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


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.

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

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.