Introduction to Contact Topology (Fall 2014)





Announcements:
  • (27-11-2014) Please hand in the third (and last) set of hand in exercises before or on Thursday, 11 December.
  • (22-10-2014) In the second part of the semester we will still meet on Mondays, 2-5 p.m., but the room will be different: 08A11 in the main building.

    • Teacher: Federica Pasquotto

    Location and time:
  • Time: Monday, 14:00-17:00
  • Location: Room HGB 08A11 - Vrije Universiteit, Hoofdgebouw


    • Organization:
  • Lectures: 2 times 45 minutes
  • Exercise session: 1 hour per week (immediately after the lectures).


    • Exam:
    There will be three sets of hand-in problems. They will count for 50 percent of your grade. At the end of the course there will be an oral exam.


    Prerequisites:
    The standard basic notion that are tought in the first course on Differential Geometry, such as: the notion of manifold, smooth maps, immersions and submersions, tangent vectors, Lie derivatives along vector fields, the flow of a tangent vector, the tangent space (and bundle), the definition of differential forms, DeRham operator (and hopefully the definition of DeRham cohomology).



    Homework:
    Please do all the homework exercises. The hand-in problems will be also posted on this page.

  • Exercise sheet for week 1
  • Exercise sheet for week 2
  • Exercise sheet for week 3
  • Exercise sheet for week 4
  • Exercise sheet for week 5
  • Exercise sheet for week 6
  • Exercise sheet for week 7
  • Exercise sheet for week 8
  • Exercise sheet for week 9
  • Exercise sheet for week 10
  • Exercise sheet for week 11
  • Exercise sheet for week 12
  • Exercise sheet for week 13


    • The schedule week by week (here we will try to add, after each lecture, a description of what was discussed in the lectures):
  • Week 1 : distributions, integrability, and the theorem of Frobenius (this can be found in any good textbook on differentiable manifolds, for instance Lee or Warner)
  • Week 2 : contact structures and contact forms, Reeb vector field of a contact form, the Reeb flow and the geodesic flow on the unit (co-)tangent bundle of a Riemannian manifold
  • Week 3 : symplectic manifolds, Liouville vector fields and hypersurfaces of contact type; the space of contact elements and contact transformations; definition of (strict) contactomorphism
  • Week 4 : Gray stability; contact vector fields and Hamiltonian functions; Darboux's theorem; submanifolds of contact manifolds
  • Week 5 : Standard neighbourhood model and istopy extension theorem for Legendrian knots in 3-dimensional contact manifolds; please read the general statements about neighbourhoods of isotropic and contact submanifolds (Theorem 2.5.8 and Theorem 2.5.15, respectively).
  • Week 6 : Legendrian and transverse knots; front and Lagrangian projection; approximation by Legendrian and transverse knots; the Thurston-Bennequin invariant of a homologically trivial Legendrian knot in a contact three-manifold.
  • Week 7 :Thurston -Bennequin invariant of a Legendrian knot from the front projection, Legendrian Reidemeister moves, rotation number, Dehn surgery and Martinet's theorem
  • Week 8 : Characteristic foliations, overtwisted disks, tight and overtwisted contact structures
  • Week 9 : Existence of overtwisted contact structures (Section 4.5, in particular Prop. 4.5.4); Surfaces in 3-dimensional contact manifolds (Section 4.6.1, up to Proposition 4.6.11)
  • Week 10 : Convex surfaces (Subsection 4.6.2) and Convex surface theory (section 4.8, please read up to Definition 4.8.3 and Examples (1) and (2) on the same page); if you are curious about the notion of convexity for a contact manifold and how it interacts with convex surfaces, please have a look at these notes by Emmanuel Giroux. You might also look up Example 3.2 and Section 4-C in these notes if you are confused about the second exercise for these week.
  • Week 11 : Dividing set of a convex surface and contact structure in a neighborhood of the surface (this is still Section 4.8, please read until Theorem 4.8.11). Next week the lecture will focus on Giroux's criterium (Proposition 4.8.13) and the classification of tight contact structures on $S^3$ (Theorem 4.10.1)
  • Week 12 Giroux's criterion and the classification of contact structures on the 3-sphere (see above); definition of weak and strong symplectic filling of a contact manifold (par. 5.1), you may want to read the definition of a symplectic cobordism as well (Def. 5.2.1)
  • Week 13 Symplectic cobordisms, (topological) surgery along spheres, surgery in terms of handle attaching; please do this week's homework about the model for a symplectic handle


    • Aim/content of the course:
    Contact topology originated more than three centuries ago in the work of Huygens, Hamilton and Jacobi on geometric optics. More recently, it has developed into a very active field of research, with profound connections to low-dimensional topology, symplectic topology, and Hamiltonian dynamics.
    A contact structure is a geometric structure on a smooth manifold, defined by a one-form which satisfies a condition called "maximal non-integrability". Examples of manifolds admitting a contact structure are: odd-dimensional Euclidean spaces, spheres, and tori.
    The main aim of this course is to cover the basic notions of contact topology: contact structures, characteristic foliations, families of contact structures and Moser's method, the tight-overtwisted dichotomy. We also hope to address topics like special knots in contact manifolds, convex surfaces and open book decompositions.

    Literature:
    The main reference for this course is:
  • H. Geiges, "An Introduction to Contact Topology", Cambridge University Press (2008)
    • Good references for the background material are:
  • J. Lee, "Introduction to Smooth Manifolds", Springer
  • F. Warner, "Foundations of Differentiable Manifolds and Lie Groups", Springer
    • Last update: 27-11-2014