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
- Simplicial and singular homology
- Exact sequences and excision
- Mayer-Vietoris sequence
- Eilenberg-Steenrod axioms
- CW complexes
- Cellular homology
- Cohomology
Keywords
Homology, cohomology, cell complexes
Learning Prerequisites
Required courses
- Metric and topological spaces
- Topology
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
Expected student activities
Attending the course, homework assignments, participating atively in the course and the exercise
Assessment methods
Exam written
Supervision
| Office hours | Yes |
| Assistants | Yes |
| Forum | Yes |
In the programs
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Topology III - Homology
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
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, Lab, other