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I've currently finished chapter 2 of his book and done all the exercises of in chapter 0, 1 and 2. Was wondering when I finished reading this book what book do I read next in algebraic topology?

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Well what do you find interesting in Hatcher and want to learn more about? –  Carl Mar 7 '12 at 18:59
    
Homotopy of spheres. –  weylishere Mar 7 '12 at 19:13
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You've done all of the exercises!? I'm impressed! –  Fredrik Meyer Mar 7 '12 at 20:00
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May's Concise Course... is a nice follow-up / compliment to Hatcher's book. –  user641 Mar 8 '12 at 4:52
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4 Answers

After reading Hatcher's book, the next topics you would want to learn to do serious work in algebraic topology are as follows.

  1. Characteristic classes of vector bundles. Milnor's book on the subject is fantastic. Hatcher has notes on this on his webpage, but I think that Milnor's book is much, much better.
  2. The Serre spectral sequence (and maybe some of the other spectral sequences that show up in algebraic topology, like the Eilenberg-Moore spectral sequence). I'm a big fan here of the notes on spectral sequences on Hatcher's webpage.
  3. Topological K-Theory. Probably Atiyah's book is the best source, though I also like Karoubi's book and the notes on Hatcher's webpage.
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A friend is currently reading Atiyah's book. I haven't looked at it myself, but he constantly complains about poor explanations and poor notation. (Take this anecdote with a grain of salt, mind.) –  Neal Mar 7 '12 at 21:09
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@Neal : The typesetting is pretty bad, but I remember finding the explanations quite good and the notation unobjectionable (years ago when I read it). My one criticism is that Atiyah's "elementary" proof of Bott periodicity is long and unenlightening. I don't understand why most books on the subject insist on using it (probably to avoid prerequisites). There are many nice proofs of complex Bott periodicity -- the two that I recommend to graduate students are Bruno Harris's proof in "Bott periodicity via simplicial spaces" and (continued) –  Adam Smith Mar 7 '12 at 21:38
    
the proof (maybe due to Atiyah and Bott?) using Kuiper's theorem that can be found in the beautiful book "Topology and Analysis" by Booss and Bleecker. –  Adam Smith Mar 7 '12 at 21:39
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Milnor's book is called "Characteristic Classes". Stasheff is listed as a coauthor (I believe that he took the notes when Milnor taught a course on the subject in the late '50's). –  Adam Smith Mar 7 '12 at 22:12
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@Mathemagician1234 : Your name dropping is annoying and boorish. Many of us have met (and even written papers with) "famous" mathematicians -- you are not impressing anyone by constantly bringing up ones you have interacted with. It just sounds kind of sad and desperate. –  Adam Smith Mar 8 '12 at 4:03
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If you are interested in the homotopy groups of spheres, after you have finished reading the rest of Hatcher's book (especially chapter 4) you should try some of the following (this is a supplement to Adam Smith's list, which you should also read!)

  • You should try and learn some homological algebra at some stage (derived functors for example). You could try Rotman's Homological Algebra book

  • Stable Homotopy and Generalised Homology by Adams. This is a funny little book. It has three parts, and strangely enough, you should start in Part 3! The first few chapters in Part 3 give a nice introduction to generalised cohomology theories and spectra. A warning though, I wouldn't go through in great detail the construction of smash product, etc, as a lot of this has been superceded now. (This is also covered nicely in Chapter 2 of Hatcher's spectral sequnce book, if I recall correctly)

  • You can try and add to your knowledge of spectral sequences with A User's Guide to Spectral Sequence by McCleary. This is kind of the ultimate book on spectral sequences, but for me personally I don't find it particuarly useful to learn from, but it is an excellent reference

  • I can definitely recommend Mosher and Tangora's Cohomology Operations and Applications in Homotopy Theory. Excellent introduction to cohomology operations and the Steenrod algebra - and as an aside it is an excellent introduction to spectral sequences.

  • If you are still intrested in homotopy groups of spheres after all this you probably want to explore the connection between formal groups laws and generalised cohomology theories. This is not a particularly easy topic! This is covered in a number of places
    • Part 2 of the afermentioned book by Adams
    • Chapter 4 of Complex Cobordism and Stable Homotopy Groups of Spheres by Ravenel (you will also need Apendix 2 on formal group laws). Note that this is kind of the ultimate book on the Adams-Novikov spectral sequence and all things related, but it is a difficult read. This will also give you an introduction to the spectra MU and BP, etc
  • There are several good course notes on the internet:

That should keep you busy for a while!

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I want to second the recommendation of Haynes Miller's notes. They're great! –  Adam Smith Mar 7 '12 at 22:55
    
+1 for those terrific lecture notes.In fact,I've found in preparing for my Master's qualifying exams,that some of the very best sources on algebraic topology are lecture notes available over the Web for free.When I have time,I'll compile a list and post it here with corresponding links. –  Mathemagician1234 Mar 8 '12 at 3:45
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J.Peter May and Kate Ponto's More Concise Course In Algebraic Topology is a follow-up to May's introductory book containing cutting edge topics such as spectral sequences, model categories,basic homotopy theory,localization and completion of spaces and more. I haven't seen the whole book,but what I have seen looks terrific.

Homotopy theory is another natural place to go after learning basic algebraic topology. There's a wonderful new book that looks outstanding for this purpose: Jeffery Strom's Modern Classical Homotopy Theory. The book is beautifully written,very modern and geometric at the same time and it's written in the form of a directed set of exercises with no explicit proofs.It takes a lot of guts to write a problem course in algebraic topology and Strom's looks fabulous. It's definitely worth a look after Hatcher.

Also, you might want to check out the back appendix of May's A Concise Course In Topology -he gives many excellent recommendations for directions for further study.

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While Peter May's book does things faster and more abstractly, it doesn't really cover more ground than Hatcher's book (in fact, for many topics Hatcher's book covers more). There's no reason to read a second book on the same topics... –  Adam Smith Mar 7 '12 at 20:08
    
And Bott and Tu's book is nice for what it covers, but it's also not really a second course in algebraic topology. It just does the material from a first course in a different way (using de Rham cohomology). –  Adam Smith Mar 7 '12 at 20:15
    
@Adam I edited my post extensively. There's some debate over the "right" way to use Bott/Tu. A lot of people feel without some experience with the basics of algebraic topology,the De Rham cohomology approach isn't very useful. Also,Bott/Tu is considered rather specialized in it's choice of topics,so using it as a first course is going to be problematic for students planning to go on in topology. Regardless of all that-it's clearly a classic in the subject and a must read. –  Mathemagician1234 Mar 7 '12 at 21:00
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I removed the downvote; however, I am rather skeptical that you have seriously read any of the books you recommended. –  Adam Smith Mar 7 '12 at 21:43
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@Adam That's ok-your opinion means very little to me. –  Mathemagician1234 Mar 7 '12 at 23:07
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Lecture Notes in Algebraic Topology by Davis and Kirk was a great post-Hatcher book for me. It covers characteristic classes, obstruction theory, spectral sequences, fiber bundles, etc.

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+1 for a very nice book with a geometric slant. –  Mathemagician1234 Mar 7 '12 at 21:01
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I have heard they are preparing a second edition which will correct typos, fix unclear explanations, and add more material and exercises. You might consider waiting for it. –  Neal Mar 7 '12 at 21:08
    
@Neal: Where have you heard this? Are you sure? And when might it be coming? –  Leon Lampret Aug 2 '13 at 13:12
    
@LeonLampret In the eponymous class, they have used drafts of the second edition. I do not know the time frame they're working on. –  Neal Aug 3 '13 at 15:08
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