres and for knots that are relevant for the study of analytical
invariants of Milnor fibres of 3-manifolds that arise as links of
complex singularities.
- MARIANTY IONEL, McMaster University, Hamilton
Special Lagrangian submanifolds in the cotangent bundle of
the sphere
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The cotangent bundle of the sphere T* Sn has a Ricci-flat Kahler
metric discovered by Stenzel. The case n=1 is the Eguchi-Hanson
metric and the case n=2 was found earlier by Candelas and de la
Ossa. In this talk I will present examples of cohomogeneity one
special Lagrangian 3-folds in the case n=3. Some of these examples
can be generalized for any dimension of the sphere. We will also
address the asymptotic behaviour of these submanifolds and their
topology. This is work in progress with M. Min-Oo.
- ERNESTO LUPERCIO, Cinvestav-IPN (Center for Research and Advanced Studies,
Mexico)
Orbifold String Topology
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In this talk I will introduce first the string topology product of
Chas and Sullivan on the homology of the free loop space of a manifold
LM. Then I will explain the Cohen-Jones theory that gives a homotopy
theory interpretation of the product and finally I will close by
explaining our construction of the string topology ring for a orbifold
(real differentiable smooth Deligne-Mumford Stack). I will hint at
the relation to the Chen-Ruan orbifold cohomology product, and its
relations to non-commutative geometry.
This is a report of my work in progress with B. Uribe and
M. A. Xicotencatl.
- JOSEPH MASTERS, SUNY Buffalo
Quasi-Fuchsian surfaces in hyperbolic knot manifolds
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We show that every 1-cusp, finite-volume hyperbolic 3-manifold admits
an essential, quasi-Fuchsian immersion of a closed, hyperbolic
surface.
Joint work with Xingru Zhang.
- BRENDAN OWENS, Cornell University, Ithaca, NY 14853
Unknotting information from Heegaard Floer homology
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I will describe Ozsvath-Szabo's obstruction to a knot having
unknotting number one, and a generalisation to higher unknotting
numbers. This leads to the completion of the table of unknotting
numbers for prime knots with 9 crossings or less.
- STEPHAN TILLMANN, Université du Québec à Montréal
Angle structures and geometric splittings
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The study of angle structures on an ideally triangulated 3-manifold
(M,T) goes back to Casson, who noticed that the existence of such a
"linear hyperbolic" structure has strong topological consequences:
M is irreducible, atoroidal and its ends are tori or Klein bottles.
In this talk I describe topological consequences in the case where a
degenerate angle structure (some angles may be 0) is dual to a normal
surface. The surface turns out to be essential, and the associated
splitting of a uniquely associated Dehn-Thurston surgery of M is
geometric: the pieces are Seifert fibered spaces or "linear
hyperbolic" cone-manifolds. I will also explain how this illustrates
the Culler-Shalen machine in special cases.
- STEFANO VIDUSSI, Department of Mathematics, 138 Cardwell Hall, Kansas State
University, Manhattan, KS 66506
Alexander polynomials and symplectic S1 xN
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We will present some results and speculations concerning the
connection between the study of the Alexander polynomials of a three
manifold N and the problem of determining when S1 ×N admits
a symplectic structure.
- MEI-LIN YAU, Michigan State University, East Lansing, MI 48824, USA
A holomorphic 0-surgery model with application to cylindrical
contact homology
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It is known that a contact structure on a closed 3-manifold M is
supported by an open book representation of M. By applying
0-surgery to every connected component of of the binding one gets a
new manifold M¢ which is a mapping torus of a closed surface. Both
RxM and RxM¢ can be endowed with R-invariant symplectic
structures as well as R-invariant compatible almost complex
structures.
In this talk we give, in the complex plane, a simple holomorphic model
of the 0-surgery. This model allows explicit relations between
pseudoholomorphic curves in RxM and pseudoholomorphic curves in
RxM¢. As an application, we use it to compute the cylindrical
contact homology of open books resulting from a positive Dehn twist on
a torus with boundary. These are the first examples of cylindrical
contact homology via open books with nontrivial monodromy.
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