TITLE: What algebraic topology book to read after Hatcher's?
QUESTION [29 upvotes]: 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?

REPLY [20 votes]: 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warning though, I wouldn't go through in great detail the construction of smash product, etc, as a lot of this has been superceded now.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:

Haynes Miller's notes on cobordism
Some course notes from Mike Hopkins - but some of the 'stacky' stuff can probably wait
Some notes from Hal Sadofsky on formal group laws in Algebraic Topology


That should keep you busy for a while!