*Instructor:* Dr. Szilárd SZABÓ

*Text:*
Allan Hatcher: Algebraic Topology Chapters 0, 2, 3 and Appendix

*Prerequisites:*

Algebra: finitely presented abelian groups, subgroup,
homomorphism, kernel, image, factor group.
Rudiments of point-set topology: open/closed sets, continuity, connectedness,
compactness.

*Course description:*

Course description: Homology groups give in some sense an algebraic measure of the complexity
of a topological space. The aim of the course is to introduce the notion of singular homology
groups and develop some intuition to working with them. We illustrate their relevance with
classical applications such as fixed-point theorems and invariance of dimension.
In spite of containing classical material, the course often relies on subtle topological
and algebraic tools, that we develop in parallel with the core of the topic.

*Topics covered:*

- Basic constructions with topological spaces, homotopy type, retraction
- CW-complexes, simplicial complexes, topological properties
- Simplicial homology
- Singular homology
- Homotopy invariance
- Relative homology, excision, invariance of dimension
- Degree
- Cellular homology
- Mayer--Vietoris long exact sequence
- Applications: Borsuk--Ulam theorem, Lefschetz fixed-point theorem
- Cohomology, universal coefficient theorem