Algebraic topology

Teacher(s)

Dos Santos Santana Forte Vaz Pedro; Lambrechts Pascal;

Language

English

Prerequisites

LMAT1131 - linear algebra (first year Bachelor of Mathematical Sciences) or equivalent course.
LMAT1231 - multilinear algebra and group theory (second year Bachelor of Mathematical Sciences) or equivalent course.
LMAT1323 - topology (Second Year B.Sc. Mathematics) or equivalent course.

Main themes

Classification of surfaces.
Euler's characteristic.
Fundamental group.
Coating.
Homology.

Learning outcomes

At the end of this learning unit, the student is able to :
1	Contribution of the course to the learning outcomes of the master's program in mathematics. At the end of this activity, the student will have progressed in his/her ability to : - Acquire independently and exploit new knowledge - Demonstrate abstraction, reasoning and critical thinking skills. In particular, they will have developed their ability to -- read a demonstration and recognize its steps, key arguments and structure -- appreciate the simplicity, clarity, rigor and originality of a demonstration and of a mathematical or logical reasoning and detect any flaws. - Fundamental disciplinary knowledge and skills, including: -- His/her knowledge of the fundamental concepts of important current mathematical theories and will be able to establish the main links between these theories. -- His/her expertise in fundamental computational tools and their use in mathematical problems. - Scientific communication, especially structuring an oral presentation, highlighting key elements, distinguishing techniques and concepts, and adapting the presentation to the level of expertise of the audience. Specific learning outcomes of the course. At the end of this activity, the student will be able to : - Recognize, classify and construct surfaces. - Compute on simple examples classical invariants of algebraic topology: fundamental group, Euler class, homology group. - Deduce some topological properties of spaces from invariants of algebraic topology. - Develop in detail an element of algebraic topology theory.

Content

This activity is a first course in algebraic topology. It is highly recommended to take lmat2215 "homological algebra" in parallel or to have already taken a course in homological algebra.
The following content is covered in this course :
- Basic notions: homotopy, construction of topological spaces, reminder of varieties, reminder of the classification of surfaces.
- Degree of a continuous application of the circle in itself and applications.
- Fundamental group : definition and methods of calculation including the Seifet-Van Kampen theorem.
Presentation of a group by generators and relations.
- Coating: definitions, examples, links with the fundamental group; universal coatings; raising theorems; classification theorems.
- Homology of spaces: definition of simplicial homology and examples of calculations; applications.
If time permits, more advanced applications of homology of spaces.

Teaching methods

Combination of lectures, directed readings and exercises to prepare.

Evaluation methods

The evaluation will consist of exercises to be prepared during the year according to detailed procedures on moodle as well as a written and oral exam after the quadrennium.
Due to health conditions, there will be no oral presentation during the year.

Online resources

Course web page on moodle

Bibliography

La bibliographie sera précisée sur la page moodle du cours.
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The bibliography will be specified on the moodle page of the course

Teaching materials

• matériel sur moodle

Faculty or entity

MATH