1) Prerequisites
- CW-complexes and singular homology to the extent treated in the Algebraic Topology 1 course taught in the fall; see the lectures notes by Steffen Sagave available at https://www.math.ru.nl/~
- Point set topology and the fundamental group to the extent these topics were already used in the Algebraic Topology 1 course.
- Basis knowledge about algebra and categories to the extent that one is familiar with the concepts of rings, modules, their tensor product, functors and natural transformations; see for example the lecture notes "Modules and Categories" by Lenny Taelman, available at https://staff.fnwi.uva.nl/l.d.
2) Aim of the course
This course covers advanced topics in Algebraic Topology, which may vary from year to year and build on the foundations provided by the Algebraic Topology 1 course from the fall.
The main subject of this edition of Algebraic Topology 2 is the homotopy theory of topological spaces, mostly using (co)homological tools. We start with introducing cohomology, a dual version of homology, which has a ring structure. Afterwards, we study homotopy groups, fibrations, and cofibrations. Combining these basics, we will develop then the Serre spectral sequence. In itself, it is foremost a powerful tool to compute the (co)homology of spaces. We will use it, however, to learn more about the homotopy groups of spheres and other spaces, both proving structural results and providing concrete calculations.
3) Rules about Homework/Exam
There will be about 4 to 5 homework assignments that are supposed to be handed in and that will count as a bonus for the total grade for this course: If the grade from the final exam is at least 5.0 and the average grade from the homework assignments is higher, then the final exam counts 75 % and the homework assignments count 25 %. The same bonus rule applies to the retake. In cases where we have doubts about the authorship of solutions of exercises and for example suspect that they are prepared using generative AI, we may ask students to orally explain their solutions to us during the exercise class and adjust the grade for the exercises based on this explanation.
The final examination and the retake will be written exams.
4) Literature
We provide detailed lecture notes for the course. Besides that, most of what is covered in this course can be found in the following standard textbooks on Algebraic Topology:
- Glen E. Bredon, Topology and Geometry
- Tammo tom Dieck, Algebraic Topology
- James F. Davis and Paul Kirk, Lecture Notes in Algebraic Topology
- Allen Hatcher, Algebraic Topology, including its Chapter 5 available here: https://pi.math.cornell.edu/~
- Docent: Magdalena Kedziorek
- Docent: Steffen Sagave