Lecture Topology II

4 hours lecture + 2 hours exercise session

Marc Kegel

Winter term 2021/2022

Lecture: Tuesdays, 15:15-16:45 in room 1.013, RUD 25 and Fridays, 11:10-12:40 in room 3.007, RUD 25
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


Content: This is the continuation of the lecture Topology 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 the beginning of the course we will recall the basics on homology theory and CW-complexes and their applications from last semesters lecture. Then we will dualize homology theory, yielding the concept of cohomology groups. At a first glance cohomology does not carry more information than homology but in a closer look we will identify an additional ring structure on the cohomology groups not present in homology. We will further investigate a particularly friendly class of topological spaces, the so-called manifolds, spaces which are locally homeomorphic to Euclidean space. For manifolds, there is the so-called Poincaré duality which relates homology and cohomology much closer. At the end of the lecture we will deal with higher homotopy groups and their relations to homology and cohomology.

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.

Exercise sheets:
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)

Table of Contents:

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

Revision: Manifolds and simplicial homology:
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. 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. More on Whitehead's and Hurewicz's theorems

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

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
8.3. Lefschetz fixed point theory

Lecture notes from my course on Topology I from 2019:

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)

Lecture notes for Topology I from the 2021 course can be found here.


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.

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:

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:
A. Putman: The generalized Schoenflies theorem, available online at his homepage.

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:
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.

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.
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 on the arXiv.