Topology IV.b - cohomology rings
MATH-506 / 5 credits
Teacher:
Language: English
Remark: Pas donné en 2024-25
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
1. Singular cohomology
2. Universal coefficient Theorem
3. Cup product
4. Künneth formula
5. Group homology and cohomology
6. Computations of group (co)homology in low degrees
7. Cup product and Yoneda product
8. Classifying spaces
9. Comparison of singular cohomology and group cohomology
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
Office hours | Yes |
Assistants | Yes |
Forum | Yes |
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
In the programs
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Topology IV.b - cohomology rings
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Topology IV.b - cohomology rings
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Topology IV.b - cohomology rings
- Lecture: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Topology IV.b - cohomology rings
- Lecture: 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 | |
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Légendes:
Lecture
Exercise, TP
Project, labs, other