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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 17th 2013
• (edited Jun 17th 2013)

added a tiny bit of basics to complex oriented cohomology theory

• CommentRowNumber2.
• CommentAuthorJon Beardsley
• CommentTimeJun 17th 2013

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

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeFeb 13th 2016

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

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 15th 2016

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

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 9th 2016
• (edited May 9th 2016)

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

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMay 11th 2016

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

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMay 24th 2016

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.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJun 17th 2016
• (edited Jun 17th 2016)

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.

• CommentRowNumber9.
• CommentAuthorJon Beardsley
• CommentTimeJun 17th 2016

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 $MU$.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJun 17th 2016
• (edited Jun 17th 2016)

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

• CommentRowNumber11.
• CommentAuthorJon Beardsley
• CommentTimeJun 17th 2016

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?

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJun 18th 2016

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.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeJul 7th 2016

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

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeJul 8th 2016

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

• CommentRowNumber15.
• CommentAuthorTim Campion
• CommentTimeApr 30th 2019

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.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeMay 1st 2019
• (edited May 1st 2019)

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.