Content of the Class
We all know that a donut has a hole. But what exactly does that mean, how can we prove it, and what are the consequences?
It is the goal of algebraic topology to answer such questions – and not just for donuts, since holes in topological spaces can be of many different types. There are many ways to approach this question: using combinatorics, algebra, topology, or even analysis with its (multivariate) differential and integral calculus. This is what this class is about. We will construct and study the “hole-detecting” homology and cohomology groups of topological spaces using several entirely different methods, but arriving at the same result in the end – showing deep relations between different fields of mathematics that seemed to be unrelated at first.
Expressed in more technical terms, we will study simplicial, singular, and de Rham (co-)homology, and the various interconnections between these constructions. For the de Rham theory we will need the notion of smooth manifolds, which will also be introduced in the class. Finally, we will discuss various applications of our theory in different areas of mathematics. Just to name a few: Simplicial complexes form a straightforward generalization of graph theory, a byproduct of our work will be the multivariate integral theorems from vector analysis such as the divergence theorem of Gauss in arbitrary dimension, and topological data analysis uses algebraic topology in the form of persistent homology as its backbone.
Taken together, this class covers a substantial amount of material – but as usual in topology most statements should be very intuitive from the many pictures that we can draw. Also, we will usually refrain from discussing every topic in full breadth and generality, include many examples, and keep the prerequisites to a minimum.
Prerequisites
Beyond a first-year mathematics class on algebra and analysis, we will only assume the knowledge from an introductory course on topology.
News
Any news will be sent through the News tab of the OLAT course. You can configure OLAT so that these messages will also be sent to you by e-mail.
Dates and Times
- Lecture: Wed 8:15-9:45 (48-208) and Fri 12:15-13:45 (48-538), starting April 15
- Example class: Fri 14:00-15:30 (48-538), starting April 17
- Holidays: May 1 (Tag der Arbeit)
Credits
You will get a certificate for this class if you attend the example classes regularly and score at least 1 out of 4 points in at least 70% of the homework problems.
It is also possible to take this class as a seminar by replacing the oral exam by a seminar talk; details will be discussed in class.
Registration for the Class
If you want to take part in this class (or seminar) please register for it until noon on April 17 in the URM system from the university network (or using VPN).
Homework assignments
Homework problems will be assigned every Friday, starting on April 17. They can be downloaded below and are due on Mondays 10 days later (at any time). You can put your solutions in Diego's mailbox next to 48-210 or upload them as a PDF file in the Submissions tab of the OLAT course. You can submit them in groups of up to 3 students. Of course, in this case everyone should have their fair share both in finding and writing down the solutions; it is expected that everyone can explain their solutions on the blackboard in the example sessions. If you submit your solutions online as a group, only one of you should upload them and indicate the names of all participants on the solution.
Class Notes
There will be notes for this class that can be downloaded below (if you access this page from the OLAT course).
Disclaimer: As it is the first time this class is offered, this is also the very first version of the notes. There will be errors and inconsistencies. The notes will be changing throughout the semester as material is added, reordered or updated – not drastically, but noticeably. Do not count on the numbers of definitions or propositions being constant in time. The complete notes will not be available from the beginning as they are not finished yet, but I hope to have at least a usable initial version of every chapter ready before it is discussed in class. It should work to download every chapter individually right before it is covered in class, and then stick to this version in case you plan to make annotations for yourself and do not want to replace it later by an updated version.
Of course, I am always happy to get feedback on the notes. So if you find errors or have suggestions on how to improve the notes, please let me know!
Literature
There should be no need to use any further literature besides the notes above. Also, this class does not follow any particular book. However, most of the material covered is quite standard, and it should be easy to find references for it. The following list is a small selection for the beginning:
- A. Hatcher: Algebraic Topology (2021)
- G. Bredon: Topology and Geometry, Springer Graduate Texts in Mathematics 139 (1993)
- R. Bott and L. Tu: Differential Forms in Algebraic Topology, Springer Graduate Texts in Mathematics 82 (1982)
Questions?
If you have any questions – about the organization of the course, the lecture, the homework problems or whatever – do not hesitate to contact us! You can write an e-mail or in the forum, ask Diego (bargans@rptu.de, 48-518), talk to me after the lecture or just come at any time to my office (andreas@rptu.de, 48-517).