# Fiches de cours 2017-2018

## Homology and cohomology

Raum Sven

English

#### Summary

This course introduces to homology and cohomology of topological spaces and groups as well as their relation via the classifying space of a group.

#### Content

1. Simplicial homology
2. Singular homology
3. Cellular homology
4. Abstract homology theories
5. Cohomology
6. Group homology and cohomology
7. Classifying spaces

#### Keywords

algebraic topology, group cohomolgy, homological algebra, classifying spaces

#### Learning Prerequisites

##### Required courses

• Topology (MATH-225)
• Rings and fields (MATH-215)

##### Recommended courses

• Rings and modules (MATH-311)
• Group theory (MATH-211)

#### Learning Outcomes

By the end of the course, the student must be able to:
• Compare (co)homology theories of spaces
• Use basic algebraic homological algebra
• Choose appropriate methods to compute (co)homology
• Compute (co)homology
• Characterize low degree (co)homology of groups
• Compute models for classifying spaces

#### Transversal skills

• Continue to work through difficulties or initial failure to find optimal solutions.
• Demonstrate the capacity for critical thinking
• Access and evaluate appropriate sources of information.
• Write a scientific or technical report.
• Use both general and domain specific IT resources and tools
• Take feedback (critique) and respond in an appropriate manner.
• Give feedback (critique) in an appropriate fashion.

#### Teaching methods

Ex-cathedra course with exercises in the classroom and at home

#### Expected student activities

• Participate in the course and the exercise sessions
• Solve regular exercises
• Prepare one LaTeX hand-in on examples illustrating the course content
• Give peer-feedback on this LaTeX hand-in
• Prepare one LaTeX hand-in on a short piece of mathematics acquired independently

#### Assessment methods

Written exam, exercises, LaTeX hand-ins and peer-feedback. In case Art. 3 al. 5 of the regulations of the section apply to some student, the exam form will be decided by the teacher and communicated to the student.

#### Supervision

 Office hours Yes Assistants Yes Forum No

#### Resources

##### Bibliography

• Allen Hatcher. Algebraic topology. ISBN-13: 978-0-521-79540-1
• Kenneth S. Brown. Cohomology of groups. ISBN-13: 3-540-90688-6
• Charles A. Weibel. An introduction to homological algebra. ISBN-13: 0-521-55987-1

### Semaine de référence

LuMaMeJeVe
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
En construction

Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand