Lecture Topology II
4 hours lecture + 2 hours exercise session
Winter term 2020/2021
Lecture: Wednesdays 13:00-14:30 and Fridays 9:15-10:45, online via Zoom
First lecture: 04.11.2020
Exercise session: Fridays 11:00-12:30, online via Zoom
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.
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, 01.03.2021, 9:00-12:00, online as take home exam.
Second exam: The second 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)
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
3.5. The Jordan-Brouwer splitting theorem
3.6. The Eilenberg-Steenrod axioms
Revision: Manifolds and simplicial homology:
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. 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
9. Differential forms and de Rham's theorem:
9.1. Differential forms and de Rham's cohomology
9.2. Smooth singular (co-)homology
9.3. De Rham's theorem
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)
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
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 beautiful argument which is not too hard to understand. A proof can be found (apart from the original works) for
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
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 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
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.