Thanks. That was the content of #11 to #13 above, but I see now that the concluding remark had remained unfixed. Thanks for catching this.

]]>Corrected a misconception: a complex orientation gives rise to a map of homotopy commutative rings out of $MU$, but *not* necessarily an $E_\infty$ ring map.

I have added more details to the section on formal group laws in $c_1^E$, here

]]>I have written this out in more detail now, at *universal complex orientation on MU*.

Ah, thanks, now I see. Thanks for catching that. I have briefly fixed the wording. I will be getting back to editing on this point more in detail next month.

]]>Sure, sorry about that. On the complex oriented cohomology theory page, Proposition 1 states that there is a bijection between equivalence classes of complex orientations of $E$, and $E_\infty$-maps $MU\to E$. However, there is already a bijection between maps of ring spectra $MU\to E$ and complex orientations of $E$, so this seems to be tantamount to stating that every map of ring spectra between $MU$ and $E$ can be lifted to a map of $E_\infty$-ring spectra. Is that true?

]]>Sorry, could you say which page you are looking at? And maybe which paragraph you are looking at?

]]>I think there is a problem with this page. I don’t believe that there is a bijection between complex orientations of and $E_\infty$-maps $MU\to E$. In particular, I think Lurie’s theorem in those notes is not showing this for $E_\infty$-rings, but just homotopy commutative. In general, if I understand correctly, the universal oriented $E_\infty$-ring is actually not $MU$.

]]>I have added pointer to the note

- Riccardo Pedrotti,
*Complex oriented cohomology – Orientation in generalized cohomology*, 2016 (pdf)

which spells out the proof that $E^\bullet(\mathbb{C}P^\infty) \simeq \pi_\bullet(E)[ [ c_1^E ] ]$ in complete detail, including all the steps that are usually glossed over.

]]>I have added both to *complex oriented cohomology* (here) and to *complex projective space* (here) a remark about the possible ambiguity in interpreting $E^\bullet(\mathbb{C}P^\infty)$ as the polynomial ring or as the power series ring in one generator.

I have added to the proof here a paragraph making more explicit why the extension problem indeed trivializes to give the conclusion.

]]>I have written out here a detailed proof of the statement that for complex oriented $E$ then $E^\bullet(B U(1))\simeq E^\bullet(\ast)[ [ c_1 ] ]$.

]]>I have added some more basics to the Properties-section at *complex oriented cohomology*. Not done yet.

I have expanded a little more the definition at complex oriented cohomology theory (in terms of generalized first Chern classes).

In the course of this I also added a bit more basics to *generalized cohomology theory* at *Relation between reduced and unreduced cohomolohy*

Urs! You’re putting up all the chromotopy stuff! Awesome!! :)

]]>added a tiny bit of basics to *complex oriented cohomology theory*