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]: Cellular homology and examples [H, 2.2]. Tensor products are right exact.
• Lecture 10 [08/10, 3A-G11]: Free resolutions and fundamental theorem of homological algebra. Right exactness of tensor products, Tor and properties. Universal coefficient theorem for homology. References: To stick with [H] we can look at 3.A with some help from section 3.1, namely Lemma 3.1. A reference that presents things in our order is [HA2, 21, 22, 24], see also [HA2, 20] for recollections on tensor products.
• Lecture 11 [14/10, IMT Salle Pellos]: UCT applied to RPn. Cohomology and universal coefficient theorem for cohomology [H, 3.1] or [HA2, 27]. Cup product. [H, Beginning of 3.2]
• Lecture 12 [15/10, 3A-G11]: Cohomology ring of RP2. Higher homotopy groups [H, 4.1].
• Lecture 13 [03/11, 3A-G11]:
• Lecture 14 [05/11, 3A-G11]: I think we will have to move this class to a bit earlier on the same day. We will talk on monday november 3rd. The plan is to finish the material and do some exercises.

Suggested exercises

• Lectures 1+2: Exercise sheet 1. Hints, half-solutions and comments.
• Lectures 3+4: Exercise sheet 2. Hints, half-solutions and comments.
• Lectures 5+6: Exercise sheet 3. Hints, half-solutions and comments.
• Lectures 7+8: Exercise sheet 4
• Lecture 9+10: Exercise sheet 5
• Lectures 11+12: Exercise sheet 6

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.
• An elementary illustrated introduction to simplicial sets: Available for free at this link.
• A Concise Course in Algebraic Topology by Peter May. Available for free at this link.
• [HA2]: Lectures on Algebraic Topology. 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 3 hour exam, taking place on of november 25th at 14h00 in a room to be determined.