2 hours lecture + 1 hour exercise session

(In der Prüfungsordnung erscheint diese Vorlesung unter Modul M39: Spezielle Themen der Mathematik)

Summer term 2021

Veranstaltungsnummer: 3314437

First lecture: 16.04.2021

**Exercise session:** Fridays, every other week, 11:00 - 12:30, online via Zoom

Veranstaltungsnummer: 3314437

First exercise session: 23.04.2020

**Office hours / further discussion:** after the lectures/exercise sessions

There will be no Moodle page for this lecture. Instead the Zoom links will appear at this webpage and anyone interested in the lecture may join directly from here.

**Zoom meeting info:** click here for Zoom meeting info

**Mailling list:** If you would like to receive information about this course by email via a mailing list, please send me a short email so I can add you to
the mailing list.

*19.06.2020*: This will be a BMS course and the language of the lecture will be English.*26.01.2021*: The first lecture will take place in the week of 12th of April. The lectures will be conducted via Zoom. Everyone is welcome to participate. Further details and information on this will appear on this webpage as soon as they are available to me.

First we will examine the same procedure in three dimensions. This will then lead to the notion of a Heegaard diagram of a 3-manifold, a 2-dimensional diagram, in which all the information of the 3-manifold is encoded.

One dimension higher we will display all the information of a smooth compact 4-manifold (or its 3-dimensional boundary) in a so-called Kirby diagram. The term Kirby calculus is then generally used to describe the modifications of such diagrams that do not change the diffeomorphism type of the corresponding 4-manifolds (or their 3-dimensional boundaries).

This lecture is aimed at all students of mathematics with basic knowledge in topology and can also be used as preparation for a thesis in the field of topology.

Criterion for admission to the final examination: For this lecture an admission restriction to the final examination is not permitted. 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)

2.1. Manifolds

2.2. Handle decompositions

3.1. Heegaard diagrams

3.2. Handle slides and stabilizations

4.1. Kirby diagrams

4.2. Linking numbers and framings

4.3. The intersection form and the homology of 2-handlebodies

4.4. Topological 4-manifolds

5.1. Handle slides

5.2. Handle cancellations

6.1. Surgery and handle bodies

6.2. Dehn surgery

6.3. Kirby's theorem

9.1. The cobordism ring

9.2. The h-cobordism theorem and the proof of the higher dimensional Poincaré conjecture

10.1. Slice knots

10.2. The Bennequin bound

10.3. Exotic copies of R

10.4. The adjunction inequality

10.5. Compact exotic 4-manifolds

We will mainly follow chapters 4 and 5 from the standard reference by Gompf and Stipsicz:

Below are some more references for further reading. More references on the Kirby calculus:

With applications in contact and symplectic geometry:

3-manifolds, Heegaard splittings and surgery descriptions:

For more information on handle decompositions of surfaces you may also consult:

General on 4-manifolds:

Trisections:

Morse theory:

Differential topology:

Algebraic topology: