Topology III - Homology
Summary
Homology is one of the most important tools to study topological spaces and it plays an important role in many fields of mathematics. The aim of this course is to introduce this notion, understand its properties and learn how to compute it. There will be many examples and applications.
Content
- CW complexes
- Simplicial and singular homology
- Exact sequences and excision
- Mayer-Vietoris sequence
- Eilenberg-Steenrod axioms
- Cellular homology
- Cohomology
Keywords
Homology, cohomology, cell complexes
Learning Prerequisites
Required courses
- Metric and topological spaces
- Topology
Recommended courses
- Group Theory
- Rings and Modules
Learning Outcomes
By the end of the course, the student must be able to:
- Define the main concepts introduced in the course
- state the theorems covered in the course and give the main ideas of their proofs
- apply the results covered in the course to examples
- compute the homology groups of CW complexes
- prove easy topological facts
- express topological arguments
Teaching methods
lectures, exercise classes, a major part of the course will be taught in the flipped classroom format.
Expected student activities
Attending the course, doing the weekly assignments, participating acclassestively in the course and the exercise
Assessment methods
Assignments, written exam
Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés
Supervision
Office hours | Yes |
Assistants | Yes |
Forum | Yes |
Resources
Bibliography
Algebraic Topology », Allen Hatcher
Algebraic Topology », Tammo Tom Dieck
Ressources en bibliothèque
Moodle Link
In the programs
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Topology III - Homology
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
Reference week
Mo | Tu | We | Th | Fr | |
8-9 | |||||
9-10 | |||||
10-11 | |||||
11-12 | |||||
12-13 | |||||
13-14 | |||||
14-15 | |||||
15-16 | |||||
16-17 | |||||
17-18 | |||||
18-19 | |||||
19-20 | |||||
20-21 | |||||
21-22 |
Légendes:
Lecture
Exercise, TP
Project, other