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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
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.
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”?
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…’).
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…’).
Thanks! I’ll have a look.
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.
Sure, it’s MR2633919, and should be downloadable from proquest
Thanks. Hm, I get no hits for “MR2633919”. Do you have the title?
@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.
Thanks for the file. We should just upload to this to the nLab server.
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)
Douglas Ravenel, section 6.5 of: Complex cobordism and stable homotopy groups of spheres
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)
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?
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?
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
added pointer to
added brief mentioning (here) of the example of $KU\big(B U(1)\big)$ and its alternative computation via the Atiyah-Segal completion theorem.
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