Algebraic Topology — ALT

Instructor: Dr. Szilárd SZABÓ

Text:  Allan Hatcher: Algebraic Topology and class notes

Prerequisites:
In order to be able to follow the course the students should have some background in topological spaces (e.g. Hausdorff-spaces, continuous maps, homeomorphisms) and basic notions of algebra (groups, rings and homomorphisms between them, normal subgroups). It is also expected that the students go over some proofs and work out some examples on their own.
Course description:
This course aims to give an elementary introduction into a fairly intuitive (but at the same time completely rigorous) part of mathematics that saw a huge development thoughout the 20th century and whose progress continues to go on, with more and more interaction with other parts of mathematics. It is intended for students interested in the concept of algebraic structures associated to topological spaces, and in general students who would like to get a glimpse of an actively developing topic with many applications. Parallelly to expanding the topological theory, we will get familiar with the necessary homological algebra machinery: amalgamated products of groups, exact sequences (short and long), free resolutions, Tor- and Ext-groups, categories.
Topics:

• Basic notions: cell complexes, homotopy, deformation retract, etc.
• Fundamental group: van Kampen's theorem, covering spaces, group actions on spaces
• Singular homology: excision, Mayer--Vietoris exact sequence, cellular homology
• Singular cohomology: universal coefficient theorem, cup-product, Poincaré-duality, Künneth-formula