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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

    • CommentRowNumber15.
    • CommentAuthorTim Campion
    • CommentTimeApr 30th 2019

    Corrected a misconception: a complex orientation gives rise to a map of homotopy commutative rings out of MUMU, but not necessarily an E E_\infty ring map.

    diff, v43, current

    • 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.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2020

    added a remark (here) expanding on complex orientation by extensions and their obstructions. Currently it reads as follows:


    In terms of classifying maps, Def. \ref{ComplexOrientedCohomologyTheory} means that a complex orientation c 1 Ec_1^E in EE-cohomology theory is equivalently an extension (in the classical homotopy category) of the map Σ 21:P 1Ω E\Sigma^2 1 \,\colon\, \mathbb{C}P^1 \longrightarrow \Omega^\infty E (which classifies the suspended identity in the cohomology ring) along the canonical inclusion of complex projective spaces

    labelComplexOrientationAsExtensionP 1 Σ 21 E Ω 2E c 1 E P \label{ComplexOrientationAsExtension} \array{ \mathbb{C}P^1 & \overset{ \Sigma^2 1_E }{ \longrightarrow } & \Omega^{\infty - 2} E \\ \big\downarrow & \nearrow \mathrlap{ {}_{c_1^E} } \\ \mathbb{C}P^\infty }

    Notice that the complex projective spaces form a cotower

    *=P 0P 1P 2P 3P =limP \ast \,=\, \mathbb{C}P^0 \hookrightarrow \mathbb{C}P^1 \hookrightarrow \mathbb{C}P^2 \hookrightarrow \mathbb{C}P^3 \hookrightarrow \cdots \hookrightarrow \mathbb{C}P^\infty \,=\, \underset{\longrightarrow}{\lim} \mathbb{C}P^\bullet

    where each inclusion stage is (by this Prop.) the coprojection of a pushout of topological spaces (or rather: of pointed topological spaces) of the form

    D 2n+2 P n+1 (po) S 2n+1 h 2n+1 P n \array{ D^{2n+2} & \overset{}{\longrightarrow} & \mathbb{C}P^{n+1} \\ \big\uparrow &\mathclap{^{_{(po)}}}& \big\uparrow \\ S^{2n+1} &\underset{h^{2n+1}_{\mathbb{C}}}{\longrightarrow}& \mathbb{C}P^n }

    (where h 2n+1h^{2n+1}_{\mathbb{C}} is the complex Hopf fibration in dimension 2n+12n+1) hence of a homotopy pushout of underlying homotopy types (rather: of pointed homotopy types) of this form:

    * P n+1 (hpo) S 2n+1 h 2n+1 P n \array{ \ast & \overset{}{\longrightarrow} & \mathbb{C}P^{n+1} \\ \big\uparrow &\mathclap{^{_{(hpo)}}}& \big\uparrow \\ S^{2n+1} &\underset{h^{2n+1}_{\mathbb{C}}}{\longrightarrow}& \mathbb{C}P^n }

    Therefore, a complex orientation by extension (eq:ComplexOrientationAsExtension) is equivalently the homotopy colimiting map of a sequence

    (Σ 21=c 1 E,0,c 1 E,1,c 1 E,2,) \big( \Sigma^2 1 \,=\, c_1^{E,0} ,\, c_1^{E,1} ,\, c_1^{E,2} ,\, \cdots \big)

    of finite-stage extensions

    * P n+1 c 1 E,n+1 Ω 2E (hpo) c 1 E,n S 2n+1 h 2n+1 P n. \array{ \ast & \overset{}{\longrightarrow} & \mathbb{C}P^{n+1} & \overset{ c_1^{E,n+1} }{\longrightarrow} & \Omega^{\infty -2} E \\ \big\uparrow &\mathclap{^{_{(hpo)}}}& \big\uparrow & \nearrow \mathrlap{ {}_{c_1^{E,n}} } \\ S^{2n+1} &\underset{h^{2n+1}_{\mathbb{C}}}{\longrightarrow}& \mathbb{C}P^n \,. }

    Moreover, by the defining universal property of the homotopy pushout, the extension c 1 E,n+1c_1^{E,n+1} of c 1 E,nc_1^{E,n} is equivalently a choice of homotopy which trivializes the pullback of c 1 E,nc_1^{E,n} to the 2n+1-sphere:

    * Ω 2E c 1 E,n+1 c 1 E,n S 2n+1 h 2n+1 P n. \array{ \ast & \overset{}{\longrightarrow} & \Omega^{\infty - 2} E \\ \big\uparrow & {}_{ c_1^{E,n+1} } \seArrow & \big\uparrow \mathrlap{ ^{_{ c_1^{E,n} }} } \\ S^{2n+1} &\underset{ h^{2n+1}_{\mathbb{C}} }{\longrightarrow}& \mathbb{C}P^n \,. }

    This means, first of all, that the non-triviality of the pullback class

    (h 2n+1) *(c 1 E,n)E˜ 2(S 2n+1)E 2n1 \big( h^{2n+1}_{\mathbb{C}} \big)^\ast ( c_1^{E,n} ) \;\in\; \widetilde E^2 \big( S^{2n+1} \big) \;\simeq\; E_{2n-1}

    is the obstruction to the existence of the extension/orientation at this stage.

    It follows that if these obstructions all vanish, then a complex EE-orientation does exist. A sufficient condition for this is, evidently, that the reduced EE-cohomology of all odd-dimensional spheres vanishes, hence, that the graded EE-cohomology ring E E_\bullet is trivial in odd degrees.


    \,

    Does anyone discuss the structure obtained if one truncates this extension process at some finite stage, hence if one asks for “unstable complex orientation”?

    diff, v51, current

    • CommentRowNumber18.
    • CommentAuthorDylan Wilson
    • CommentTimeNov 24th 2020

    Yes. These are related to ’buds’ of formal groups and the spectra X(n), which are the Thom spectra of ΩSU(n)ΩSU=BU\Omega SU(n) \to \Omega SU = BU. See, e.g., Proposition 6.5.4 of Ravenel’s complex cobordism book, or Hopkins’s Northwestern thesis (’stable splittings…’).

    • CommentRowNumber19.
    • CommentAuthorDylan Wilson
    • CommentTimeNov 24th 2020

    Yes. These are related to ’buds’ of formal groups and the spectra X(n), which are the Thom spectra of ΩSU(n)ΩSU=BU\Omega SU(n) \to \Omega SU = BU. See, e.g., Proposition 6.5.4 of Ravenel’s complex cobordism book, or Hopkins’s Northwestern thesis (’stable splittings…’).

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2020

    Thanks! I’ll have a look.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2020
    • (edited Nov 24th 2020)

    Thanks again for the pointers. Ravenel’s section 6.5 is just what I was looking for.

    But could you give a more concrete pointer to “Hopkins’s Northwestern thesis”? I am not sure which document this is.

    • CommentRowNumber22.
    • CommentAuthorDylan Wilson
    • CommentTimeNov 24th 2020

    Sure, it’s MR2633919, and should be downloadable from proquest

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2020

    Thanks. Hm, I get no hits for “MR2633919”. Do you have the title?

    • CommentRowNumber24.
    • CommentAuthorDylan Wilson
    • CommentTimeNov 24th 2020
    • (edited Nov 24th 2020)
    really? that's the mathreviews number. anyway, here's the full bibtex entry :)

    @book {MR2633919,
    AUTHOR = {Hopkins, Michael Jerome},
    TITLE = {S{TABLE} {DECOMPOSITIONS} {OF} {CERTAIN} {LOOP} {SPACES}},
    NOTE = {Thesis (Ph.D.)--Northwestern University},
    PUBLISHER = {ProQuest LLC, Ann Arbor, MI},
    YEAR = {1984},
    PAGES = {96},
    MRCLASS = {Thesis},
    MRNUMBER = {2633919},
    URL =
    {https://www.proquest.com/docview/303306354},
    }

    (edited after David's correction- I had no idea that if you use the 'bibtex entry' function on mathscinet it automatically puts in the university's proxy you're using to access the site... I guess all my bibtex files have junk urls hidden in them!)
    • CommentRowNumber25.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 24th 2020

    @Dylan that url contains a Harvard proxy and is basically useless. Here’s the real url: https://www.proquest.com/docview/303306354

    @Urs I’ve emailed you a copy.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2020

    Thanks for the file. We should just upload to this to the nLab server.

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2020

    Have now compiled the following list of references on finite-stage complex orientations (here):

    On complex orientation at finite stage and Ravenel’s spectra X(n)X(n) and T(m)T(m):

    diff, v53, current

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2020
    • (edited Nov 25th 2020)

    I have a followup question.

    It needs a tad of notation before i can state it:

    Consider 𝕂{,,}\mathbb{K} \,\in\, \big\{ \mathbb{R}, \mathbb{C}, \mathbb{H} \big\} and write

    ddim (𝕂). d \;\coloneqq\; dim_{\mathbb{R}}(\mathbb{K}) \,.

    Given a multiplicative cohomology theory EE, let me write

    𝕂P 1G dE d \mathbb{K}P^1 \overset{\;\;\;G_d\;\;\;}{\longrightarrow} E_d

    for the canonical representative of the dd-fold suspended EE-ring unit:

    [G d]=Σ d(1 E)E˜ d(𝕂P 1). \big[ G_d \big] \;=\; \Sigma^d (1^E) \;\;\; \in \widetilde E^d \big( \mathbb{K}P^1 \big) \,.

    Then given a 𝕂\mathbb{K}-orientation c Ec^E for EE-cohomology, its “first extension stage” c E,1c^{E,1}, in the sense of the following diagram on the left, is equivalently a homotopy HH as shown on the right:

    * 𝕂P 2 c E,1 E d (hpo) G d S 2d1 h 𝕂P 1* E d H G d S 2d1 h 𝕂P 1. \array{ \ast & \overset{}{\longrightarrow} & \mathbb{K}P^{2} & \overset{ c^{E,1} }{\longrightarrow} & E_d \\ \big\uparrow &\mathclap{^{_{(hpo)}}}& \big\uparrow & \nearrow \mathrlap{ {}_{ G_d } } \\ S^{ 2 d - 1 } &\underset{ h }{\longrightarrow}& \mathbb{K}P^1 } \;\;\;\;\;\; \simeq \;\;\;\;\;\; \array{ \ast & \overset{}{\longrightarrow} & E_d \\ \big\uparrow & {}_{ H } \seArrow & \big\uparrow \mathrlap{ ^{_{ G_d }} } \\ S^{2d - 1} &\underset{ h }{\longrightarrow}& \mathbb{K}P^1 \,. }

    But we also have a canonical homotopy of the following form, by degree reasons:

    0G 2d1G dG d, 0 \overset{ G_{2d-1} }{\Rightarrow} G_d \cdot G_d \,,

    Finally, combining these two homotopies to a loop, we get the following class:

    [Hh *G dh *G 2d1]E 2d1(S 2d1). \big[ H \cdot h^\ast G_d \,-\, h^\ast G_{2d-1} \big] \;\;\;\in\; E^{2d-1}\big( S^{2d-1} \big) \,.

    When E=HAE = H A is ordinary cohomology, then this is the homotopy Whitehead integral formula for the Hopf invariant of hh.

    So for general 𝕂\mathbb{K}-oriented EE, we have an “EE-Whitehead integral” class induced from any choice of 𝕂\mathbb{K}-orientation. I suppose.

    Has this been considered anywhere?

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2020

    Oh, now I see. For E=KUE = KU this “EE-Whitehead integral” gives the Hopf invariant in K-theory as in the proof by Adams-Atiyah of the Hopf-invarariant-one theorem.

    So I should maybe better ask: Does anyone discuss the generalization of the constructions in Adams-Atiyah’s proof of the Hopf invariant, now with K-theory replaced by other oriented cohomology theories?

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2020
    • (edited Dec 29th 2020)

    Following up on #27 above, I am giving the list of references for finite-rank complex E-orientation its own bare entry finite-rank complex orientation and MΩSUn – references, to be !include-ed into relevant entries, for ease of synchronization

    diff, v55, current

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeJan 3rd 2021

    added pointer to

    diff, v57, current

    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeJan 24th 2021

    added a remark (here) making fully explicit that/how the universal EE-Chern class c 1 Ec^E_1 is identified with a Thom class on the universal complex line bundle

    diff, v60, current

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