4 hours lecture + 2 hour exercise session

Winter term 2023/24

- There will be no lectures / exercise sessions on the following dates:
- Tuesday 24.10.23 RTG lecture
- Tuesday 31.10.23
- Thursday 2.11.23
- Tuesday 7.11.23 RTG lecture
- Thursday 16.11.
- Tuesday 21.11.23 RTG lecture
- Tuesday 5.12.23 RTG lecture
- Tuesday 16.1.24 RTG lecture
- Tuesday 30.1.24 RTG lecture
- Thursday 31.1.24 exams

- Instead of the above dates, we will have optional (i.e. not necessary for the exam) discussion sessions, in which we can speak about anything you want, including questions on the lecture or exercises, background material that you want to recall, topics that are not covered in the lecture (and will not be relevant for the exam) but might still interest you or anything else you might want to discuss.
- The dates for the discussion sessions are (we are meeting in my office):
- Thursday, 9.11.23, 13:00-14:00
- Wednesday, 15.11.23, 11:00-13:00
- Thursday, 23.11.23, 13:00-14:00
- Wednesday, 6.12.23, 13:00-14:00
- Wednesday, 13.12.23, 11:00-13:00
- Wednesday, 18.01.24, 13:00-14:00
- The RTG lectures might be interesting to some of you as well. At least my lecture about the Poincare conjecture should be accesible for participants of the contact geometry lecture as well. More information on the RTG Lectures can be found here and here.
- On Thursday 19.10.23 we will have an additional lecture instead of an exercise session.
- We have changed the exercise and lecture slots: The Exercise will take place on Wednesday and the lecture on Thursday.
- On Wednesday, 25.10.23 there will be a research talk in the symplectic geometry seminar about Legendrian knots in contact manifolds. Further informations can be found here.

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.

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 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)

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

3.1. 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

4.1. 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

5.1. 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

6.1. Symplectic manifolds

6.2. Fillable contact manifolds

6.3. Weinstein handles and other types of fillings

6.4. Holomorphic curves

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

8.1. Open books

8.2. The Thurston-Winkelnkemper construction

8.3. The Giroux correspondence

8.4. Tightness and fillability in open books

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.

More on convex surface theory can be found here:

More on the contact homology invariants of Legendrian knots can be found here:

The full proof of the classification of overtwisted contact structures can be found here:

More about symplectic fillings can be found for example in:

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.)

Background and further information on algebraic topologie can be found here:

Background and further information on knot theory can be found here: