What algebraic topology book to read after Hatcher's?
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?
After reading Hatcher's book, the next topics you would want to learn to do serious work in algebraic topology are as follows.
- 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.
- 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.
- 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.
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warning though, I wouldn't go through in great detail the construction of smash product, etc, as a lot of this has been superceded now.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!
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.