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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 EE then E (BU(1))E (*)[[c 1]]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 (P )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 (P )π (E)[[c 1 E]]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 E_\infty-maps MUEMU\to E. In particular, I think Lurie’s theorem in those notes is not showing this for E E_\infty-rings, but just homotopy commutative. In general, if I understand correctly, the universal oriented E E_\infty-ring is actually not MUMU.

    • 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 EE, and E E_\infty-maps MUEMU\to E. However, there is already a bijection between maps of ring spectra MUEMU\to E and complex orientations of EE, so this seems to be tantamount to stating that every map of ring spectra between MUMU and EE can be lifted to a map of E 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 Ec_1^E, here

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