Advanced algebraic topology topics overview

Question

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).

Answer 1

By no means do I speak for algebraic topologists, but as a geometric topologist who wishes to formulate problems in some form of algebra.

(1) One recent trend in low dimensional topology has been to translate diagrammatic manipulations into algebraic equations. The conversion of the braid relation $s_1s_2s_1=s_2s_1s_2$ into the Yang-baxtger relation is one example. More generally, the category of tangles is a braided monoidal category. Invariants of knots can be found by finding functors to other braided monoidal categories.

In a tortuous route, from Jones, to bracket, to Khovanov, through pre-sheaves, Turner has related knot invariants to invariants derived from stable homotopy. His idea illustrates many trends in the application of algebraic topology to low dimensional questions.

(2) Again in the low dimensional realm, I would consider problems that are concerned with the the computation of quandle homology to be very important. But the desire for such computation is quite personal. Quandle homology is intimately related to knotting. The most recent works on this by Clauwens and independently by Nosaka represent great use of the existing machinery to study geometric problems. Both works rely heavily on the ground breaking ideas of Fenn, Rourke, and Sanderson. Still algebraic techniques such as spectral sequences and/or projective resolutions are far from being used uniformly.

(3) To expand upon item (1) a bit further, I think that there is much to be done in understanding an interpreting diagrammatic descriptions of knots, manifolds, embeddings and immersions. In particular, the idea of turning diagrammatic equivalences into algebra is extremely fruitful. On the other side, the ideas of categorification from the point of view of Khovanov's work lead to diagrammatic algebra. This new algebra is not algebraic topology as it was practiced in the 20th century, but a new type of algebraic geometry. Still it will be informed by the constructions of loop spaces, classifying spaces, suspensions, and fibrations.

Answer 2

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.

Answer 3

This an extended comment inspired by Aaron's excellent answer.

First, as far as models of spectra are concerned, I would recommend you check out the articles by John Greenlees on the topic. He wrote an article for a summer school that I found very helpful.

There is something called the Kan seminar at MIT, they have in it a reading list of papers, you could start there as well. They pick "easy" to read articles that they have students present.

A lot of early algebraic topology was interested in understanding different geometric objects through their invariants. Now the best invariant is homotopy, but we can not even compute that on the easiest example $S^n$. This is one of the biggest issues, and a lot of brilliant people are working on it.

K-theory is another cohomology theory that is really cool, it can tell you a lot about the geometry of the space you care about. The elements of $K^0(X)$ are virtual vector bundles, so every actual bundle gives you some element of K-theory. This is how the geometry starts to show up.

Sheaf cohomology I am super unfamiliar with. It is of a bit of a different flavor than the cohomology you might be familiar with, it keeps track of a lot of local information. There are a lot of other questions around where people have been asking about them.

Spectral Sequences are tools for computing things. Computing really hard things, usually. They are mostly a tool for getting hard to reach information.

It seems to me that K-theory might be something you would be most interested in given your background. You can;t really beat May's list of references though, and the book is free.

Answer 4

The book which seems to me would best suit your needs is Switzer's Algebraic Topology- Homology and Homotopy, which gives very nice motivation, and leads right up to the 1970's state of the art (cobordisms, Steenrod algebra,...)

Then, the next book I would look at would recommend would be Rudyak's On Thom Spectra, Orientability, and Cobordism. Despite its fearsome title, this book seems to me like a really good volume 2 to Switzer, explaining clearly basic things like the relationship between homology and cobordism in low dimensions, which one doesn't find in other textbooks. Because TQFT is a functor from a bordism category, and especially in light of ETQFT work such as Hopkins-Lurie which is heavily algebraic-topological, my opinion is that these two books provide a bee-line the front lines of what you might in the future be interested in.