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# NUS Math Module Review: MA5209 Algebraic Topology

Assessment Structure:

30% Homework (4 sets total)

20% Presentation

50% Final (Online, Open Book, Open Internet)

This is a follow-up of MA4266 Topology, obviously in the direction of algebraic topology. The content includes fundamentals of algebraic topology (e.g. homotopy and CW complexes), fundamental groups (including van Kampen theorem and covering spaces), and homology (including simplicial homology, singular homology, and some basic computation techniques). The textbook of reference is Algebraic Topology, Allen Hatcher. The topics above are basically whole of chapter 0, whole of chapter 1, and chapter 2.1.

All lectures were held via zoom (and were of course recorded). Lecture attendance was not compulsory. Like most graduate modules, this module has no tutorial classes. Prof. Han’s lectures mostly follow the textbook (skipping some examples in the middle), inserting some additional explanations in the middle where he deems necessary. He also inserted a few lectures on category theory in around week 3-4.

Algebraic topology can be a rather intuitive subject, but a lot of effort is needed to make it happen (unfortunately I did not put in the required effort to make it happen). Although personally weak at algebra, I did not find the algebra part of this module too disastrous (compared to say MA5211 Lie Theory), as they mostly stem from the basic/intuitive part of group theory (e.g. very basic representation theory/generators, free groups etc). The most difficult part of this module, however, is applying the concepts taught to solve the exercises. There were many instances which I thought I understood the theorems (and even the proofs) well, but I was completely lost when I turned to the exercises. This is worsened by the fact that Hatcher’s exercises are known to be very difficult. Another issue with algebraic topology is that it is not rigorously taught, with many “proofs” provided being just proof by pictures (e.g. the “proof” that suspension of S^n is S^{n+1} is usually provided as follows: geometrically clear for the case of n = 1 and n = 2, and “therefore” true for all n). Students must be prepared to either 1) black box many things and accept the intuition or 2) spend lots of time making everything rigorous. Personally, I suggest the first option.

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