Duration
30h Th, 10h Pr, 20h Mon. WS
Number of credits
| Master in mathematics (120 ECTS) | 8 crédits | |||
| Master in mathematics (60 ECTS) | 8 crédits |
Lecturer
Language(s) of instruction
French language
Organisation and examination
Teaching in the first semester, review in January
Schedule
Units courses prerequisite and corequisite
Prerequisite or corequisite units are presented within each program
Learning unit contents
This course constitutes an introduction to algebraic topology and homological algebra.
One starts by studying the singular homology of topological spaces in general and by treating explicitly a few significant examples (sphere, torus, sphere with handles, projective spaces, ...). To illustrate the usefulness of the preceding theory, we conclude this first part by establishing Jordan's theorem in arbitrary dimension.
In the second part, the study of product spaces leads us to define and study tensor products of complexes and to introduce the "Tor" functors. We conclude this part with Künneth's theorem.
The third part of the course is devoted to the duality between cohomology and homology and motivates the introduction and study of the "Ext" functors.
The course ends with a brief study of the homology and cohomology of topological manifolds and of Poincaré duality.
Learning outcomes of the learning unit
At the end of the course, the students should have a good idea of what algebraic topology is and of how its study leads naturally to homological algebra and derived functors.
Prerequisite knowledge and skills
A good knowledge of basic algebra and topology is essential.
Planned learning activities and teaching methods
The course consists of blackboard lessons, exercises sessions and a personal work.
During the lessons, the main theoretical results are introduced, established and illustrated with examples.
During the exercises sessions, the students are trained to solve by themselves various problems using the results considered in the lessons
The personal work consists of the preparation of a short paper presenting and establishing a result related to the course but not considered during the lessons.
Mode of delivery (face-to-face ; distance-learning)
Face-to-face course.
Recommended or required readings
Lecture notes are in preparation and a list of reference works is available.
Assessment methods and criteria
An examination comprising an oral part on the theory and a presentation of the personal work is organized during the first session. A similar examination is organized during the second session.
Work placement(s)
Organizational remarks
The course is given during the first quadrimester of even academic years. It is therefore not given in 2019-2020.
Contacts
Jean-Pierre Schneiders Département de Mathématique (Bât. B37, Bureau 1/60) Allée de la Découverte, 12 - 4000 Liège (Sart-Tilman) Phone: (04) 366.94.01 - E-Mail: jpschneiders@ulg.ac.be Web page: http://www.analg.ulg.ac.be/jps/
Adaptation of teaching commitments following the COVID-19 pandemic for the May-June 2020 session
Teaching methods implemented : distance-learning
Assessment subjects
Assessment methods
Contacts
Adaptation of teaching commitments following the COVID-19 pandemic for the Aug-Sept 2020 session
Assessment subjects
Assessment methods
Contacts
Items online
Course web page
Web page giving access to various informations on the course and to the electronic version of the notes.