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added a tiny bit of basics to complex oriented cohomology theory
Urs! You’re putting up all the chromotopy stuff! Awesome!! :)
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
I have added some more basics to the Properties-section at complex oriented cohomology. Not done yet.
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 to the proof here a paragraph making more explicit why the extension problem indeed trivializes to give the conclusion.
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 pointer to the note
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 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 $MU$.
Sorry, could you say which page you are looking at? And maybe which paragraph you are looking at?
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?
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.
I have written this out in more detail now, at universal complex orientation on MU.
I have added more details to the section on formal group laws in $c_1^E$, here
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