4 hours lecture + 2 hours exercise session

Winter term 2020/2021

(1) If I do not have your e-mail address, I cannot send you any e-mails.

(2) If I cannot send you e-mails, I cannot send you the exam.

(3) If I cannot send you the exam, you do not know the exam questions.

(4) If you do not know the exam questions, you cannot answer them.

This means: If you want to take the exam, you should send me your e-mail address as soon as possible.

Veranstaltungsnummer: 3314431

First lecture: 04.11.2020

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

Veranstaltungsnummer: 33144311

First exercise session: 06.11.2020

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

**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*: The first lecture will take place on Wednesday 04.11.2020.*19.06.2019*: This course is a BMS course and is therefore held in English.*04.12.202*: Today's lecture had to be canceled halfway through due to internet problems in the HU network. That is why today's exercise could not take place. We will continue next week at the usual times.

**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 basics of homology theory).

**Exam:** The exam will take place at Monday, 01.03.2021, 9:00-12:00, online as take home exam.

**Second exam:** The second exam will take place at Wednesday, 07.04.2021, 9:00-12:00, online as take home exam.

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 4 (a) has been corrected.

Sheet 9 (pdf)

Sheet 10 (pdf)

Sheet 11 (pdf)

Sheet 12 (pdf)

Sheet 13 (pdf)

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

3.1. 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

R.1. Manifolds

R.2. The connected sum

R.3. Classification of surfaces

R.4. Simplicial complexes

R.5. A proof sketch for the classification theorem of surfaces

R.6. Simplicial homology

R.7. The degree of a map

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. More on Whitehead's and Hurewicz's theorems

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

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

7.1. 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

8.1. The intersection form and classification of manifolds

8.2. Tangent spaces and transverse intersections

8.3. Algebraic intersection numbers and the intersection form

8.3. Lefschetz fixed point theory

9.1. Differential forms and de Rham's cohomology

9.2. Smooth singular (co-)homology

9.3. De Rham's theorem

Carolin Wengler has made the effort to format her lecture notes 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 of topology (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.

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 beautiful argument which is not too hard to understand. A proof can be found (apart from the original works) for example in:

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 webpage.

For further reading I suggest the above mentioned script 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:

For the use of homology theory in applied mathematics (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.

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.