Roadmap to Hill-Hopkins-Ravenel

Question

How does one go from an understanding of basic algebraic topology (on the level of Allen Hatcher's Algebraic Topology and J.P. May's A Concise Course in Algebraic Topology) to understanding the paper of Hill, Hopkins, and Ravenel on the Kervaire invariant problem?

I have read that some understanding of chromatic homotopy theory and equivariant homotopy theory (neither of which I am familiar with, aside from their basic idea) is required, but it seems that there are still quite a few steps from basic algebraic topology to these two subjects and from these two subjects to the Hill-Hopkins-Ravenel paper.

I do have some idea of a few topics beyond what is discussed in the books of Hatcher and May, such as model categories on the level of Dwyer and Spalinski's Homotopy Theories and Model Categories, as well as some idea of spectral sequences (both of which are required reading for the paper, I believe), but it would be nice if we could come up with some sort of roadmap (with an ordered list of subjects, and even better if we had references) from the Hatcher and May books to the Hill-Hopkins-Ravenel paper. For that matter, I don't even know how spectral sequences and model categories fit in, so it would be nice I guess if there's also an explanation as to how each of these prerequisite subjects fit into the paper.

I'm looking for something like the very nice answer to this query regarding the Norm Residue Isomorphism theorem.

Answer 1

There is one major topic that is missing from your list: spectra. Spectra (and $E_\infty$-ring spectra) are the basics of modern stable homotopy theory and are not treated, if not very cursorily, in the references you've looked at. Seriously, spectra are the bread and butter of a homotopy theorist nowadays and their absence from the standard references is as embarrassing as the absence of schemes would be from the standard material for algebraic geometry.

That said, let me try to put together a list of references that should help you getting started.

• The first place to look at is Adams' amazing Stable homotopy and generalized cohomology. This is a collection of lecture notes held at the university of Chicago and, famously, has to be read in reverse order (first part III, then part II and optionally part I) since they arranged the lectures chronologically rather than logically. Part III will give you a basic understanding of spectra, Part II will introduce the very basics of chromatic homotopy theory. Be warned that the treatment is very old (in fact if I'm not mistaken it is the first published construction of the stable homotopy category) and in particular the construction of the smash product of spectra is very lacking from a modern perspective, so you could skip it and just take the basic properties. In addition, the treatment of localizations has a couple of imprecisions, I suggest you complement that section with Bousfield's The localization of spectra with respect to homology.

• To get a more modern perspective, you cannot go much wrong with Schwede's Symmetric spectra. This is a modern treatment of a different (and better) model for spectra, which has the advantage of introducing you to $E_\infty$-ring spectra too, one of the basic structures that play a very important role in Hill-Hopkins-Ravenel (and all the rest of modern homotopy theory).

• Next you should get some familiarity with equivariant homotopy theory. A nice starting point is Adams' Prerequisites (on equivariant homotopy theory) for Carlsson's lecture. Again it is a bit old fashioned but having read it helps hitting the ground running.

• Since we're talking about equivariant homotopy theory, how could I not mention Schwede's Lectures on stable equivariant homotopy theory? The model he uses for $G$-spectra is slightly different from the one in Hill-Hopkins-Ravenel, but again having read it will certainly help.

• Hill-Hopkins-Ravenel's paper itself has a nice self-contained introduction to stable homotopy theory, that I strongly encourage you to read. Maybe skip the proof that the norm is homotopically meaningful in appendix B (it is very technical and while it has some very interesting ideas the only thing you really need is the statement of the theorem).

• Chromatic homotopy theory is surprisingly less present in the actual solution of the Kervaire invariant one problem (you basically need only to know about the Adams-Novikov spectral sequence and the statement of a few highly nontrivial computations), but if you want to learn it (you should if you're interested in this circle of ideas), I can recommend Ravenel's Nilpotence and periodicity in stable homotopy theory and Lurie's lecture notes on the subject. If you are interested on how to do those computation that are needed for the HHR paper, your best friend is Ravenel's Complex cobordism and stable homotopy groups of spheres (credits to Lennart Meier for prompting me to add this). Also reading Ravenel's previous paper on the odd Kervaire invariant problem is helpful for understanding how the main HHR paper connects with chromatic homotopy theory.

• Since you mentioned spectral sequences, I think you will generally pick them up as you work through the above material. However a paper that really helped me with them is Boardman's Conditionally convergent spectral sequences. Highly recommended.

I am sure my list has some glaring omissions, but this should at least get you started (and keep you busy for a while I suspect). Good luck and feel free to hang around the homotopy theory chatroom if you have questions!

Answer 2

Hill, Hopkins, and Ravenel have recently published a new book designed to teach readers (with a background like the OP's) everything they need to know to understand the proof. It goes carefully through spectra, equivariant orthogonal spectra, model categories, spectral sequences, etc. The last chapter is the proof. The authors explicitly write that such a book was needed because the prerequisite theory did not exist when they first made the key calculation that the proof hinges on. They point out that this book is self-contained and the goal was to have everything in one place for readers to follow.