4 hours lecture + 2 hours exercise session

Winter term 2021/2022

Veranstaltungsnummer: 3314429

First lecture: 19.10.2021

**Exercise session (by Shubham Dwivedi):** Wednesdays, 15:00-16:30 in room 1304, RUD 26

Veranstaltungsnummer: 33144291

First exercise session: 20.10.2021

**Office hours / further discussion:** after the lectures

*07.06.2021*: This course is a BMS course and is therefore held in English.*07.06.2021*: The lecture and tutorial will both take place in person.*20.10.2021*: We agreed on the following slightly modified lecture times: Tuesdays 15:15-16:45 and Fridays 11:10-12:40.*02.03.2022*: At Friday, 4th of March, 14:00, we will have a quick zoom meeting (at the usual zoom link) to speak about the grades and the exam.*08.04.2022*: At Wednesday, 13th of April, 15:00, we will have a quick zoom meeting (at the zoom room: https://hu-berlin.zoom.us/j/63593510001) to speak about the grades and the exam.

**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, 21. February 2022, 10:00-12:00, at RUD 26, 1`304.

**Second exam:** The second exam will take place at Tuesday, 5. April 2022, 10:00-12:00, at RUD 26, 1`304.

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) Shubham has written solutions to Exercise 4 and the bonus exercise. I suggest verifying in detail that the map induced by H is really continuous, where H is the map that was used to show that the cone is contractible.

Sheet 3 (pdf)

Sheet 4 (pdf)

Sheet 5 (pdf)

Sheet 6 (pdf)

Sheet 7 (pdf)

Sheet 8 (pdf)

Sheet 9 (pdf)

Sheet 10 (pdf)

Sheet 11 (pdf) Shubham has written solutions to Exercise 3(c). Note, that you can also use directly Corollary 6.8.(2) to deduce that g induces a non-trivial map on first cohomology.

Sheet 12 (pdf)

Sheet 13 (pdf)

Sheet 14 (pdf)

Sheet 15 (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

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

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

4.1. 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

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:

Some animations parallel to the constructions via pictures in lecture can be found here:

A visualization of the Hopf fibration.

Some interesting 3D-animations (mostly fitting to last semesters topics) can be found on Neil Strickland's webpage.

Turning a sphere inside-out: wikipedia and youtube.

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:

An animation visualizing the construction of Alexander's horned sphere.

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 by Polterovich, Rosen, Samvelyan and Zhang.