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 4th, 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.
The course of December 4 will be given by João.

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).

Pictures

You can find here the pictures of the various boards of the course of November 27 on on the free product of groups, free groups, amalgamated sum of groups, and presentations of groups.

Lecture Notes

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

Schedule

• Decembre 11: Exam 3: Surfaces and Seifert-Van Campen theorem.
• Decembre 8: Exercise session on Seifert-Van Campen theorem and applications.
• Decembre 4: Course (João). Seifert-Van Campen theorem.
• December 1: Exercise session. Show that S_3 (the symmetric group on 3 letters) is not free. Show that the free product of groups is unital, associative, and symmetric. Prove that the free group on n elements is given by the free product of n copies of Z (the group of integers). Exercises 3.1 & 3.2 of the book. Find presentations for the dihedral group D_n and the symmetric group S_n. Show that the free product <S_1;R_1> * <S_2;R_2> of two groups with presentations is isomorphic to the group with presentation <S_1, S_2;R_1, R_2>. Same question with the free product with amalgamation over the free group <S>. Determine the free product of Z/4Z with Z/6Z amalgamated over Z/2Z. Describe the pushout construction for vectors spaces and for abelian groups.
• November 27: Course. Free products of groups, free group, group quotient, presentation of a group, and amalgamated sums of groups.
• November 24: Exercise session. Finish Exercise 2.9, prepare Exercise 2.10. Do the worksheet on the Euler characteristic.
• November 20: Course. Surfaces: proof of the classification theorem.
• November 17: Exercise session. Prove that any simplicial complex of dimension 2 is a cellular complex made up of points, intervals, and discs. Give a triangulation of the torus; give a minimal triangulation of the torus. Exercices 2.7, 2.8, 2.9, 2.10.
• November 13: Course. Surfaces: definition of topological manifold of dimension k, planar representation of a surface, connected sum, classification of surfaces (without boundary, statement only), simplicial complex, triangulation.
• November 13: Exam 2. Chapter 2: Product topology, quotient topology, topological group action, cellular complexes.
• November 10: Exercise session. Exercices 2.1, 2.4, 2.5. Give a finite cellular decomposition of the torus. Show that any finite CW-complex is path-connected as soon as its 1-skeleton is path-connected. Show that any finite CW-complex is compact.
• November 6: Course. End of topological group action, cellular spaces.
• November 3: Exercices 2.3, 2.6, 2.12. P^n(C) is homeomorphic to S^{2n+1}/S^1 and the complex projetive spaces are compact, separated and connecte. 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:  
• Exam 2:  
• Homework 1:  
• Worksheet (November 24th):  

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 2025).

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)

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Last updates: November 30, 2025.