6
$\begingroup$

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.

$\endgroup$
  • 2
    $\begingroup$ Hatcher's algebraic topology is free online and is considered a good text by many. $\endgroup$ – James Jun 16 '15 at 10:01
  • $\begingroup$ Thanks. I am aware that Hatcher's book is widely recommended. What I am looking for is, if possible, someone can suggest a list of sections which should be read and which can skipped. $\endgroup$ – caffeinemachine Jun 16 '15 at 10:04
  • $\begingroup$ The best part about Hatcher is the exercises, you can have a look at them and see if any of them interest you. $\endgroup$ – James Jun 16 '15 at 10:10
  • 2
    $\begingroup$ To add, I have read Munkres and I didn't find it appealing. In my opinion, his exercises are weak, and he goes through a rigor-first approach. Munkres also doesn't talk about CW-complexes (it introduces 2-dimensional CW-complex in one of the chapters, but I don't recall having seen it anywhere else), so it's recommended you learn Hatcher's chapter 0. I think it won't be quite so much as a waste of time as you think to read chapter 1 and 2 too, although you have background in fundamental groups and covering spaces. $\endgroup$ – Balarka Sen Jun 16 '15 at 10:37
  • 4
    $\begingroup$ The bread and butter for a working knowledge of algebraic topology should include the fundamental group and (co)homology groups, with all the machinery that go along with these gadgets (van-Kampen, classification of covering spaces, Mayer-Vietoris, cup products, Kunneth formula, Poincare duality, universal coefficients). These comprise of most of the content in any advanced undergraduate course. After that, there are many directions you can go, closing in on graduate topics - (co)fibrations, higher homotopy groups, spectral sequences, classifying spaces, principal bundles... $\endgroup$ – Dan Rust Jun 16 '15 at 13:34
13
$\begingroup$

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 $\S$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 $\S$1.2, though you can skim the more detailed discussion/examples of covering space actions on a first read. The appendices $\S$1.A-1.B are fun and have plenty of applications but are not strictly necessary. (But do read the definition of a $K(G,1)$ and statement of Proposition 1B.9.)

Exercises: $\S$1.1 -- 5, 6, 8, 10, 13, 16; $\S$1.2 -- 4, 6, 8, 9, 10, 15, (+22 if you like knots); $\S$1.3 -- All great, so do what time permits.

Chapter 2: Homology. Almost nothing to skip in $\S$2.1. Many people skim the details of barycentric subdivision/Proposition 2.21 in the section on the Excision Theorem, but you definitely need to understand the statement and the way it is used to prove the main theorem. Lots of courses skip many of the examples in $\S$2.2. You should read about cellular homology, but your goal should be computational intuition and not mastery of the details, such as why it is equivalent to the other theories. Learn Mayer-Vietoris very well, homology with coefficients well, and get a feel for Euler characteristic. Some people glaze over $\S$2.3, but it helps you organize your thoughts about homology and provides an introduction to category theory -- something you need sooner or later. For $\S$2.A, at least internalize the central statement: $H_1(X)$ is the abelianization of $\pi_1(X,x_0)$. Many courses skip $\S$2.B-2.C.

Exercises: $\S$2.1 -- 4, 5, 7, 11, 13, 15, 16, 20, 22, 27, 29, 30; $\S$2.2 -- 4, 12, 14, 15, 31, 32, 41; ($\S$2.3 -- Try all, if you read the section.)

Chapter 3: Cohomology. In a first reading of $\S$3.1, you can skip the details of the discussion on pages 191-195 beginning after "Our goal is to show that the cohomology groups..." and ending before "Summarizing, we have established...". For $\S$3.2, some people only learn the statement of the Künneth formula during a first reading. Skip all but the first theorem in "Spaces with Polynomial Cohomology" if you'd like. As for $\S$3.3, there's much debate about how much detail a first reading about Poincaré duality should include. I suggest reading all of the introduction and "Orientations and Homology", plus all of "The Duality Theorem" up to (and including) the statement of Poincaré duality for closed manifolds. Also learn the results (but not necessarily the proofs) in "Other Forms of Duality". For $\S$3.A, just learn the major results and computational propositions/corollaries. Same, but less so, for $\S$3.B. Skip $\S$3.C-3.E and $\S$3.G-3.H for now, but maybe check out $\S$3.F for limits.

Exercises: $\S$3.1 -- 5, 6, 8, 13; $\S$3.2 -- 1, 3, 7; $\S$3.3 -- 2, 3, 5, 7-10, 16, 24, 30, 31, 32, 33

If you have time: Learn the basics of homotopy theory. Try to read $\S$4.1 (which is full of really useful ideas), plus some of the other fundamentals like the Hurewicz theorem, fibrations and fiber bundles, and the connection between singular cohomology and Eilenberg–MacLane spaces.

$\endgroup$
  • $\begingroup$ Thank you so much. This is exactly what I was looking for. I don't have any specific goals in mind since I do not even know what is there is algebraic topology except for the fact that the machinery can be used to prove really crazy things. I also know that I am going to use algebraic topology a lot since I am interested in Mechanical Linkages and Applied Topology. At any rate, I am very much interest in geometry so algebraic topology has to mastered well. But doing things without guidance was a mistake I didn't want to repeat. And thanks a bunch for even mentioning the exercises! $\endgroup$ – caffeinemachine Jun 16 '15 at 17:27
  • $\begingroup$ The answer doesn't mention some exercises which I find interesting, so here's a list: 0.16(this tells you $\Bbb RP^\infty \cong K(\Bbb Z/2, 1)$), 0.18(helps you visualize Hopf fibration). 1.2.22 is recommended (helps you compute knot groups), 1.3.6 (standard counterexample), 1.3.25 (relates to the galois short exact sequence for galois theory of fields), 1.3.28. 2.1.7, 2.1.8 (computes homology of lens spaces), 2.1.10 (relates to Hatcher's paragraph on bordism somewhere in 2.1). 2.2.11, 2.2.26, 2.2.36 and 2.2.29 (algorithm for computing homology of 3-manifolds, composed with Heegaard splitting) $\endgroup$ – Balarka Sen Jun 16 '15 at 19:19
  • $\begingroup$ The above list consists of the exercises I find interesting, not the only ones you just should do. I recommend doing all of them. Also, I recommend you to read the chapter on Van Kampen theorem thoroughly : I don't think Munkres works a lot through fundamental group of cell complexes. 1.3 is an excellent read. Also, make sure you understand the proof of excision theorem. Learn about cellular homology well, as it's a computational tool. But try to appreciate the theory too, as you'd find similar ideas used in spectral sequences later on as you'll learn more algebraic topology. $\endgroup$ – Balarka Sen Jun 16 '15 at 19:31
  • $\begingroup$ @BalarkaSen Thank you so much. $\endgroup$ – caffeinemachine Jun 17 '15 at 6:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.