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GEOMETRIC TOPOLOGY

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.

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GEOMETRIC TOPOLOGY

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  1. GEOMETRIC TOPOLOGY MAIN GOAL: TO PROVE TOPOLOGICAL RESULTS ABOUT SMOOTH MANIFOLDS BY ENDOWING THEM WITH ADDITIONAL GEOMETRIC STRUCTURES

  2. GEOMETRIC TOPOLOGY OF LOW DIMENSIONAL MANIFOLDS SYMPLECTIC FOUR DIMENSIONAL MANIFOLDS CONTACT THREE DIMENSIONAL MANIFOLDS

  3. Property P

  4. CONTACT THREE DIMENSIONAL MANIFOLDS

  5. Frobenius Theorem

  6. Contact forms

  7. Contact structure

  8. Legendrian curve A curve in a contact 3-manifold is called Legendrian if it is everywhere tangent to the contact planes.

  9. Overtwisted Disk

  10. Tight versus overtwisted

  11. Tight versus overtwisted

  12. Darboux’s Theorem

  13. Contact Topology

  14. Global structure

  15. Global structure

  16. 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.

  17. Classification of tight contact structures?

  18. Convex surfaces

  19. 2002 Giroux’s ICM talk in Beijing

  20. Open books

  21. Complement of the Hopf link in the 3-sphere fibers over the circle

  22. Abstract open books

  23. Mapping torus M

  24. Stabilization of an open book

  25. Stabilization of an open book

  26. Open books and contact structures

  27. Etnyre’s Lemma

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