Topology IV.b - cohomology rings

Summary

Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative algebra. We study an algebraic version, namely group cohomology, and compare both approaches.

Content

Keywords

Cohomology, cup product, extensions, Yoneda product, classifying space

Learning Prerequisites

Required courses

Topology, Algebraic Topology, Group Theory, Rings and Fields

Recommended courses

Rings and modules

Important concepts to start the course

Homology, homological algebra, exact sequences, cell complex

Learning Outcomes

By the end of the course, the student must be able to:

• Manipulate chain complexes
• Compute cohomology groups and products
• Compare singular with group cohomology
• Define the concepts from the course
• Prove important properties of cohomology
• Apply the concepts to examples

Transversal skills

• Make an oral presentation.
• Write a scientific or technical report.
• Communicate effectively, being understood, including across different languages and cultures.

Teaching methods

ex-cathedra teaching, exercise classes, project in pairs

Expected student activities

Attend the lectures and exercise sessions, solve exercises, hand in homework, prepare a presentation

Assessment methods

The final grade will be assigned based on:

20% - homework in small groups

20% - oral presentation during an exercise session

60% - 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

Resources

Virtual desktop infrastructure (VDI)

No

Bibliography

Algebraic Topology, Allen Hatcher

Algebraic Topology, Edwin Spanier

Modern Classical Homotopy Theory, Jeffrey Strom

Algebraic Topology, Tammo Tom Dieck

Cohomology of groups, Kenneth S. Brown

Cohomology of finite groups, Alejandro Adem and R. James Milgram

Ressources en bibliothèque

• Algebraic Topology / Hatcher
• Algebraic Topology / Spanier
• Modern Classical Homotopy Theory / Strom
• Algebraic Topology / Dieck
• Cohomology of groups / Brown
• Cohomology of finite groups / Adem