Lecture Topology II

4 hours lecture + 2 hours exercise session

Marc Kegel

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)
Veranstaltungsnummer: 3314430
First lecture: 16.10.2019

Exercise session: Fridays 11:00-12:30 in room 1.013 (RUD 25)
Veranstaltungsnummer: 33144301
First exercise session: 18.10.2019



Announcements:

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.



Exercise sheets:
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
3.4. Excision
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)



Literature:

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 homepage.
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 example in:
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.




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