Math 273, Contact geometry in 3 dimensions

• Spring 2012, WF 1:30-3, Science Center 113
• Office hours: Science Center 242h, Th 1-3 or by appointment

Suggested references:

• Hansjörg Geiges, An introduction to contact topology
• Burak Özbağcı and András Stipsicz, Surgery on contact 3-manifolds and Stein surfaces
• Course notes by John Etnyre, Ko Honda, and Patrick Massot
• Expository articles by John Etnyre: Introductory lectures on contact geometry, Legendrian and transversal knots, open book decompositions and contact structures, and contact geometry in low-dimensional topology
• Images of characteristic foliations and convex surfaces by Patrick Massot

More references will be added as we discuss them in class; I will also post notes or point to a single source covering each lecture, and occasionally I will link to papers of interest below. The notes are based on a combination of the above references and any extra papers I might mention, and may contain mistakes that weren't in the original sources.

Topics covered:

• 1/25: Basic definitions; Darboux's theorem, Gray's theorem, existence of contact structures; Legendrian and transverse knots, parallel parking. (notes)
• 1/27: Lutz twists, Pontryagin-Thom construction; every plane field is homotopic to an overtwisted contact structure. (notes; see Geiges §4.2-4.3)
• 2/1: Characteristic foliations, convex surfaces, dividing sets. (notes; see Geiges §4.6.1-4.6.3)
  • Reference: Giroux, Convexité en topologie de contact or the English translation by Daniel Mathews
• 2/3: More on convex surfaces, Giroux flexibility, Stein fillable implies tight and filling by holomorphic disks, Cerf's theorem (not covered in class). (notes)
• 2/8: Legendrian Realization Principle, Giroux's criterion for tightness, finitely many Euler classes of tight contact structures. (notes)
• 2/10: Classical invariants of Legendrian knots, Thurston-Bennequin inequality, classification of topologically trivial Legendrian knots in a tight contact structure. (notes)
  • Reference: Eliashberg–Fraser, Topologically trivial Legendrian knots (arXiv)
• 2/15: Edge rounding, bypasses, examples of bypasses (the imbalance and right-to-life principles), families of convex surfaces. (notes)
  • Reference: Honda, On the classification of tight contact structures I (arXiv), §3
• 2/17: More on families of convex surfaces, Poincaré-Bendixson property, retrograde connections; classification of tight contact structures on S² × I, B³, S³, S¹ × S², R³. (notes)
  • Reference: Giroux, Structures de contact en dimension trois et bifurcations des feuilletages de surfaces (arXiv)
• 2/22: Classification of tight contact structures on T³. (notes)
  • Reference: Kanda, The classification of tight contact structures on the 3-torus (MathSciNet)
• 2/24: Attaching bypasses to tori and the Farey tessellation, minimal twisting, basic slices. (notes)
  • Reference: Honda, On the classification of tight contact structures I (arXiv), §4.1-4.3
• 2/29: Basic slices and neighborhoods of bypasses; continued fractions and bypasses; tight contact structures on S¹ × D² and minimally twisting structures on T² × I. (notes)
  • Reference: Honda, On the classification of tight contact structures I (arXiv), §4.3-4.5
• 3/2: The classification of tight contact structures on lens spaces; contact surgery. (notes)
  • Reference: Honda, On the classification of tight contact structures I (arXiv), §4.6
• 3/7–9: Contact-type surfaces in symplectic manifolds, Weinstein handles, Legendrian surgery preserves fillability, weakly fillable implies strongly fillable for rational homology spheres, (S³, ξ_st) has a unique Stein filling. (notes; see Özbağcı–Stipsicz §7)
• 3/21–23: The Poincaré homology sphere with opposite orientation does not have any tight contact structures. (notes)
  • Ref. 1: Etnyre–Honda, On the nonexistence of tight contact structures (arXiv)
  • Ref. 2: Lisca–Stipsicz, Ozsváth-Szabó invariants and tight contact three-manifolds, II (arXiv)
• 3/28: Contact (+1)-surgery is inverse to Legendrian surgery, often overtwisted, and can be used to perform Lutz twists; every contact structure comes from Legendrian surgery on one link and contact (+1)-surgery on another in (S³, ξ_st). (notes)
  • Reference: Ding–Geiges, A Legendrian surgery presentation of contact 3-manifolds (arXiv)
• 3/30: Capping off symplectic fillings, converting contact Dehn surgeries into contact ±1 surgeries. (notes)
• 4/6: Classification of overtwisted contact structures on S² × [0,1] up to stable isotopy. (notes)
  • Reference: Huang, A proof of the classification theorem of overtwisted contact structures via convex surface theory
• 4/11: Classification of overtwisted contact structures up to isotopy. (notes)
  • Reference: Huang, A proof of the classification theorem of overtwisted contact structures via convex surface theory
• 4/13: An open book supports a unique contact structure; Milnor open books for S³ support the tight contact structure ξ_st. (notes)
  • Reference: Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures (arXiv)
• 4/18: Every contact structure is supported by an open book. (notes)
• 4/20: Open books supporting the same contact structure are related by positive stabilizations. (notes)
  • Reference: van Koert, Lecture notes on stabilization of contact open books
• 4/25: The contact invariant in Heegaard Floer homology. (notes)
  • Reference: Ozsváth and Szabó, Heegaard Floer homology and contact structures (arXiv)