The whole plethora of topology

Question

In his answer to a recent MO question, Johannes Ebert sketches the proof of a very nice result (implying that homotopy spheres are parallelizable) which, as he says, involves the whole plethora of topology of the 1950s: Bott periodicity, Hirzebruch signature formula and Adams's results on the Hopf invariant and the J-homomorphism...

What is the equivalent plethora of topology of the end of the XXth century?

Answer 1

Both John and Charles are right, there is a lot more topology now then there was then. New areas have popped up, older areas have expanded and forked even. I don't know about most of the different areas, but Charles and John haven't offered an answer yet, so here we go. Also, everything below is just what I have gleaned from conversations, personal reading and inference. That being said I will not include the "It seems to me..." etc. but you should certainly feel free to include it in your reading.

The material that Johannes referred to is a lot of hardcore algebraic topology from the 1950's. A lot of this stuff was concerned with proving theorems in geometry that could be attacked using homotopy theory, think vector fields on spheres and Hopf invariant one. These proofs were culminations of an immense amount of work. A lot of very hard computational work went into this. Specifically, a deep understanding of the structure of the Steenrod algebra and its cohomology. The Adams spectral sequence was invented around this time, and it organized a lot of the homotopical work into one frame, but not necessarily the geometry implied by the homotopy theory.

Jumping off from the Adams SS we have Novikov "taking Adams suggestion" and working with $MU$ instead of $H\mathbb{F}_2$ as a base for a Adams SS. I am not familiar with Novikov's work, but after Quillen proved his theorem relating $\pi_*MU$ to the Lazard ring a lot of cool things started happening (this is where number theory comes in!). Quillen also proved that $MU_p$ (localized at p) splits as a wedge of suspensions of $BP$. As everyone knows, you have to do homotopy theory one prime at a time now, so we should be working with the Adams-Novikov SS based on $BP$ (from know on some prime $p$ is fixed). Miller, Ravenel, and Wilson did a lot of hard work on this. I am not sure about the timing, but at some point Morava had some preprints floating around that were a huge influence on the work to come. $E_2$ of this ANSS is really hard to compute, so you need to (like May did) have a spectral sequence converging to it. This is called the Chromatic spectral sequence. Anyway, the upshot is that people started finding families of elements in the homotopy groups of spheres, the greek letter elements for example.

This at some point Ravenel saw the work of Bousfield on localization and wrote a paper on "Localization with respect to certain periodic homology theories." This paper ends with the Ravenel Conjectures which shaped a lot of modern homotopy theory. All but one of the conjectures were solved by Devinatz, Hopkins, and Smith (the unsolved conjecture is called the Telescope conjecture, which people believe to be false nowadays). These were solved in the early 90's. I think most people that understand any of this stuff understand a lot more than I do, so I won't say anymore about that.

After the solution to the Ravenel conjectures (at least historically) people started to make computations in the $K(n)$-local setting ($K(n)$ is like a graded field in homotopy theory, $\pi_*(K(n))=\mathbb{F}_p[v_n,v_n^{-1}]$). These theories are called Morava $K$-theories and they have the effect of isolating type $n$ phenomena (type 0 is what rational homotopy can see, type 1 is what $p$-local $K$ theory can see...). There are also Morava $E$-theories (also called Lubin-Tate theories because of their coefficients and denoted $E_n$ for chromatic level n) and Johnson-Wilson $E$-theories being used (denoted $E(n)$ for chromatic level n), they see more of the chromatic picture, in that $E_n$ should see all phenomena of degree $n$ and lower (I am not really clear on the difference between the two, but it seems fashionable to spend more time talking about the Morava $E$-theories these days, for good reasons that I do not know). It is a theorem that the $K(n)$ (or maybe $E_n$) localizations of a finite CW complex tell you what you want to know about the $p$-localization of a finite CW complex. So the program was initiated to understand the $p$-local sphere by first understand the its localizations with respect to $K(n)$ for each $n$.

The construction of these theories, prior to a good smash product on the category of spectra, were all a little ad hoc. If I have a formal group law over a ring $R$ then I get a map from $\phi: R \to pi_*MU$. Now the Landweber exact functor theorem will tell me when $MU_*(X)\otimes_{\phi} R$ is a cohomology theory. Then Brown representability gives me a spectrum. This is not such a good construction, because the spectrum is coming "out of the vacuum". But along with a good construction of the smash product we got a good construction of spectra related to $MU$. (There is also the Baas-Sullivan construction that is related to manifolds with singularity, but I know even less about this). Using this perspective though, about formal group laws, we can say that $\pi_*E_n$ is the ring that classifies the deformations of the universal height $n$ formal group law over $\mathbb{F}_p$ (this ring comes out of work of Lubin and Tate on formal group laws, hence the alternate name). This height $n$ formal group law has automorphisms and so we get an action of that automorphism group, called the Morava stabilizer group $\mathbb{S}_n$ on $\pi_*E(_n$. Let $\mathbb{G}$ be a maximal finite subgroup of $\mathbb{S}_n$, then one can lift that action to get an action the spectrum $E_n$. Hopkins and Miller were able to show that the the spectra involved were $A_{infty}$. Goerss and Hopkins were then able to improve this to get the spectra to be $E_{\infty}$ (I think this is also related to the action of the Morava stabilizer groups acting by ring maps, but I am no sure). If we take the homotopy fixed points of the above action we get what are called higher real $K$ theories (the name comes from the fact that $KO$ can be gotten from $KU$ by looking at the (homotopy) fixed points of $KU$ under the obvious $C_2$ action).

The above construction gives you some very interesting cohomology theories, but it is hard to understand what geometric implication they might have, and any higher order structure they might have. For example, $BP$ has only recently been shown to be $E_4$ by Basterra and Mandel. Also, Niles Johnson and Justin Noel have recently shown that the natural complex orientation of $BP$ can not be an $E_\infty$ map. Anyway, the point is that a more "geometric" construction is needed here, and this is where TMF and TAF enter the picture. And that is a little sliver of what one might call the modern plethora of topology.

Let me briefly clarify what I mean by geometric construction: it is not that we can relate the spectra above to geometry in the sense of manifolds or vector bundles (there is a construction of "$K$"-theory of two vector bundles that is $v_2$ periodic (chromatic level 2) but I do not think that this theory is complex orientable, or it has some other deficiency that keeps it from being an example of an "elliptic" cohomology theory). We do have the construction directly in the sense that it does not come from an abstract existence result. Tyler is certainly right when he says we do not know (we the community, maybe individuals have yet to post preprints) what representatives of elements of, for example, $TMF^*(X)$ look like.

Also, I feel like I should mention that Davis and Mahowald have a way of getting embedding theorems out of these stable homotopy theory computations. By embedding theorems I mean results concerning when you can embed $\mathbb{R}P^n$'s.

Here are some topics I did not mention: LS Category, Goodwillie Calculus or Functor Calculus in general, Rational Homotopy Theory, Equivariant homotopy theory, Embedding theory, Surgery, Models of the category of spectra with good smash products, THH, TC, and Algebraic K-theory.

Some people I did not make mention of that played a large role (I am sure I am forgetting some): Mark Behrens, Tyler Lawson, Mark Mahowald, Peter Landweber, Mike Hill, Jacob Lurie, Paul Goerss, Mark Hovey, and so many more.

I really do apologize if I have made any aweful errors, please let me know so I can fix them.