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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.

• 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 $c_1^E$ in $E$-cohomology theory is equivalently an extension (in the classical homotopy category) of the map $\Sigma^2 1 \,\colon\, \mathbb{C}P^1 \longrightarrow \Omega^\infty E$ (which classifies the suspended identity in the cohomology ring) along the canonical inclusion of complex projective spaces

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

Notice that the complex projective spaces form a cotower

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

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

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

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

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

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

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

of finite-stage extensions

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

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

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

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

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

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

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

• 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 $\Omega SU(n) \to \Omega SU = BU$. 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 $\Omega SU(n) \to \Omega SU = BU$. 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

• 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 $X(n)$ and $T(m)$:

• Douglas Ravenel, section 3 of: Localization with Respect to Certain Periodic Homology Theories, American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 351-414 (doi:10.2307/2374308, jstor:2374308)

• Michael Hopkins, Stable decompositions of certain loop spaces, Northwestern 1984

• {#DevinatzHopkinsSmith88} Ethan Devinatz, Michael Hopkins, Jeffrey Smith, Theorem 3 of: Nilpotence and Stable Homotopy Theory I, Annals of Mathematics Second Series, Vol. 128, No. 2 (Sep., 1988), pp. 207-241 (jstor:1971440)

• Doug Ravenel, The first Adams-Novikov differential for the spectrum $T(m)$, 2000 (pdf, RavenelAdamsNovikovForTm.pdf:file)

• Ippei Ichigi, Katsumi Shimomura, The Modulo Two Homotopy Groups of the $L_2$-Localization of the Ravenel Spectrum, CUBO A Mathematical Journal, Vol. 10, No 03, (43–55). October 2008 (cubo:1498)

• Xiangjun Wang, Zihong Yuan, The homotopy groups of $L_2 T(m)/\big(p^{[\tfrac{m}{2}]+2}, v_1 \big)$ for $m \gt 1$, New York J. Math.24 (2018) 1123–1146 (pdf)

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

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

Given a multiplicative cohomology theory $E$, let me write

$\mathbb{K}P^1 \overset{\;\;\;G_d\;\;\;}{\longrightarrow} E_d$

for the canonical representative of the $d$-fold suspended $E$-ring unit:

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

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

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

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

$0 \overset{ G_{2d-1} }{\Rightarrow} G_d \cdot G_d \,,$

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

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

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

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

Has this been considered anywhere?

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeNov 26th 2020

Oh, now I see. For $E = KU$ this “$E$-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

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTimeJan 3rd 2021

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeJan 24th 2021

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

• CommentRowNumber33.
• CommentAuthorUrs
• CommentTimeJun 21st 2021
• (edited Jun 21st 2021)

added brief mentioning (here) of the example of $KU\big(B U(1)\big)$ and its alternative computation via the Atiyah-Segal completion theorem.