Instructor: Dr. Szilárd SZABÓ
Text: Allan Hatcher: Algebraic Topology Chapters 0, 2, 3 and Appendix
Algebra: finitely presented abelian groups, subgroup, homomorphism, kernel, image, factor group. Rudiments of point-set topology: open/closed sets, continuity, connectedness, compactness.
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.
- 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
- Cellular homology
- Mayer--Vietoris long exact sequence
- Applications: Borsuk--Ulam theorem, Lefschetz fixed-point theorem
- Cohomology, universal coefficient theorem