Answers to this sort of question obey an uncertainty principle: The more precisely I outline a manageable "syllabus", the less likely it is to match your specific needs and interests. That said, here's a rough outline based on Hatcher's Algebraic Topology that can be adjusted as needed.
General advice: Most of Hatcher's sections begin with a paragraph or two regarding the usefulness of or intuition for the topic at hand. I strongly recommend revisiting these gems of wisdom as you move through the text; they offer a review of the central ideas and a sort of road map for the subject. Also, given the volume of new ideas, I emphasize the standard advice of skimming proofs during an initial read of each subsection and then going back to understand the details of the arguments. Finally (but first, really), read the preface! It's an instruction manual for how to read the book and understand its place in the broader literature.
Chapter 0: Some Underlying Geometric Notions. Read the whole thing at first, but don't expect yourself to memorize the details of every construction in one read. Since it sounds like you know the basics of homotopy of maps, homotopy equivalence, contractibility, et cetera, your goal here should be to make sure you're comfortable with CW complexes and their homotopy properties (e.g. homotopy equivalence under collapsing contractible subcomplexes, equivalence of homotopic attaching maps, CW pairs possessing the homotopy extension property).
Exercises: 1-6, 10, 17, 19
Chapter 1: The Fundamental Group. If you feel pretty good about fundamental groups, van Kampen's theorem, classification of covering spaces, just skim §1.1-1.3 and review the statements of theorems/propositions/corollaries and definitions typeset in bold. Read the examples and "Applications to Cell Complexes" in §1.2, though you can skim the more detailed discussion/examples of covering space actions on a first read. The appendices §1.A-1.B are fun and have plenty of applications but are not strictly necessary. (But do read the definition of a
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Best Answer
2 years ago
I want to start self studying algebraic topology and I am looking for guidance regarding the same. In the past I have made the mistake of trying to learn a mathematical subject by reading fat books cover to cover, which almost always never happened.
I think it would be a good idea to follow Munkres' Elements of Algebraic Topology but I am open to suggestions.
It would be great if someone can point out which sections I can leave out and which I most definitely cannot in the first reading. Reading each and every section would be a waste of time (according to my past experience).
I apologize if this post is inappropriate for MSE.
My Background: I have a decent background in Algebra (esp. Groups and Field Theory/ Galois Theory) and strong background in Linear Algebra. I am quite comfortable with general topology. Last semster I had a course on differential geometry and I am starting to get comfortable with vector bundles and their tensor products etc. As of now, I am a little bit scared of differential forms. I just touched upon algebraic topology from Munkres' Topology Part II. I have covered the following topics: Quotient Spaces, Covering Spaces, Fundamental Groups, Path and Homotopy Lifting Properties and saw some applications like the Brouwer's Fixed Point Theorem and Borsuk Ulam Theorem.
Thank you.