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I hope here is the best place to ask this, I will begin my master degree very soon, I've already attended the regular undergraduate courses included Real Analysis, Analysis on manifolds, Abstract Algebra, Field Theory, point-set topology, Algebraic Topology, etc... I like very much algebraic topology and I found it really beautiful, I would like to know which areas of algebraic topology are the most interesting to begin to work with and which books I can study with my background in order to get the prerequisites to begin to study this subject.

I want as soon as possible has a "taste" of a current research field in algebraic topology, and I know that an algebraic topologist can give me a "shortest way" while I attend the regular courses of my master degree.

Thank you

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closed as off topic by Andy Putman, Dan Petersen, Bill Johnson, Felipe Voloch, Steven Sam Dec 1 '12 at 20:34

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Algebraic Topology is a huge field, and it's virtually impossible I think to say "which areas of algebraic topology are the most interesting to begin to work with" -- if you are looking toward an eventual PhD, you will probably have to embark on an apprenticeship stage where you learn many basic tools and calculations, which can become quite elaborate. But you could start by reading May's A Concise Course in Algebraic Topology, or Allen Hatcher's Algebraic Topology: math.cornell.edu/~hatcher/AT/ATpage.html But this site is mainly for specific, highly focused research questions. –  Todd Trimble Dec 1 '12 at 17:39
    
I think it is honest and reasonable question, but on mathoverflow some part of members are quite strict to "soft-questions", many of them get closed, some does not. Let me give some examples of similar spirit questions which were not get closed, may be you can use them in future. There are several "Learning roadmap" questions: mathoverflow.net/questions/2755/… mathoverflow.net/questions/1291/… and many more - search on "roadmap", BUT beware of closed : mathoverflow.net/questions/40076 –  Alexander Chervov Dec 2 '12 at 13:12
    
continued... There are also "recomend text-book" questions: mathoverflow.net/questions/67736/… mathoverflow.net/questions/2446/… mathoverflow.net/questions/7834/… etc, search on "text-book"... In particular see this on algebraic topology mathoverflow.net/questions/51066/… , mathoverflow.net/questions/18041/… –  Alexander Chervov Dec 2 '12 at 13:22
    
continued... your question contains several subquestions, in particular "would like to know which areas of algebraic topology are the most interesting to begin to work with " I am afraid this is mostly the reason why the question has been closed... To my taste it is reasonable, but there can be other opinions, people may say that "interesting" is "subjective and argumentative" and close it. At the moment I cannot give advise how to reformulate it... –  Alexander Chervov Dec 2 '12 at 13:28
    
PS concerning books I did not read carefully Bott, Tu book, but from browsing it and friend's opinions, it is quite good. There are some books in Russian like A.S. Mishenko "Vector bundles and K-theory" which is quite not bad, but probably this is useless to mention. –  Alexander Chervov Dec 2 '12 at 13:32

2 Answers 2

If you want to learn about algebraic topology, you can begin by very classical readings. When I was a Ph-D student, I first read Milnor Stasheff's book on "Characteristic classes", here you will learn a lot of differential and algebraic topology. There are so many good books to read, J.-F. Adams "Infinite loop spaces" or his blue book on "stable homotopy and generalised homologies", J. Milnor on Morse theory.

I highly recommand Andrew Ranicki's homepage where you will find a lot of cool stuff about algebraic and geometric surgery, PL-topology, exotic spheres. Jacob Lurie also has some very good notes of his courses on his homepage. Dan Freed is giving a course on the cobordism and his notes are very nice, and you will find plenty of references here. You can also look at H. Miller notes "Notes on cobordism" and "Vector fields on spheres" (just google it). And J. P. May has also a list of very good books on his homepage.

And overall, read classical papers by Adams, Pontryagin, Quillen, Serre, Sullivan, Thom...John Francis has a list of classical papers for the Kan seminar on his homepage.

I am sure my list is too long and I have forgotten plenty of good references (homepages, notes of courses and books).

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I've already studied the whole Lee's book Introduction of topological manifolds and some books of simplicial homology. Do you think I can understand the Milnor Stasheff's book, for example? Thank you for your answer. –  user26832 Dec 1 '12 at 18:17
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My recommendation is to go to a math library and glance through the books and papers that are being recommended to you. See which ones seem readable to you. Try one. If it works, great. Otherwise, try another one. You'll get better at figuring out what works and what doesn't. Any book with Milnor as an author or co-author is worth considering. Another beautiful book is Bott-Tu's Differential Forms in Algebraic Topology. –  Deane Yang Dec 1 '12 at 18:31
    
If you know simplicial homology, cohomology and some basics in differential manifolds, Milnor Stasheff is quite readable. I completely agree with Deane Yang's comment, Bott-Tu is very worth reading. –  David C Dec 1 '12 at 19:19
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I'd second Bott-Tu's differential forms –  Mozibur Ullah Dec 1 '12 at 19:26

For your first research problem, I recommend that you find an adviser in your department. If there is no algebraic topologist in your department, find some other adviser, and ask to suggest an interesting problem.

It is very unlikely that, as a master student, you will be able to find and solve a reasonable research problem yourself, without a help from an experienced adviser. In this site, people can give you only reading recommendations, and this is probably not enough to begin your own research. But of course, there were rare exceptions in history when self-taught mathematicians did good research.

Here is an outstanding problem in algebraic topology on this site:

fedja (mathoverflow.net/users/1131), Two commuting mappings in the disk, Two commuting mappings in the disk (version: 2009-11-25)

To understand the statement of the problem, little knowledge is required. What does one need to learn to solve this problem, nobody knows:-)

I wish you luck.

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in fact, I'm yet an undergraduate student, I will begin my master classes in March. Thank you for your answer, I'm going to see this problem, thank you very much for your answer. –  user26832 Dec 1 '12 at 19:13

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