Master I Lecture

Algebraic Topology

(September-December 2024)

Abstract

The goal of this lecture will be to use linear algebra and group theroy in order to understand topological spaces up to continuous deformations.

Organisation

The course will take place every Tuesday from September 17th to December 10th, 2024. The exercise session will take place from 8.30am to 11.45am and the lectures will take place from 1.45pm to 5.pm. Until October 15th (included), Bruno Vallette will teach the exercise session and after November 5th (included) it will be Vadim Lebovici.

Main reference

We will mainly follow the excellent, concise and complete book "Topologie Algébrique" by Yves Félix and Daniel Tanré published by Dunod (256 pages).

Lecture Notes

The recollection notes on the definition of topological spaces and the product topology is available here:  

Schedule

• December 10th: Presentations of groups, Seifert-Van Kampen theorem, fundamental group of surfaces and the end of the classification of compact connected surfaces (survey of Chapter 3).
• December 3rd: Free product of groups (beginning of Section 3.1).
• November 26th: Classification of compact connected surfaces (end of Section 2.4).
• November 19th: Definition of surfaces, planar representation, connected sum, simplicial complex of dimension 2 (beginning of Section 2.4).
• November 12th: cellular complexes (Section 2.2).
• November 5th: end of the topological group action (Section 2.3).
• October 21st: Construction of topological spaces (Chapter 2), quotient topology and beginning of topological group action (Section 2.1 and beginning of 2.3).
• October 15th: Lecture (Fundamental group of the circle (Section 1.3) and Applications (Section 1.4)) and exercise section (Proof of the invariance of the dimension, Exercices 1.11 and 1.12).
• October 11th: Exercices from the course (fundamental group of a product of spaces, commutativity in the fundamental group of the torus), Exercices 1.8 and 1.9 Page 25.
• October 8th: Exercices from the course, Exercices 1.5, 1.6, and 1.7, properties of the fundamental group (Section 1.2) and the fundamental group of the circle (Section 1.3).
• September 24th: Exercices 1.1, 1.2, and 1.3 on page 25.
• September 17th: Homotopy (Section 1.1) and Fundamental group (Section 1.2)

Exams, homeworks and worksheets

• Exam 1:  
• Exam 2:  
• Exam 3:  
• Homework 2:  
• Worksheet (December 3rd):  

Beyond the lectures

• Analysis situs: great website dedicated to the works of Poincaré on the algebraic topology of manifolds.
• Voyages aux pays des maths: episode of this great series (watch the other ones!) dedicated to the Poincaré conjecture.

Other references

• Algebraic Topology: A First Course, Marvin J. Greenberg and John R. Harper, Mathematics Lecture Note Series, 58, The Benjaming/Cummings Publishing Company.
• Algebraic Topology, Allen Hatcher, Cambridge University Press, 2001.
• A concise course in Algebraic Topology, J. Peter May, Chicago Lectures in Mathematics, 1999.

Follow-up

• Homotopy theories (Master 2 Mathematics,Fundamental Courses II, November-December 2024).

Seminars

• Seminar of the Algebraic Topology team (Université Sorbonne Paris Nord).

Teachers

• Bruno Vallette (lectures/exercise sessions until October 15th)
• Vadim Lebovic (exercise sessions from November 5th)

Back to main page

Last updates: December 11, 2024.