Course teacher(s)
Andriy Haydys (Coordinator)ECTS credits
5
Language(s) of instruction
english
Course content
• 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.
Objectives (and/or specific learning outcomes)
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.
Prerequisites and Corequisites
Required and Corequired knowledge and skills
MATH-F-211 Topologie
Teaching methods and learning activities
• Lectures, including remote lecture on Teams if lecturing in person will not be possible.
References, bibliography, and recommended reading
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 to the teaching profile
• 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.
Other information
Additional information
It is intended to provide written lecture notes.
Contacts
Andriy Haydys, andriy.haydys@ulb.be
Campus
Plaine
Evaluation
Method(s) of evaluation
- Oral examination
Oral examination
Oral examination
Language(s) of evaluation
- english