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]:
- Lecture 3 [08/09, 3A-G11]:
- Lecture 4 [10/09, 3A-G11]:
- Lecture 5 [15/09, ??]:
- Lecture 6 [17/09, 3A-G11]
- Lecture 7 [29/09, 3A-G11]:
- Lecture 8 :
- Lecture 9:
- Lecture 10:
- Lecture 11:
- Lecture 12:
- Lecture 13:
- Lecture 14: Exercise class
Suggested exercises
- Lectures 1+2: Exercise sheet 1.
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.
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.
- 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.