Recently I became very much intrigued by algebraic topology and am spending quite some time learning it. My reasons are three-fold:
- it's a beautiful theory;
- it gives geometric justification to (or perhaps rather an application of) many purely algebraic structures; and
- it has fascinating applications in quantum field theory and condensed matter theory.
Nevertheless, what I am familiar with currently are just basics: various homology and cohomology theories, homotopy theory and some standard applications (Brouwer, Borsuk-Ulam, etc., etc.). While these are of course interesting of and by themselves (and I expect spending a great amount of time on understanding all of this properly), I guess it is more or less understood for some fifty years now, so supposedly people work on topics far more advanced than this (or at the very least they use far more advanced tools to understand standard but hard problems).
So, I'd also like to know what the field is about from the modern perspective (some interesting problems and research topics, advanced tools, etc.) so that I can see a little where will the study of the subject lead me in the long run.
Sorry if the question is too broad but I am not sure where else to look (I've more or less browsed through all general articles on AT at wikipedia and tried to search MO too). I've heard few magic words like K-theory, sheaf cohomology, various spectral sequences, etc. but I don't understand these at all yet; more importantly my motivation to learn these things is lacking since I have no idea how or when these magic words are used (although I am pretty sure they are used a lot).
Best Answer
If you want a more modern perspective on algebraic topology, Peter May's "A concise course in algebraic topology" phrases a lot of things in terms of fiber/cofiber sequences, takes an axiomatic approach to (co)homology, and discusses the bare bones of some more advanced topics like K-theory. He has written a sequel with Kate Ponto, "More Concise Algebraic Topology: Localization, completion, and model categories" though it has not yet been published. It covers additional topics such as localizations, model categories, and spectral sequences.
Another nice introductory textbook might be tom Dieck's book, although I don't have a lot of familiarity with it.
If you want to learn more about spectral sequences and their applications (mostly to algebraic topology), you should look at McCleary's "A user's guide to spectral sequences".
Another thing you might be interested is spectra and the stable category, although I don't know a good reference. Something like the paper/book by EKMM is probably not a good introduction to the subject. I'm sure that someone on MO will have an opinion if you want to learn the material.
There are a ton of books on topics like K-theory, and I don't have any recommendations.
Characteristic classes are wonderful, and if you want more than May or tom Dieck include, the book by Milnor and Stasheff is a classic.
At the end of May's concise course, he has a list of books for further reading. Although the list is a few years old, it might prove a useful place to start, if these aren't quite what you are looking for.