Topology of 3-Manifolds

Topology of 3-Manifolds

2 hours lecture + 1 hour exercise session

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

Summer term 2020

Lecture: Fridays 09:15 - 10:45 in room 1.114 (RUD 25) temporarily online via Zoom
Veranstaltungsnummer: 3314435

Exercise session: every other Friday 13:00 - 14:30 in room 1.114 (RUD 25) temporarily online via Zoom
Veranstaltungsnummer: 33144351

Office hours / further discussion: after the lectures/exercise sessions via Zoom

Zoom meeting info: click here for Zoom meeting info

Announcements:

25.04.2020: We will shift the exercise session slightly. The exercise session will take place every other week Fridays 13:00-14:30.
25.04.2020: The next two Fridays are holidays in Berlin ('Tag der Arbeit' and 'Jahrestag der Befreiung vom Nationalsozialismus'). I would not like the lecture to be cancelled for two weeks in a row. Therefore, I will offer lectures and exercises at the following alternative dates:
Lecture: Thursday 30th of April 15:00-16:30
Exercise: Thursday 30th of April 17:00-18:30
Lecture: Thursday 7th of May 15:00-16:30
In the replacement lectures we will speak about knot invariants and in particular define and analyze the Jones polynomial. Since I am not sure, if everyone of you is fine with this, I will make the topic about the Jones polynomial optional for the exam (if you don't want to be ask about it I will not do so). But I think it is a very interesting topic and I hope many of you can make it to the Thursday replacement lectures. In the lecture Fridays on 15th of May we will start discussing manifolds. If you cannot come to the lectures on the Jones polynomial you can join again at that date. (The rest of the lecture will not really depend on the Jones polynomial and can be heard independently.)
11.06.2020: The UT Austin is organising online summer mini courses this year. Maybe some of them are interesting for you. Have a look at https://web.ma.utexas.edu/SMC/Minicourses.html.
Related to this lecture is the course on "Heegaard splittings and dehn surgery" (which should be parallel in parts to what we did in lecture) and the course on "Kirby calculus and 4-manifold" (of which you can think in some sense as "Heegaard diagrams of 4-manifolds").
10.07.2020: I will offer an additional lecture (not mandatory for the exam) on Friday 17.7. at 13:00 where we will discuss the proof of the Hilden-Montesinos theorem, saying that every 3-manifold arises as a 3-fold branched covering over the 3-sphere branched along a knot. Additionally, I will shotly discuss how some of the topics from this lecture can be generalized to 4-manifolds.
At Wednesday, 22.7.2020, at 9:15 I will offer an additional office hour, where we can discuss any questions you might have. (Includding questions on the exercise sheet 7.)

Content: In this lecture, we will introduce some basic results on 3-manifolds, i.e. topological spaces locally modeled on Euclidean 3-space. There are two classical ways to study 3-manifold: By their embedded submanifolds of dimension 1 (knots) or dimension 2 (surfaces). Therefore, we will first study knots and surfaces on its own.
Then we will move on to so-called structure theorems of 3-manifolds, which say that we can present and analyze 3-manifolds in simple combinatorial 2-dimensional graphics. In particular, we will prove that any 3-manifold admits a Heegaard splitting along a surface and an open book decomposition. Moreover, we will prove that any 3-manifold can be obtained by surgery along a link and as a 3-fold branched cover along a knot.

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

Due to this special situation, I will keep the lecture much closer to the book by Prasolov and Sossinsky (see literature below) than I had planned. (Some topics that might be a bit too short because of this, we will discuss in more detail in the lecture on 4-manifolds, which is planned for the summer semester 2021). If there is any confusion, you can consult the book as well. After each lecture I will also upload the notes. However, I strongly recommend (not only for educational reasons) that you make your own notes, as I will not write down everything I say.

Prerequisites: Prerequisites are the introductory lectures (Analysis I, II and Linear Algebra I, II) and basic notions from point set topology (covered in the module Topology I). Results from algebraic topology (fundamental group, homology theory) and differential topology are useful, but they are not needed for the understanding of the lecture.
Parallel to the lecture I recommend attending the seminar on knot theory by Klaus Mohnke.

Important: Please send me a short email if you are interested in attending this course so that I can contact you with important information (like the Zoom link to the lectures) and check this website regularly.
Everyone (also from outside the HU) is very welcome to participate, just send me a short email that I can contact you.

Exam: There will be oral exams at Tuesday, 28. July 2020 and Tuesday, 27 October 2020.
Due to the special situation I also offer you to examine the content of the book by Prasolov and Sossinsky instead of the lecture, especially since I don't know if everybody has the necessary infrastructure to really follow the lectures online. In this case please contact me by e-mail.
Criterion for admission to the final examination: For this lecture an admission restriction to the final examination is unfortunately 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 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)

Table of Contents: (tentative)

0. Overview

1. Knots and links:
1.1. Knot projections and Reidemeister moves
1.2. Knot invariants
1.3. Knot polynomials

2. Manifolds and handle decompositions:
2.1. Top - PL - Diff
2.2. Handle decompositions

3. Heegaard splittings:
3.1. Existence of Heegaard splittings
3.2. Kirby calculus of surfaces
3.3. Heegaard diagrams
3.4. Lens spaces
3.5. Handle slides and stabilizations

4. The mapping class group of surfaces
4.1. Generators
4.2. Relations
4.3. Baer's theorem

5. Dehn surgery:
5.1. Surgery and handlebodies
5.2. Surgery desccription of 3-manifolds
5.3. Surgery coefficients and linking numbers
5.4. The Poincaré homology sphere
5.5. Integer surgery and the Lickorish-Wallace theorem
5.6. The Rolfsen twist and Kirby's theorem

6. Branched coverings:
6.1. Branched covers of surfaces and the Riemann-Hurwitz formula
6.2. Branched covers of 3-manifolds

7. Outlook: smooth 4-manifolds - Kirby calculus and trisections

Lecture notes: The notes of the lecture can be viewed via the following OneNote link or as pdf file. (Obviously, these notes do not contain everything that was said in the lecture and therefore do not replace in any way the attendance of the lecture and an own transcript).

Notes from the exercise session: The notes from the exercise sessions can be viewed via the following OneNote link or as pdf file. (These notes are even more sketchy than the lecture notes and will not replace your own solutions of the exercises).

Evaluation: The results of the evaluation can be view here: lecture and exercise session.

Literature:
Most material mentioned here is available online at the linked pages (some sourches only if you connect to the HU network via VPN, if you do not know how to do it tell me in the first lecture).

Most of the material from the lecture is covered in the classic but still highly recommended book by Rolfsen. A more elementary approach is chosen by the newer book of Prasolov and Sossinsky. We will follow roughly the book by Prasolov and Sossinsky. I recommend having regularly a look in both books.
V. Prasolov and A. Sossinsky: Knots, Links, Braids and 3-Manifolds, AMS, 1997, available online here.
D. Rolfsen: Knots and Links, Publish or Perish, 1976, available online here. Before you look into Rolfsen's book I also recommend reading the following review on it.

For more background on handle decompositions I recommend:
H. Geiges: How to depict 5-dimensional manifolds, Jahresbericht der DMV, 2017, available online at the ArXiv.
R. Gompf and A. Stipsicz: 4-Manifolds and Kirby Calculus, AMS, 1999, available online here.
For more information on handle decompositions of surfaces you may also consult:
C. Wengler: Skript zur Topologie I, 2019, available online here.

For more background knowledge about some topics that we will only cover marginally in the lecture, but which might be interesting parallel to the lecture, I recommend:

Knot theory:
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.

Morse theory:
Y. Matsumoto: An Introduction to Morse Theory, American Mathematical Society, 2002, available online here.
J. Milnor: Morse Theory, Princeton University Press, 1963, available online here.

The mapping class group:
B. Farb and D. Margalit: A Primer on Mapping Class Groups, Princeton University Press, 2012, available online here.

Differential topology:
T. Bröcker and K. Jänich: Einführung in die Differentialtopologie, Springer. (A translation to English is also available.) I could not find the book by Bröcker and Jänich at a legal online source. (If you do please let me know, that I can link it here.) However, most of the material about differential topology that I will mention in the lecture can be easily found online.

Algebraic topology:
S. Friedl: Lecture notes for Algebraic Topology I-IV, available online at his homepage.
A. Hatcher: Algebraic topology, available online at his homepage.
C. Wendl: Lecture notes for Topologie I and II, available online at his homepage.

back