Room 208
Vivatsgasse 7
53111 Bonn
Vivatsgasse 7
53111 Bonn
Steven Sivek
Imperial College Department of MathematicsMath 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)
- 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, ξst) 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, ξst).
(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)
- 4/11: Classification of overtwisted contact structures up to isotopy. (notes)
- 4/13: An open book supports a unique contact structure; Milnor open
books for S3 support the tight contact structure
ξst.
(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)