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 $MU$.

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 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 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 $TMF$ down below and $MSpin$, $MSU$, $MSp$, $MString$, etc. above $MU$.

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 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 $MU$.

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?