*Instructor:* Dr. Szilárd Szabó

*Texts based on:*

- W. Massey: A basic course in Algebraic Topology (GTM 127)
- W. Fulton: Algebraic Topology, a first course (GTM 153)
- Allan Hatcher: Algebraic Topology

*Prerequisites*:
basic abstract algebra (groups), some point-set topology
(e.g. understanding of compactness, connectedness)

*Course description:*

In the first part of the semester we study the fundamental group of topological spaces and use it to classify their covering spaces. In the second part we define the notion of a surface and give the classification of compact surfaces. In the last part we introduce the singular homology and cohomology groups of topological spaces and describe their most important properties. Throughout the semester we will illustrate the results by examples.

*Topics:*

- Fundamental group: functoriality, homotopy invariance, Seifert--van Kampen theorem.
- Covering spaces: lifting property, deck transformations, classification.
- Surfaces: triangulations, classification, Euler characteristic.
- Singular homology: functoriality, relative version, Mayer--Vietoris exact sequence, universal coefficient theorem, Künneth theorem.
- Singular cohomology: cup-product, Poincaré-duality.