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?



Has anyone seen a nice map of multiplicative cohomology theories?
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](http://www.mit.edu/~sanathd/talbot-notes.pdf), 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](http://math.harvard.edu/~lurie/papers/Elliptic-II.pdf).) 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](https://mathoverflow.net/questions/259844/the-most-outrageous-or-ridiculous-conjectures-in-mathematics/259918#259918)
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](https://arxiv.org/abs/1702.05033) 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](https://arxiv.org/pdf/math/0702719.pdf).


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](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf), but
it's hard to work through. There's an overview in Chapter 10 of the TMF book
(see [here](http://math.mit.edu/conferences/talbot/2007/tmfproc/)), and some
notes in Appendix A.3 of Eric Peterson's [book
project](http://www.math.harvard.edu/~ecp/teaching/Spring2016/278/).


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](https://chat.stackexchange.com/transcript/message/36647423#36647423).)
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](http://math.harvard.edu/~lurie/252x.html) from eight years ago and
[COCTALOS](http://web.math.rochester.edu/people/faculty/doug/otherpapers/coctalos.pdf).
For more references, check out [this
page](https://chromotopy.org/teaching/Fall2013/msri/). I hope this helps;
let me know if there's something I should add/talk more about.


