# Coursebooks

## Algebraic topology

#### Lecturer(s) :

Urech Christian Lucius

English

#### Summary

The aim of this course is to introduce the notions of homology and cohomology of topological spaces and to learn tools to compute them.

#### Content

- Cell complexes and simplicial complexes
- Simplicial and singular homology
- Exact sequences and excision
- Mayer-Vietoris sequence
- Eilenberg-Steenrod axioms
- Cohomology
- Universal coefficient theorem
- Cup product
- Poincaré duality

#### Keywords

Homology, cohomology, cell complexes

Topology

Group Theory

#### 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 simplicial complexes
• apply excision and Mayer-Vietoris

#### Teaching methods

ex-cathedra teaching, exercise classes

#### Expected student activities

Attending the course, solving the weekly assignments, participating actively in the exercise classes

#### 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

#### Resources

##### Bibliography

Algebraic Topology », Allen Hatcher

Algebraic Topology », Tammo Tom Dieck

### In the programs

• Mathematics, 2019-2020, Bachelor semester 6
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Algebraic topology
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks

### Reference week

MoTuWeThFr
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
Under construction
Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German