Algebraic Topology (S1 2025/2026)

Topics Covered

• Lecture 1 [01/09, 3A-G11]: Homotopy Equivalence [H, 0.1], Fundamental group [H,1.1], language of categories [M,L], π₁ as a functor [L, 1.2.5].
• Lecture 2 [03/09, 3A-G11]: Homological algebra (examples, homotopy) [HA]. Singular chains and homology [H, 2.1].
• Lecture 3 [08/09, 3A-G11]: More singular chains and homology [H, 2.1], Homotopy invariance [H, 2.1]
• Lecture 4 [10/09, 3A-G11]: Long exact sequence of a short exact sequence [H, 2.1] or [HA, 3], Relative homology [H,2.1].
• Lecture 5 [15/09, 3A-G25]: Small chains and excision [H,2.1].
• Lecture 6 [17/09, 3A-G11]:Five-Lemma, H(X,A) is sometimes equal to H(X/A), examples of computations of homologies [H,2.1] and [H,2.2].
• Lecture 7 [29/09, 3A-G11]: Examples of computations of homologies, Mayer-Vietoris sequence [H,2.1] and [H,2.2]. (Co)products, pullbacks and pushouts in general and in the categories we care about [L,M].
• Lecture 8 [01/10, 3A-G11]: (Co)products, pullbacks and pushouts in general and in the categories we care about [L,M]; CW complexes: properties and examples [H, Chapter 0] Cellular homology [H, 2.2].
• Lecture 9 [06/10, 3A-G11]:
• Lecture 10 [08/10, 3A-G11]:
• Lecture 11 [14/10, IMT Salle Pellos]: This is unusually on a Tuesday at 17:00. The class is at Room Pellos at the IMT. Building 1R2, top floor right up the elevator. If you have been to Maxime Wolff's office, it's on the same corridor.
• Lecture 12 [15/10, 3A-G11]:
• Lecture 13 [To determine, To determine]: There is some doubt on whether the class will take place on monday the 3rd of november (if so, it's in 3A-G11 at 13h30 as usual) or whether we need to postpone it to tuesday at a time and placeto determine.
• Lecture 14 [05/11, 3A-G11]: Exercise class?

Suggested exercises

• Lectures 1+2: Exercise sheet 1. Hints, half-solutions and comments.
• Lectures 3+4: Exercise sheet 2.
• Lectures 5+6: Exercise sheet 3.
• Lectures 7+8: Exercise sheet 4

These exercises are for you to practise what we have learned in class. There is no evaluation based on these exercises. The final exam will account for 100% of the final grade.

If you're thirsty for more exercises, [H] has an infinite supply of exercises at the end of each section. Bredon's book also has a bunch of exercises. If you would like more exercices on a specific topic, let me know.

Questions about any exercise can be asked directly after class (but I might not be able to come up with a solution on the spot), or by e-mail (which on the other hand risks a delayed answer). If you want me to check your solutions, feel free to send me them by e-mail.

Prerequisites

Basic knowledge of point-set topology, group theory, and abstract algebra is required. Attendance of the M1 course Topologie et Algèbre or equivalent is quite helpful but not required. We will review essential materials as needed.

References

• [H]Algebraic Topology by Allen Hatcher. Available for free at this link.
• Topology and Geometry by Glen E. Bredon.
• [M]: Categories for the Working Mathematician by Saunders Mac Lane.
• [L]: Basic Category Theory by Tom Leinster. Available for free at this link.
• [HA]: A Beginner's Guide to Homological Algebra: A Comprehensive Introduction for Students. Available for free at this link.
• A Concise Course in Algebraic Topology by Peter May. Available for free at this link.

Comment on the references

Globaly, [H] is a good reference to follow. We will be starting with Chapter 2. Two big differences to keep in mind: 1) Hatcher avoids the language of category theory for a very long time, while I prefer to introduce the categorical language from the start. 2) Hatcher works over the integeres, while we work with an arbitrary R-module.
Bredon is a good alternative (some proofs are presented differently), but it is writted for people coming from a more differential geometrical background. And finally, May's course is very good, but quite concise, so while it's good to get a well structured overview, sometimes you might want to supplement it with another reference.

Grading

The grade will be determined by a single written exam, taking place on the week of november 24th.