Has anyone seen a nice map of multiplicative cohomology theories?

I have seen lots of descriptions of this map in the literature but never seen it nicely drawn anywhere.

I could try to do it myself but I really lack expertise, hence am afraid to miss something or do it wrong.

Let me just provide some glimpses, and maybe somebody can nicely tie them together.

At the "initial end" there is the stable (co)homotopy, corresponding to the sphere spectrum.

At the "terminal end" there is the rational cohomology (maybe extending further to real, complex, etc.)

From that terminal end, chains of complex oriented theories emanate, one for each prime; the place in the chain corresponds to the height of the formal group attached. Now here I am already uncertain what to place at each spot - Morava $K$-theories? Or $E$-theories? At the limit of each chain there is something, and again I am not sure whether it is $BP$ or cohomology with coefficients in the prime field.

Next, there is complex cobordism mapping to all of those (reflecting the fact that the complex orientation means a $MU$-algebra structure). But all this up to now only happens in the halfplane. There are now some Galois group-like actions on each of these, with the homotopy fixed point spectra jumping out of the plane and giving things like $KO$ and $TMF$ towards the terminal end and $MSpin$, $MSU$, $MSp$, $MString$, etc. above $MU$. Here I have vague feeling that moving up from $MU$ is closely related to moving in the plane from the terminal end (as $MString$, which is sort of "two steps upwards" from $MU$, corresponds to elliptic cohomologies which are "two steps to the left" from $H\mathbb Q$) but I know nothing precise about this connection.

As you see my picture is quite vague and uncertain. For example, I have no idea where to place things like $H\mathbb Z$ and what is in the huge blind spot between the sphere and $MU$. From the little I was able to understand from the work of Devinatz-Hopkins-Smith, $MU$ is something like homotopy quotient of the sphere by the nilradical. Is it correct? If so, things between the sphere and $MU$ must display some "infinitesimal" variations. Is there anything right after the sphere? Also, can there be something above the sphere?

How does connectivity-non-connectivity business and chromatic features enter the picture? What place do "non-affine" phenomena related to algebroids, etc. have?

There are also some maps, like assigning to a vector bundle the corresponding sphere bundle, which seem to go backwards, and I cannot really fit them anywhere.

Have I missed something essential? Or all this is just rubbish? Can anyone help with the map, or give a nice reference?

• I cannot figure out what is your question. Can you focus it a little bit? I assume you are aware of the important role the moduli stack of formal groups plays in organizing all this. – Denis Nardin Jul 6 '17 at 14:37
• @DenisNardin Just that - aware. I honestly don't know how does it affect the picture, except that of course all these "directions" in the map are not unique, there are whole families of maps between some of these spots on the map (in other words, it is a category). Still, I just want to know whether there is sense in some sort of map that might picture this category. Roughly like the diagram of extensions of a field. There are automorphisms at each vertex, but otherwise it is much like a poset, is not it? This is what I want here - a sketch, a picture simplified as much as possible. – მამუკა ჯიბლაძე Jul 6 '17 at 16:02
• There are spectra T(n) (or X(n) over integers) between the sphere and MU. As to HZ, you can consider it as BP<0> (and HZ/p as BP<-1> – user43326 Jul 6 '17 at 16:17
• @user43326 This is exactly the precious "secret" information that I am up to. I'd love to see it all in an answer – მამუკა ჯიბლაძე Jul 6 '17 at 19:46
• This isn't an answer to your entire question but Paul Balmer has a nice picture of the (tensor triangulated) spectrum of the stable homotopy category on page 16 of this article on his webpage: math.ucla.edu/~balmer/Pubfile/TTG.pdf (this is essentially just a nice way of packaging the contents of the nilpotence theorem) – James Cameron Jul 6 '17 at 21:12

I'm not sure I understand what "the" map is here, but I'll attempt to answer the questions that were asked in the body of the question. Sorry if I'm just saying things that you already know. $\newcommand{\Sp}{\mathrm{Sp}}\newcommand{\Mfg}{\mathscr{M}_{\textbf{fg}}}\newcommand{\QCoh}{\mathrm{QCoh}}\newcommand{\Eoo}{\mathbf{E}_\infty}$

Quillen's theorem says that the Lazard ring $L$ (which classifies formal group laws over rings, so that a fgl over a ring $R$ is a ring map $L\to R$) is canonically isomorphic to $MU_\ast$ via the universal complex orientation $MU\to MU$. The key idea driving chromatic homotopy theory is that there's a functor $\Sp \to \QCoh(\Mfg)$, given by sending a spectrum $X$ to its $MU$-homology, which is naturally a $(MU_\ast, MU_\ast MU)$-comodule. The stack $\mathscr{M}_{(MU_\ast, MU_\ast MU)}$ associated to the Hopf algebroid $(MU_\ast, MU_\ast MU)$ is exactly $\Mfg$. Now, if $(A,\Gamma)$ is a Hopf algebroid, then $\QCoh(\mathscr{M}_{(A,\Gamma)})$ is exactly the category of $(A,\Gamma)$-comodules. All of this tells us that the $MU$-homology of a spectrum is a quasicoherent sheaf over $\Mfg$.

Chromotopists have adopted the philosophy that this functor is a rather good approximation of $\Sp$. Morava $K$-theories and $E$-theories come from this philosophy. The main tool utilized here is the Landweber exact functor theorem, which can be phrased as follows: if $\text{Spec }R\to \Mfg$ is a flat map, then the functor $X\mapsto MU_\ast(X)\otimes_{MU_\ast} R$ is a cohomology theory. This is reasonable, since for that functor to be a cohomology theory, we don't need $\mathrm{Tor}_{MU_\ast}(R,N)$ to vanish for every $MU_\ast$-module $N$ --- we just need it to vanish for $(MU_\ast,MU_\ast MU)$-comodules.

A theorem of Lazard's says that over an algebraically closed field $k$ of characteristic $p$ (for some prime $p$ that'll be fixed forever), there is a unique (up to isomorphism) formal group law of height $n$ for each $n$. People call (a choice of) such a formal group law the Honda formal group law of height $n$. At height $1$, the multiplicative formal group law $x+y+xy$ provides an example. (Over a field of characteristic $0$, everything is isomorphic to the additive formal group law; this is what the logarithm does). In particular, there's a unique geometric point of $\Mfg$ (over $k$) for each $n$.

We'd like to use the Landweber exact functor theorem to produce a cohomology theory from this geometric point (corresponding to the integer $n$, say) --- but the inclusion of a geometric point into something is rarely ever flat. Instead, we can look at the infinitesimal neighborhood of this point, and consider its inclusion into $\Mfg$. The structure of this infinitesimal neighborhood was determined by Lubin and Tate: it is (noncanonically) isomorphic to $\text{Spf }W(k)[[u_1,\cdots,u_{n-1}]]$. The ring $W(k)[[u_1,\cdots,u_{n-1}]]$ is complete local, with maximal ideal $\mathfrak{m}$ generated by the regular sequence $p, u_1, \cdots, u_{n-1}$. The map $\text{Spf }W(k)[[u_1,\cdots,u_{n-1}]]\to \Mfg$ satisfies the hypotheses of Landweber's theorem, providing us with a spectrum $E_n$, called Morava $E$-theory, with $\pi_\ast E_n \simeq W(k)[[u_1,\cdots,u_{n-1}]][\beta^{\pm 1}]$, where $\beta$ is a class living in degree $2$. For instance, when $n=1$, Morava $E$-theory is precisely $p$-adic complex $K$-theory $KU^\wedge_p$.

A priori, there's no reason for $E$-theory to be a multiplicative cohomology theory (i.e., an $\Eoo$-ring spectrum). But Goerss, Hopkins, and Miller proved with what's known as Goerss-Hopkins obstruction theory (I livetexed notes from this year's Talbot workshop here, which was on obstruction theory, but you should check the Talbot website for the official and edited notes) that $E_n$ really is an $\Eoo$-ring spectrum! (It seems appropriate to remark here that Lurie has recently given an alternative moduli-theoretic proof of this result; see here.) They also proved something more: if $\mathbf{G}_n$ denotes the profinite group of automorphisms of the geometric point, then there is a lift of the action of $\mathbf{G}_n$ to an action on $E$-theory via $\Eoo$-ring maps. Moreover, $\mathrm{Aut}(E_n) \simeq \mathbf{G}_n$. (For instance, at height $1$, the group $\mathbf{G}_1 \simeq \mathbf{Z}_p^\times$, and the action of $\mathbf{G}_1$ on $E_1 = KU^\wedge_p$ is given by the Adams operations.)

We can now realize the geometric point itself, by quotienting out the ideal $\mathfrak{m}$. This is a general procedure that you can do in homotopy theory: if $R$ is a ring spectrum, and $I\subseteq \pi_\ast R$ is an ideal generated by a regular sequence, you can form the quotient $R/I$ (by taking iterated cofibers). But if $R$ is an $\Eoo$-ring, there's no guarantee that $R/I$ will also be an $\Eoo$-ring: this is true with Morava $E$-theory and the ideal $\mathfrak{m}$. The quotient $E_n/\mathfrak{m}$ is denoted $K(n)$, and is called Morava $K$-theory. (For instance, when $n=1$, Morava $K$-theory is essentially $K$-theory modulo $p$.) The spectrum $K(n)$ is not an $\Eoo$-ring --- it is only an $A_\infty$-ring, i.e., an $\mathbf{E}_1$-ring spectrum. Note, also, that $K(n)$ isn't complex-oriented. I should mention here that I'm really talking about the 2-periodic versions of all these cohomology theories, but this'll suffice for now.

Why do chromotopists care, though? For this, we need to embark on a brief detour. The moduli stack $\Mfg$ admits a filtration by height. If $\Mfg^{\geq n}$ denotes the moduli stack parametrizing formal groups of height at least $n$, we have an exhaustive filtration of closed substacks $$\cdots\subset \Mfg^{\geq 2}\subset \Mfg^{\geq 1}\subset \Mfg.$$ Note that the complement of each of these inclusions is open, hence flat. It follows from the Landweber exact functor theorem that there's a spectrum corresponding to $\Mfg^{<n}\hookrightarrow \Mfg$. This spectrum turns out to be intimately related to Morava $E$-theory (for instance, they have the same Bousfield class).

It turns out that we can replicate this filtration in the category of spectra via the functor $\Sp\to \QCoh(\Mfg)$ described above. This is the content of the Ravenel conjectures. Let's write $L_n X$ for the Bousfield localization (I wrote another answer here that might be useful) of $X$ with respect to $E$-theory, and $L_{K(n)} X$ for the Bousfield localization of $X$ with respect to Morava $K$-theory. When you work in the $K(n)$-local stable homotopy category, the action of $\mathbf{G}_n$ on $E$-theory becomes a continuous action.

There are four remarkable theorems relating the structure of the stable homotopy category to $\Mfg$.

• Chromatic convergence: Let $X$ be a finite $p$-local spectrum. Then $X$ is the (homotopy) limit of its chromatic tower $$\cdots\to L_2 X\to L_1 X\to L_0 X.$$

• The thick subcategory theorem: There's an exhaustive filtration of "thick subcategories" (i.e., a subcategory that's closed under retracts, finite limits, and finite colimits) $$\cdots\subset \mathscr{C}_2\subset \mathscr{C}_1\subset \mathscr{C}_0 = \Sp^\omega,$$ such that any thick subcategory of the category of spectra is one of the $\mathscr{C}_k$. Moreover, each of the subcategories $\mathscr{C}_n$ is defined to contain those spectra for which the $K(m)$-homology is zero for $m>n$. Note the similarity to the height filtration! (The similarity is not unexpected, since a spectrum is in $\mathscr{C}_k$ when its associated sheaf is supported on $\Mfg^{\geq k}$.)

• Chromatic fracture: There's a (homotopy) pullback square $$\require{AMScd} \begin{CD} L_n X @>>> L_{K(n)}X \\ @VVV @VVV\\ L_{n-1} X @>>> L_{n-1}L_{K(n)}X. \end{CD}$$

• The Devinatz-Hopkins fixed points theorem: the continuous homotopy fixed points $E_n^{h\mathbf{G}_n}$ of the $\mathbf{G}_n$-action on $E_n$ is equivalent to $L_{K(n)} S$. This gives rise to a homotopy fixed point spectral sequence (sometimes called the Morava spectral sequence) $$E_2^{s,t} = H^s_c(\mathbf{G}_n,\pi_t E_n) \Rightarrow \pi_{t-s} L_{K(n)} S.$$

Combining all this, we see that the first step in computing $\pi_\ast S$ would be to compute $\pi_\ast L_{K(n)} S$, which'd follow from the Morava spectral sequence. It turns out that this is exceedingly hard, but (as usual) height $1$ is manageable. See Henn's notes on the arXiv, which works out this case.

Instead of attempting to compute the group cohomology of this huge profinite group, we can try to detect classes by looking at homotopy fixed points with respect to smaller subgroups. If $G\subseteq \mathbf{G}_n$ is a finite subgroup, we can consider the homotopy fixed points $E_n^{hG}$, and there's a map $L_{K(n)} S\to E_n^{hG}$, which gives a composite homomorphism $\pi_\ast S \to \pi_\ast L_{K(n)} S \to \pi_\ast E_n^{hG}$. This is particularly interesting when $G$ is a maximal finite subgroup, because we recover some well-known spectra.

At height $1$ and and the prime $2$, we know that $\mathbf{G}_1 \simeq \mathbf{Z}_2^\times \simeq \mathbf{Z}_2 \times \mathbf{Z}/2$, so the maximal finite subgroup is $\mathbf{Z}/2$. The group action on $E_1 = KU^\wedge_2$ is given by complex conjugation, so $E_1^{h\mathbf{Z}/2}$ is the universally loved spectrum $KO^\wedge_2$. At height $2$, I recall reading somewhere that the fixed points $E_2^{hG}$ (for $G$ a maximal finite subgroup of $\mathbf{G}_n$) is related to $TMF$ via $$L_{K(2)} TMF \simeq \prod_{\# S_p}E_2^{hG},$$ where $S_p$ is the set of isomorphism classes of supersingular elliptic curves over $\overline{\mathbf{F}_p}$. This follows essentially by construction; an analogue at higher chromatic height is described in Chapter 14 of Behrens-Lawson.

But $KO$ and $TMF$ are not complex-oriented! Instead, they admit orientations from $MSpin$ and $MString$: there are $\Eoo$-maps $MSpin \to KO$ and $MString \to TMF$ that lift the Atiyah-Bott-Shapiro orientation and the Witten genus. This is in Ando-Hopkins-Rezk, but it's hard to work through. There's an overview in Chapter 10 of the TMF book (see here), and some notes in Appendix A.3 of Eric Peterson's book project.

Let me now try to answer some questions in your eighth paragraph. The nilpotence theorem says that elements in the kernel of $\pi_\ast R \to MU_\ast R$ are nilpotent. (A simple corollary is Nishida's nilpotence theorem: if $R=S$, then everything in $\pi_\ast S$ is torsion, and since $MU_\ast$ is torsion-free, the kernel of $\pi_\ast S\to MU_\ast$ is the whole of $\pi_\ast S$, so anything in $\pi_\ast S$ is nilpotent.) The proof of this theorem goes by filtering the map $S\to MU$, which is presumably what you mean by "things between $S$ and $MU$". (I'm not sure what you mean by "above" the sphere: it is the initial object in the category of spectra.)

We have a sequence of maps $\ast\to \Omega SU(2) \to \cdots\to \Omega SU \xrightarrow{\sim} BU$ (the last equivalence is thanks to Bott periodicity). Consequently, we get maps $\Omega SU(n) \to BU$ for every $n$, and the Thom spectrum of the corresponding complex vector bundle over $\Omega SU(n)$ is denoted $X(n)$. For instance, $X(1) = S$ and $X(\infty) = MU$. This is a homotopy commutative ring spectrum, but since the map $\Omega SU(n) \to BU$ is a $2$-fold loop map, it is at best (for $n\neq 0,\infty$) an $\mathbf{E}_2$-ring spectrum. (It's not an $\mathbf{E}_3$-ring spectrum, see here.) Each $X(n)$ admits a canonical map from $S$ and to $MU$; moreover, the map $X(n) \to MU$ is an equivalence below degree $2n+1$. The proof of the nilpotence theorem now reduces to showing that if the image of $\alpha$ under $h(n):\pi_\ast R \to X(n)_\ast R$ is zero, then the image of $\alpha$ under $h(n+1)$ is also zero.

I ran a seminar last month on this stuff; I wrote detailed notes at http://www.mit.edu/~sanathd/iap-2018.pdf, which expand on the discussion above. Good sources to learn this stuff are Jacob Lurie's course from eight years ago and COCTALOS. For more references, check out this page. I hope this helps; let me know if there's something I should add/talk more about.

• In your notes you say it is unfortunate that a lot of people use ASS to refer to the Adams spectral sequence. Actually, a lot of us scrupulously say Adams ss to avoid this ugly abbreviation. – Robert Bruner Jul 11 '17 at 16:27
• Many thanks for your brilliantly illuminating answer (for me, so it is hardly flattering, as it is not so difficult to illuminate somebody as submerged into darkness as me). And even more thanks for many links to extremely informative sources. I will try to study them. – მამუკა ჯიბლაძე Jul 11 '17 at 21:56
• @მამუკაჯიბლაძე Sorry; I'd understood the question differently. There are nontrivial maps $X\to S$, of course. For instance, there's a map $\Sigma^\infty_+\mathbf{RP}^\infty\to S$, called the Kan-Priddy map. (This is defined via the composition of the maps $\mathbf{RP}^{n-1}_+ \to O(n)$ sending a line to the reflection it defines and $O(n)\to \Omega^n S^n$, which adjuncts to $\Sigma^n_+ \mathbf{RP}^{n-1}\to S^n$. Each of these maps is nullhomotopic --- but when we send $n\to \infty$, we get a nontrivial map $\Sigma^\infty_+ \mathbf{RP}^\infty\to S$.) – skd Jul 11 '17 at 22:36
• There is a very interesting ring above S: the Waldhausen K-theory, A(*) = K(S) augments onto S. – Robert Bruner Jul 12 '17 at 5:43
• The comment about the Kahn-Priddy map (aka the transfer associated to the inclusion of the trivial group into the group of order 2) says something that is very very wrong: each of those maps is definitely not nullhomotopic. Also, restricted to the unit, it is multiplication by 2, so it isn't a map of ring spectra, as it isn't unital. – Nicholas Kuhn Jul 12 '17 at 19:58

An informal reference could be the diagram on page 2 of the lecture notes from my September 2000 lecture in Oberwolfach, where I discussed the chromatic red-shift conjecture.