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I have started studying algebraic topology with the help of Armstrong(Basic), Massey, and Hatcher. If I plan to do research in algebraic topology in future:

  1. What else should I study after completing homology(basic), cohomology(basic) and homotopy theory(basic)?

  2. After completing Hatcher how far I would be (in terms of time and effort) from tackling a research problem?

    I have average background in Algebra and never studied Category theory in detail.However I feel comfortable working with algebra. I would like to work in those areas which require more algebraic machinery than any other area and which are more Geometric in flavour.

  3. Are there other areas to which I should switch over to like Geometric topology or algebraic geometry?

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This is best answered by someone who knows you personally. Preferably an academic advisor. –  Potato Jun 14 '12 at 9:29
    
@Potato I dont have anyone around here to answer this question. Most of the people I know are working in general topology. –  Kamran Jun 14 '12 at 9:33
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It is going to be extraordinary difficult to start doing original research in algebraic topology without the guidance of an advisor, so finding one should be your first priority. Without more specifics about you situation and background, this is really impossible to answer. –  Potato Jun 14 '12 at 9:37
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A researcher in algebraic topology could sometimes require A. Dold, lectures on algebraic geometry, because it uses very powerful techniques and covers very much in (co-)homology theory, but isn't easily read the first time — so less people like it. But if you learn how to read it, it should become a mighty tool. But be aware, of the fact that the usage will depend on what you are going to research in detail... –  Ben A. Jun 14 '12 at 11:09
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3 Answers 3

up vote 20 down vote accepted

Let me attempt to answer this question. I should mention that I am not a research algebraic topologist. In fact, I am a student of algebraic topology and I hope to one day become a researcher in the area. I am currently on the path toward this goal.

Let me begin by saying that you are definitely on the right track by reading Hatcher's textbook. I think that the most fundamental topics of algebraic topology are covered in Hatcher's textbook and a knowledge of these topics will be very useful to you as a research mathematician no matter in which area of mathematics you specialize. I will assume that you have completed Hatcher's book and you are interested in further topics in algebraic topology.

I think the next step in algebraic topology (assuming that you have studied chapter 4 of Hatcher's book as well on homotopy theory) is to study vector bundles, K-theory, and characteristic classes. I think there are many excellent textbooks on this subject.

My favorite book in K-theory is "K-theory" by Michael Atiyah although some people object because they feel that the proof of Bott periodicity in this book is not very intuitive but rather long and involved (and I agree). However, you may as well assume Bott periodicity on faith if you read this book as the techniques used in proving Bott periodicity are not used or mentioned elsewhere in the book (although minor exceptions may show this statement to be false). I think a very slick proof of Bott periodicity is discussed in the paper "Bott Periodicity via Simplicial Spaces" by Bruno Harris. I would recommend you to read this paper if you are interested in a proof of Bott periodicity.

Alternatively, you may wish to learn from Hatcher's textbook entitled "Vector Bundles and K-theory" (available free online from his webpage) or the textbook by Max Karoubi entitled "K-theory: An Introduction". Hatcher's book discusses the image of the J-homomorphism (in stable homotopy theory) which is an important an interesting application of K-theory. I don't think that this is discussed in Atiyah's textbook. Similarly, Hatcher has a more detailed description of the Hopf-invariant one problem than that of Atiyah's book. Thus a good plan would be to read Atiyah's textbook and supplement it with a reading of the Hopf-invariant one problem and the J-homomorphism in Hatcher's book. Alternatively, you could read Karoubi's book which is much lengthier than the two (combined) but is an excellent textbook as well.

If you learn vector bundles and K-theory very well, then you should also learn the theory of characteristic classes. I believe that this is discussed in some detail in Hatcher's book (the same one entitled "Vector Bundles and K-theory") and the most basic properties of characteristic classes are proved. However, a more detailed discussion of characteristic classes can be found in the book entitled "Characteristic Classes" by Milnor and Stasheff. I would recommend reading the latter book if you have time and wish to learn about characteristic classes fairly thoroughly. Otherwise, the minimal treatment of characteristic classes in Hatcher's book is also sufficient in the short-term.

A good topic to learn about at this stage is spectral sequences. Spectral sequences furnish an extremely useful and efficient computational tool in algebraic topology. I can't really recommend the good book on spectral sequences because there are many but you might wish to look at "A User's Guide to Spectral Sequenes" by John McCleary and Hatcher's book on spectral sequences (available free online on his webpage).

Finally, you should now learn homotopy theory in more depth. An excellent place to do this is "Stable Homotopy and Generalized Homology" by Frank Adams. Unfortunately, this is as far as I can advise you because this is as far as I have progressed in algebraic topology. I think once you finish the book "Stable Homotopy and Generalized Homology" by Frank Adams the next step could be to start reading research papers (which you have to do sooner or later). Of course, advice on reading research mathematics papers is long and involved so I won't go into details in this answer as we are discussing algebraic topology. But, the books I suggested should keep you busy at least in the short term.

I hope this helps!

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Do you beat him in chess :-)? –  M.B. Jun 14 '12 at 12:33
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@AmiteshDutta Thanx for taking pain in giving answer with so much details.which areas do you think are the most geometrically flavored and use algebra in bulk? –  Kamran Jun 14 '12 at 13:58
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I have to say that I think the proof of Bott Periodicity in Atiyah's book is wonderful. It is exactly the sort of proof one would hope exists, whether or not one really wants to read the details: go in, analyze the data, and see what you get! (More "magical" proofs like Harris' are also very nice but offer rather different insights.) It's worth noting that Atiyah's proof can be simplified a bit: Hatcher's notes make some simplifications, and Husemoller's book explains how to prove injectivity of the Bott map more abstractly, using Puppe sequences and the Chern character. –  Dan Ramras Jun 14 '12 at 18:37
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Nice answer. Regarding Atiyah's proof of Bott periodicity - see if you can grab a first edition of the text book. It is my understanding that this has a 'nicer' proof of Bott periodicity. –  Juan S Jun 14 '12 at 22:44
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@Kamran I think it would be best if you asked this as a separate question (and then I, or someone else, or perhaps many other people, can answer your question). I would recommend John Milnor's "Morse Theory" and John Milnor's "Lectures on the h-cobordism theorem" (in that order) as sources for differential topology. I would recommend Qing Liu's "Algebraic Geometry and Arithmetic Curves" as a source for algebraic geometry ... –  Amitesh Datta Jun 16 '12 at 7:39

Amitesh has already given an excellent answer. Continuing from Adams' "blue book" (which you should begin at chapter 3, and quit when you reach smash products!), I'd strongly recommend Switzer's "Homotopy and Homology" (or something like that). This "starts from the beginning", but both recasts the entire story in much more mature language and technology and gets quite far. (Skip his chapter on products, though, too.) And when you're done, I'd be happy to give more suggestions from there!

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@Aaron...Thanx for your answer.I will be in contact for further suggestions. –  Kamran Jun 16 '12 at 5:47

Amitesh Datta has already given a good list of next textbooks, which include topics that are widespread both in algebraic topology and in nearby fields.

Algebraic topology is a big subject. After getting past these texts, you have to make choices about the direction that you are interested in pursuing, because it's not feasible to pursue all of them. Most directions will require time investment before you know what research problems are available and feasible (this is what an advisor is for). Many of these directions will not have textbooks and you will need to make inroads into the literature.

This is not a bad thing. Steenrod supposedly said:

In your undergraduate studies the mathematics that you have read has been primarily in textbooks. But now you are ready to read original articles – articles are living mathematics, and textbooks are dead mathematics. You should read original articles, even if they’re harder and not so well written.

(The only source I can find online for this quote is here.)

The list of recommended papers for the Kan seminar at MIT is a collection of works that, because of their results or their methods, have been particularly influential. Some of the methods and language of these can be recast in more modern terms; doing so is good exercise. Some of them have outlooks that are surprisingly modern. Some of them lead to well-trod research topics, but some have spawned new subjects in their own right that are still active.

Mathematics is a human subject, and the distribution of some of the Kan seminar's list is particular to the culture of algebraic topology at MIT. However, it's a good starting point. Find something there that whets your appetite; go to MathSciNet and look up recent papers that make reference to them; go to the arXiv; find someone who can help you understand what the state of the art is or where to find it; good luck.

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