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MATH-F419
Algebraic Topology
Titulaire(s) du cours
Andriy Haydys (Coordonnateur)Crédits ECTS
5
Langue(s) d'enseignement
anglais
Contenu du cours
• Singular homology, basic definitions and properties.
• The Hairy ball theorem.
• Computations of homology groups for specific topological spaces (spheres, graphs, surfaces).
• CW complexes and cellular homology, equivalence to singular homology.
• Fundamental group, covering spaces.
• The Hairy ball theorem.
• Computations of homology groups for specific topological spaces (spheres, graphs, surfaces).
• CW complexes and cellular homology, equivalence to singular homology.
• Fundamental group, covering spaces.
Objectifs (et/ou acquis d'apprentissages spécifiques)
The idea is to associate to topological spaces algebraic objects (groups, rings etc). If this is done judiciously, one can hope for example to distinguish non-homeomorphic spaces or essentially different continuous maps (in a suitable sense). This in turn allows one to prove interesting results, for example that any continuous map from a closed ball in a finite-dimensional Euclidean space into itself has a fixed point (Brower’s theorem).
At the end of this teaching unit, a student will be able
• To compute homology groups of certain topological spaces;
• Apply homology groups for studies of topological spaces and continuous maps between them;
• Decide if a topological space is a CW space;
• Compute cellular homology groups;
• Compute fundamental groups of topological spaces;
• Describe covering spaces of a given topological space.
At the end of this teaching unit, a student will be able
• To compute homology groups of certain topological spaces;
• Apply homology groups for studies of topological spaces and continuous maps between them;
• Decide if a topological space is a CW space;
• Compute cellular homology groups;
• Compute fundamental groups of topological spaces;
• Describe covering spaces of a given topological space.
Pré-requis et Co-requis
Connaissances et compétences pré-requises ou co-requises
MATH-F211 Topologie
Méthodes d'enseignement et activités d'apprentissages
• Lectures, including remote lecture on Teams if lecturing in person will not be possible.
Références, bibliographie et lectures recommandées
J. Vick. Homology theory. An introduction to algebraic topology.
A. Hatcher. Algebraic Topology ( Chapters 1 and 2), available on line: http://www.math.cornell.edu/~hatcher/AT/ATpage.html
Contribution au profil d'enseignement
• To acquire advanced notions in certain areas of mathematics.
• Develop an abstraction process for the development of a theory.
• Write a mathematically rigorous solution or a result in a mathematical theory.
Autres renseignements
Informations complémentaires
It is intended to provide written lecture notes.
Contacts
Andriy Haydys, andriy.haydys@ulb.be
Campus
Plaine
Evaluation
Méthode(s) d'évaluation
- Examen oral
Examen oral
Examen oral
Langue(s) d'évaluation
- anglais