Lecture Topology II
4 hours lecture + 2 hours exercise session
Winter term 2019/2020
Lecture: Wednesdays 13:15-14:45 in room 1.013 (RUD 25) and Fridays 9:15-10:45 in room 1.013 (RUD 25)
First lecture: 16.10.2019
Exercise session: Fridays 11:00-12:30 in room 1.013 (RUD 25)
First exercise session: 18.10.2019
Important announcements for the second exam:
- 09.05.2020: The second exam will take place at 2. June 2020 at 8:00. You can register for the exam in the usual way (via Agnes).
The registration is now open and will close soon at 19. May 2020. If you were already registered for the original appointment, you are automatically registered for
this new appointment as well. For further information please have a look at the webpages of the 'Prüfungsbüro'
However, the exam will NOT take place in RUD 26, 1'306 as written in the 'Prüfungsplan', but will instead be hold ONLINE. The information will soon change also in the 'Prüfungsplan'.
The exact details are as follows.I will send you the exercises of the exam via E-Mail at 8:00 as well as a declaration of independence ('Selbstständigkeitserklärung').
The declaration of independence you will have to copy in your own handwriting and sign it.
Then you have 2 hours (or maybe a bit more) to solve the exercises in your own handwriting
with pen and papers without the help of others. For that you are allowed to use everything that we had discussed in lecture and exercise session. If you want to use a result that we did
not discus in lecture you have to prove it. You may also use other sources, as books, other scripts, etc. (but should of cores cite everything correctly that you are using).
It is NOT allowed to contact other persons during the time of the exam.
After the time of the exam has expired you will get another 30 minutes to scan or photograph your solutions and send them
via e-mail in a single pdf file to me.
You will need the following for taking part in the exam:
(1) A working internet connection to receive the mail with the exam and for sending your solutions back to me.
(2) Pens and paper.
(3) A device to photograph or scan your solutions. (A usual smartphone should do. I suggest not using your camera but instead directly a scan app, I heard good things about Notebloc.)
Please make yourself familiar with how to scan several handwritten sheets of paper with the smartphone and convert them into a single pdf before the exam. So that there are no problems on the day of the exam.
Please contact me if you do not have aces to one of the three items above at the date of the exam or if you have any questions.
If you did not recived this information by e-mail already, that means I do not have your e-mail. In that case please contact me by e-mail (so that I can send you the exercises at the date of the exam).
Please also share this information with your colleges who might want to write the second exam. It is important that I have the e-mail adress of everyone who wants to write the exam.
- 11.06.2019: The first lecture will take place on Wednesday 16.10.2019.
- 11.06.2019: This course is a BMS course and is therefore held in English.
- 23.07.2019: To avoide collisions with other lectures the time and place of the lecture has changed.
- 23.09.2019: Lecture notes from last semesters course on Topology I are now available further down on this page.
- 15.10.2019: On Friday 18.10. there will be no exercise session, instead there wil be an additional lecture.
On Wednesday 23.10. there will be an exercise session instead of the lecture.
- 18.10.2019: Since the old room was far too small, the Friday lecture and the exercise session will take place in room 1.013 (starting from next week).
- 29.11.2019: The lecture on Wednesday 4.12. will be cancelled and (if necessary) added at the end of the semester.
- 27.01.2020: Exam: The exam will take place at Wednesday, 4.3.2020, 09:00-11:00 in room 0'313 (RUD 26).
Please register for it in due time. You can have a look in your exam at Friday, 6.3.2020, 10:00-11:00 in room 0'318 (RUD 26).
- 14.01.2020: In the winter term 2020/2021 an advanced seminar on
selected topics of algebraic and differential topology
will be offered,
in which further topics will be discussed, which were not discussed in this lecture due to lack of time.
- 14.01.2020: In the summer term 2020 a two-hour lecture on 3-manifolds will be offered, in which we will
investigate methods to present and analyze 3-manifolds.
Content: This is the continuation of my lecture Topologie I from the summer term. It is aimed at the audience of that lecture and
other interested students with a basic knowledge of topology. In this second part we will analyze CW complexes and study higher homotopy
groups, more general homology theories and cohomology theory and discuss further applications of these theories.
Prerequisites: Prerequisites are the introductory lectures (Analysis I, II and Linear Algebra I, II),
elementary algebra (groups, homomorphisms, rings, etc.), and the lecture Topology I (in particular the fundamental group and simplicial homology).
Exam: The exam will take place at Wednesday, 4.3.2020, 09:00-11:00 in room 0'313 (RUD 26).
Second exam: The second exam takes places at Tuesday, 2.6.2020, online (see information above).
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.
Sheet 1 (pdf)
Sheet 2 (pdf)
Sheet 3 (pdf)
Sheet 4 (pdf)
Sheet 5 (pdf)
Sheet 6 (pdf)
Sheet 7 (pdf)
Sheet 8 (pdf) (A typo in Exercise 2 has been corrected. It should be the wedge sum instead of the Cartesian product.)
Sheet 9 (pdf)
Sheet 10 (pdf)
Sheet 11 (pdf)
Sheet 12 (pdf)
Sheet 13 (pdf)
Sheet 14 (pdf)
Sheet 15 (pdf)
Table of Contents: (tentative)
1. Categories and Functors
2. Homotopy groups:
2.1. Definition, functoriality and first computations
2.2. Homotopy groups of spheres
2.3. Relative homotopy groups and the long exact sequence of a pair
2.4. The Hopf fibration and the long exact sequence of a fibration
3. Singular homology:
3.1. Definition and functoriality
3.2. Relative homology groups
3.3. The long exact sequence of a pair
3.5. The Jordan-Brouwer splitting theorem
3.6. The Eilenberg-Steenrod axioms
4. Cellular homology:
4.1. CW complexes
4.2. Cellular homology groups
4.3. Equivalence of cellular homology and singular homology
4.4. The local degree of a map
4.5. Proof of Whitehead's theorem
4.6. Proof of Hurewicz's theorem
5. Homology with coefficients
5.1. Tensor products
5.2. Homology with coefficients
5.3. The Tor functor and the universal coefficient theorem
5.4. Künneth's formula and the homology of a cartesian product
5.5. Persistent homology and topological analysis of data
6. The cohomology ring:
6.1. Dualizing a chain complex
6.2. The universal coefficient theorem for cohomology
6.3. Singular and cellular cohomology
6.4. Graded Rings
6.5. The cup product
6.6. Calculations of cohomology rings
6.7. Cross products and Künneth's formula for cohomology
6.8. The dimension of real division algebras
7. Poincaré duality:
7.1. The cap product and the Poincaré isomorphism
7.2. First applications of Poincaré duality
7.3. Poincaré duality via simplicial cohomology
7.4. Cohomology with compact support
7.5. The general proof of Poincaré duality
8. Smooth manifolds and intersection forms:
8.1. The intersection form and classification of manifolds
8.2. Tangent spaces and transverse intersections
8.3. Algebraic intersection numbers and the intersection form
Lecture notes from last semesters course on Topology I:
Carolin Wengler has made the effort to format her lecture notes from the last semester lovingly with LaTeX and
kindly made them available to me. (If you find errors, including smaller typos, please report them to me, such that I can correct them.)
Carolin Wengler's lecture notes (pdf) (in German)
For the basics from the last semester (point set topology, fundamental group and simplicial homology theory) I recommend the books by M. Armstrong and K. Jänich.
A very popular textbook on (algebraic) topology is the book by A. Hatcher. In addition, I would also like to recommend you the
lecture notes by S. Friedl and C. Wendl, and the books by G. Bredon and A. Fomenko and D. Fuchs.
M. Armstrong: Basic Topology, Springer, 1983.
G. Bredon: Topology and geometry, Springer, 1993.
S. Friedl: Lecture notes for Algebraic Topology I-IV, available online at his
A. Fomenko and D. Fuchs: Homotopical topology, Springer, 2016.
A. Hatcher: Algebraic topology, available online at his homepage.
K. Jänich: Topologie, Springer, 1996.
C. Wendl: Lecture notes for Topologie I and II, available online at his homepage.
Further material in supplement or parallel to the lecture:
We did not discussed a proof of the smooth (generalized) Schoenflies theorem, since it uses differential topology and is thus not really fitting into the topic
of this lecture. However, the proof uses a beutiful argument which is not too hard to understand. A proof can be found (apart from the original works) for
A. Putman: The generalized Schoenflies theorem, available online at his homepage.
Some animations parallel to the constructions via pictures in lecture can be found here:
An animation visualizing the construction of Alexander's horned sphere.
A visualization of the Hopf fibration.
Some interesting 3D-animations (mostly fitting to last semesters topics) can be found on Neil Strickland's
For details about Boy's embedding of the real projective plane into Euclidean 3-space I suggest:
The Youtube video by Jos Leys,
the construction method by Arnaud Chéritat, and
the short article by Robion Kirby.
For more information about persistent homology I recommend the book by Edelsbrunner and Harer (assuming no prior knowledge in topology) and the book project by
Polterovich, Rosen, Samvelyan and Zhang.
H. Edelsbrunner and J. Harer: Computational Topology - An Introduction, AMS, 2010.
L. Polterovich, D. Rosen, K. Samvelyan and J. Zhang: Topological Persistence in Geometry and Analysis, available online
For further reading I suggest the above mentioned Skript by S. Friedl and the book by A. Fomenko and D. Fuchs. For information on 4-manifold the book by
A. Scorpan is a good starting point. For more general differential topology I suggest:
T. Bröcker and K. Jänich: Einführung 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.
A. Scorpan: The wild world of 4-manifolds, AMS, 2005.