Algebraic Topology (S1 2024/2025)

Topics Covered

• Lecture 1: 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: Fundamental group of S¹ and applications (disc does not retract into boundary, Brouwer fixed point theorem), π₁(Sⁿ), van Kampen' Theorem and applications [H, 1.1 - 1.2].
• Lecture 3: (Co)products, pullbacks and pushouts in general and in the categories we care about [L,M]; CW complexes: properties and examples [H, Chapter 0]
• Lecture 4 : CW complexes and their π₁, everything is a π₁ [H, 1.2]. Homological algebra [HA]
• Lecture 5: More homological algebra (examples, homotopy) [HA]. Singular chains and homology [H, 2.1].
• Lecture 6: Homotopy invariance [H, 2.1], long exact sequence [H, 2.1] or [HA, 3].
• Lecture 7: Five-Lemma, Relative homology [H,2.1], Small chains [H,2.1].
• Lecture 8 : Excision, examples of computations of homologies, Mayer-Vietoris sequence [H,2.1] and [H,2.2]
• Lecture 9: Cellular homology and examples [H, 2.2]
• Lecture 10: 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: 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: Cohomology ring of RP2. Higher homotopy groups [H, 4.1].
• Lecture 13 : Relative homotopy groups and long exact sequence. Whitehead's Theorem [H,Thm 4.5]. Statement of relation between π_n and π_n [H, 2.A] and [H, 4.2] (Remark: we would need quite some baggage to prove this for n bigger than 1)
• Lecture 14 (U6-404) : November 6th New time: 15h45. Exercise class
• Exam: November 27th, 13h30-16h30. Rules for the exam to appear here.

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 (Minor modifications on exercises 2 and 8.2)
• Lecture 9: Exercise sheet 5
• Lectures 10+11: Exercise sheet 6
• Lectures 12+13: Exercise sheet 7

These exercises are for you to practise what we have learned in class. This is a selection I found helpful, based on the questions some of you asked me in class (or afterward). There is no evaluation based on these exercises. The final exam (taking place on november 27th) 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).

Prerequisites

Basic knowledge of point-set topology, group theory, and abstract algebra is strongly recommended. Attendance of the M1 course Topologie et Algèbre will be 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. (This is the classical reference. Below a freely available alternative.)
• [L]: Basic Category Theory by Tom Leinster. Available for free at this link.
• [HA]: This can be essentially any reference that you find by googling homological algebra, or the individual concepts on wikipedia. I'm suggesting 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. Available for free at this link.
• [HA2]: One of multiple good references for tensor products, Tor, Ext, which has Algebraic Topology in mind. Lectures on Algebraic Topology. Available for free at this link.

Comment on the references

Globaly, [H] is a good reference to follow, as for the moment all topics that we will be covering are contained in there and in roughly the same order. Notable exceptions are:

• Things from [H,Chapter 0] are introduced as needed.
• We choose to take the categorical point of view from early on.
• We work with homology over a general ring, instead of the integers.

More will be added here, as we make progress in our journey to the world of algebraic topology.