Master I Lecture

Algebraic Topology

(September-December 2025)

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 Thursday 8.30am-11.45am from September 11st to December 11th, 2025.
The exercise session will take place every Monday 8.30am-11.45am from September 15th to December 8th, 2025.
Until October 6th (included), Bruno Vallette will teach the exercise session and, after October 13th (included), it will be João Lourenco.

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

• November 6: Course. End of topological group action, cellular spaces.
• November 3: Exercise session. Exercices 2.3, 2.4, 2.6, 2.12. Show that P^n(C) is homeomorphic to S^{2n+1}/S^1 and conclude that the complex projetive spaces are compact, separated and connecte. Show that P^3(R) is homeomorphic to SO(3).
• October 16: separated quotient spaces (pages 33-34), topological group action (pages 38-41 except Proposition 2.15).
• October 13: proof on the explcit form of the product topology, proof that a space X is separated iff its diagonal is closed in the product XxX, proof that the 1-dimensional real projective space P_1(R) is homeomorphic to the circle S^1, proof that X compact implies X/R compact, proof that [0,1]/{0,1} is homeomorphic to the circle S^1, proof that the quotient of the cylinder S^{n-1}x[0,1] by S^{n-1}x{0} is homeomorphic to the n-dimensional ball B^n, Exercices 2.1 and 2.2, proof that R^2/Z^2 is homeomorphic to the torus.
• October 9: Course on the construction of topological spaces (Chapter 2). General topology, product topology, quotient topology.
• October 6: Exercices 1.11, 1.12, 1.13, 1.14.
• October 2: Strike.
• September 30: Applications of the fundamental group of the circle (Section 1.4) and the fundamental group of the higher dimensional spheres (Section 1.5).
• September 29: The fundamental group of the circle (Section 1.3).
• September 25: the fundamental group of a product of topological spaces, exercises 1.8, 1.9, and 1.10 (pages 25-26).
• September 22: Fundamental group (Section 1.2).
• September 18: Strike.
• September 15: Exercices 1.1, 1.2, 1.3, 1.7 (pages 24-25).
• September 11: Homotopy (Section 1.1).

Exams, homeworks and worksheets (2025)

• Exam 1:  

Exams, homeworks and worksheets (2024)

• 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 6th)
• João Lourenco (exercise sessions from October 13th)

Back to main page

Last updates: October 16, 2025.