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# Coursebooks 2017-2018

## Homology and cohomology

#### MATH-371

#### Lecturer(s) :

Raum Sven#### Language:

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

- Simplicial homology
- Singular homology
- Cellular homology
- Abstract homology theories
- Cohomology
- Group homology and cohomology
- 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

##### Ressources en bibliothèque

### In the programs

**Semester**Spring**Exam form**Written**Credits**

5**Subject examined**

Homology and cohomology**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 |

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