Post on 21-Mar-2016
description
transcript
GEOMETRIC TOPOLOGY
MAIN GOAL:
TO PROVE TOPOLOGICAL RESULTS ABOUT SMOOTH MANIFOLDS BY
ENDOWING THEM WITH ADDITIONAL GEOMETRIC STRUCTURES
GEOMETRIC TOPOLOGY OF LOW DIMENSIONAL MANIFOLDS
SYMPLECTIC FOUR DIMENSIONAL MANIFOLDS
CONTACT THREE DIMENSIONAL MANIFOLDS
Property P
CONTACT THREE DIMENSIONAL MANIFOLDS
Frobenius Theorem
Contact forms
Contact structure
Legendrian curve
A curve in a contact 3-manifold is called
Legendrian if it is everywhere tangent to the contact planes.
Overtwisted Disk
Tight versus overtwisted
Tight versus overtwisted
Darboux’s Theorem
Contact Topology
Global structure
Global structure
Classification of overtwisted contact structures
Martinet+Lutz+Eliashberg Overtwisted contact structures are classified:
There is, up to isotopy, a unique overtwisted
contact structure in every homotopy class of oriented plane fields.
Classification of tight contact structures?
Convex surfaces
2002 Giroux’s ICM talk in Beijing
Open books
Complement of the Hopf link in the 3-sphere fibers over the circle
Abstract open books
Mapping torus M
Stabilization of an open book
Stabilization of an open book
Open books and contact structures
Etnyre’s Lemma
Our recent contribution