Contact Geometry

4 hours lecture + 2 hour exercise session

Marc Kegel

Winter term 2023/24

Lecture: Tuesdays and Thursdays 9:15 - 10:45, in room INF 205 / SR 4
Exercise session: Wednesdays, 9:15 - 10:45, in room INF 205 / SR 4
Office hours / further discussion: after the lectures/exercise sessions


Content: Contact structures are certain hyperplane fields on odd-dimensional manifolds and are in some sense opposite to foliations. (More precisely a contact structure is a maximal non-integrable hyperplane field.) While contact manifolds originated from theoretical physics, for example as hyperplanes in phase spaces (or more general as hyperplanes in symplectic manifolds), they also appear naturally in many areas of pure mathematics. In particular, the relations to the geometric topology of low-dimensional manifolds is rich and surprising.

In this lecture, we will first discuss the basics of contact geometry in arbitrary dimension and then specialize to low-dimensions and discuss some of its applications.

This lecture is aimed at all students with basic knowledge in topology.

Prerequisites: Prerequisites are the introductory lectures (Analysis I, II and Linear Algebra I, II), basic notions from point set topology and differential topology (smooth manifolds, vector fields, differential forms). Results from algebraic topology (fundamental group, homology theory) are useful, but they are not strictly needed for the understanding of the lecture.

Exam: There will be oral exams on Thursday 1.2. between 9:30 and 12:00. If you want to take part in the exam, please contact me after the lecture.
Criterion for admission to the final examination: For this lecture there is no admission restriction to the final examination. Nevertheless, I would like to encourage you to regularly work on the exercise sheets. I recommend taking the exam if at least 50% of the exercises have been solved correctly.

Exercise sheets:
Exercise sheet 1 (pdf)
Exercise sheet 2 (pdf)
Exercise sheet 3 (pdf)
Exercise sheet 4 (pdf)
Exercise sheet 5 (pdf)
Exercise sheet 6 (pdf)
Exercise sheet 7 (pdf)
Exercise sheet 8 (pdf)
Exercise sheet 9 (pdf)
Exercise sheet 10 (pdf)
Exercise sheet 11 (pdf)
Exercise sheet 12 (pdf)

Table of Contents: (tentative)

1. Overview:

2. Contact manifolds:
2.1. Hyperplane fields, foliations, and the Frobenius theorem
2.2. Contact structures and Reeb vector fields
2.3. Gray stability, the Moser trick and Darboux's theorem
2.4. Order of contact

3. Knots in contact 3-manifolds:
3.1. Neighborhood and isotopy extension theorems
3.2. The front and Lagrangian projections
3.3. Approximation theorems
3.4. Seifert surfaces and the Alexander polynomial
3.5. The classical invariants

4. Surfaces in contact 3-manifolds:
4.1. The characteristic foliation
4.2. Singularities of the characteristic foliation
4.3. Convex surfaces
4.4. Giroux's elimination lemma
4.5. The dividing set
4.6. The Legendrian realization principle and Giroux's criterion
4.7. The Bennequin bound

5. Classification of contact 3-manifolds and Legendrian knots:
5.1. Tight versus overtwisted contact structures
5.2. Classification of tight contact structures
5.3. Classification of Legendrian unknots
5.4. Classification of tangential 2-plane fields
5.5. Classification of overtwisted contact structures

6. Symplectic fillings:
6.1. Symplectic manifolds
6.2. Fillable contact manifolds
6.3. Weinstein handles and other types of fillings
6.4. Holomorphic curves

7. Denn surgery on Legendrian knots:
7.1. Contact Dehn surgery and symplectic fillings
7.2. Surgeries on the unknot
7.3. Contact Kirby moves
7.4. The homotopical invariants
7.5. The theorem of Ding-Geiges
7.6. Contact surgery numbers
7.7. Stein traces

8. Open books and the Giroux correspondence:
8.1. Open books
8.2. The Thurston-Winkelnkemper construction
8.3. The Giroux correspondence
8.4. Tightness and fillability in open books

Lecture notes:
Since some people had conflicts with other lectures, Josua Kugler has kindly agreed to make his notes of the lecture available. You can find them here.

Viktor Stanislaus Stein has extended his code to visualize contact structures and Legendrian and transverse knots in there. The code can be accessed here. Others are invited to get involved.

The standard reference in the field is the book by Geiges. Most (but not all) of the material we will discuss in the course is covered in much detail in that book. Some other useful sources can be found below. During the course I will add further references.

J. Etnyre: Legendrian and Transversal Knots, 2004, available online at the arXiv.
J. Etnyre: Lectures on open book decompositions and contact structures, 2004, available online at the arXiv.
J. Etnyre: Lectures on contact geometry in low-dimensional topology, 2006, available online at the arXiv.
H. Geiges: Contact geometry, 2003, available online at the arXiv.
H. Geiges: An Introduction to Contact Topology, Cambridge University Press, 2008.

More on convex surface theory can be found here:
J. Etnyre: Notes on convex surfaces, 2004, available online here.
J. Etnyre and K. Honda: Knots and contact geometry, 2000, available online at the arXiv.

More on the contact homology invariants of Legendrian knots can be found here:
Y. Chekanov: Differential algebras of Legendrian links, 1997, available online arXiv.
L. Ng: Computable Legendrian invariants, 2000, available online arXiv.

The full proof of the classification of overtwisted contact structures can be found here:
M. Borman, Y Eliashberg, E. Murphy: Existence and classification of overtwisted contact structures in all dimensions, 2014, available online arXiv.

More about symplectic fillings can be found for example in:
B. Ozbagci: Lectures on the topology of symplectic fillings of contact 3-manifolds, available online here.
B. Ozbagci und A. Stipsicz:b> Surgery on Contact 3-Manifolds and Stein Surfaces, Springer-Verlag, 2004.
C. Wendl: A biased survey on symplectic fillings, available online here.

Background and further information on differential topologie can be found here. The excellent book by Milnor is very short and should be known to every mathematician. (So if you have not read it I recommend doing so.)
T. Broecker and K. Jaenich: Einfuehrung in die Differentialtopologie, Springer, 1973.
V. Guillemin and A. Pollack: Differential topology, Prentice-Hall, 1974.
M. Hirsch: Differential topology, Springer, 1976.
J. Milnor: Topology from the Differentiable Viewpoint, The University Press of Virginia, 1965.

Background and further information on algebraic topologie can be found here:
M. Armstrong: Basic Topology, Springer, 1983.
S. Friedl: Lecture notes for Algebraic Topology I-IV, erhältlich online auf seiner Homepage.
A. Fomenko and D. Fuchs: Homotopical topology, Springer, 2016.
K. Jänich: Topologie, Springer, 1996.

Background and further information on knot theory can be found here:
G. Burde, M. Heusener and H. Zieschang: Knots, De Gruyter, 2013, available online here.
P. Cromwell: Knots and Links, Cambridge University Press, 2012, available online here.
P. Ozsvath, A. Stipsicz and Z. Szabo: Grid Homology for Knots and Links, American Mathematical Society, 2002, available online here.