- 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/~sagave/teaching/at-lecturenotes-2020.pdf
- Point set topology and the fundamental group to the extent these topics were already used in the Algebraic Topology 1 course
- Basic 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.j.taelman/ca.pdf
Aim of the course
This course covers advanced topics in Algebraic Topology which vary from year to year and build on the foundations provided by the Algebraic Topology course from the fall.
The main subject of this edition of Algebraic Topology II 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.
- Lennart Meier (Utrecht)
- Steffen Sagave (Nijmegen)