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
- 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).
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 will take place at Monday, 6.4.2020, 09:00-11:00 in room 1'303 (RUD 26).
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)
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. The local degree of a map
4.4. Invariance of cellular homology
4.5. Proof of Hurewitcz's theorem
4.6. Proof of Whitehead's theorem
5. Homology with coefficients
5.1. The universal coefficient theorem
5.2. Künneth's formula
5.3. 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. The cup product
6.4. The dimension of real division algebras
6.5. Homotopy and cohomology
7. Poincaré duality and intersection forms:
Further possible topics: Topological 4-manifolds, other homology and cohomology theories, de Rham's theorem, the linking 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 it to me, such that I can correct it.)
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.
M. Armstrong: Basic Topology, Springer, 1983.
S. Friedl: Lecture notes for Algebraic Topology I-IV, available online at his
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.