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I have been making trivial edits (adding references, basic statements, cross-links ) to Hopf invariant and a bunch of related entries, such as Kervaire invariant, Hopf invariant one problem, Arf-Kervaire invariant problem, normed division algebra.
added these pointer:
Discussion via differential forms/rational homotopy theory
J. H. C. Whitehead, An expression of Hopf ’s invariant as an integral, Proc. Nat. Acad. Sci. U. S. A.33 (1947), 117–123 (jstor:87688)
Dev Sinha, Ben Walter, Lie coalgebras and rational homotopy theory II: Hopf invariants (arXiv:0809.5084)
and this one:
If at the beginning the requirement is $n \gt 1$ in $\phi \;\colon\; S^{2n-1} \longrightarrow S^n$, then I guess the real Hopf fibration shouldn’t be included.
Mind you, Adams and Atiyah allow that case, so presumably change to $n \gt 0$.
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added the claim (here) that the Hopf invariant of a map $\phi$ may be read off as the unique free coefficient of the Sullivan model of $\phi$.
This follows straightforwardly, and I’d like to cite this from a canonical RHT source, if possible. But I don’t see it in the textbooks (FHT, …). If anyone knows opus, page and verse for a canonical citation of this fact, please let me know.
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You have $E_8$ in the diagram, where you want $E_{2n}$.
Does anything stop the octonionic Hopf fibration appearing in a similar diagram? “Octonionic orientation” receives precisely 0 hits.
By the way, bottom right of your new diagram you should have $\Sigma^8 \kappa$.
Thanks again, fixed now.
Regarding octonion-oriented cohomology:
There are no octonionic projective spaces beyond $\mathbb{O}P^1 and \mathbb{O}P^2$ (e.g. “Why octonions are bad” here).
Which made me wonder:
The diagrams for the $\mathbb{K}$-Hopf fibrations which I was showing involve exactly and only these two cases $\mathbb{K}P^1$, $\mathbb{K}P^2$.
So while $\mathbb{K}$-orientation in $E$-cohomology in the sense of lifts though
$\widetilde E( \mathbb{K}P^\infty ) \longrightarrow \widetilde E( \mathbb{K}P^1 )$does not make sense for $\mathbb{K} = \mathbb{O}$, what does make sense are “orientations to stage 2”, being lifts through
$\widetilde E( \mathbb{K}P^2 ) \longrightarrow \widetilde E( \mathbb{K}P^1 ) \,.$But these finite-stage orientations have received little attention, even for $\mathbb{K} = \mathbb{C}$: It looks like the list of references compiled here essentially exhausts the available literature. And these reference all focus on technicalities not going to the heart of the subject.
Another common generalization of the Hopf invariant is the ’e-invariant’. Suppose $f:S^k \to S^0$ is a stable map which vanishes in $E$-(co)homology. Then we get an extension $E^*(S^{k+1}) \to E^*(Cf) \to E^*(S^0)$ in $\mathrm{Ext}^1(E^*, E^*)$. This is an extension in, for example, the category of modules over E^*-cohomology operations, and gives an invariant for $f$ (but you could think of it as an extension in any abelian category where $E^*(S^k)$ lives, e.g. just as modules over $E^*$ if you want). When $E$ is ordinary cohomology, this is the Hopf invariant, but in general it can detect much more (e.g. the \alpha family when E is KU, say).
(Of course this is basically the beginning of the Adams spectral sequence relative to E)
I wonder if this invariant agrees with yours when they are both defined? Yours depends on a choice of ’stage 2 orientation’, but maybe that orientation gives a preferred class in Ext to compare to? Presumably the extension class for the module E^*(KP^2)?
Thanks, Dylan. That’s a great hint. I’ll think about this.
added the diagrammatic proof of the homotopy Whitehead ingegral/functional cup product-formula for the Hopf invariant (here)
Re #18, right that’s what I was thinking, that octonions get you to pass up one stage at least.
I was wondering if those section in Laughton’s thesis on quaternionic towers had anything to do with quaternionic orientation at finite stages, but I think not.
Dylan, thanks again for the hint towards the e-invariant in #19.
So I was trying to read up on e-invariants computed in other generalized cohomogy theories, beyond K-theory.
From Prop. 1 in Krueger 73 I gather that for all cohomology theories $E$ satisfying the usual assumptions for the $E$-Adams spectral sequence, the e-invariants all “agree”, under some pertinent isos between their $Ext^1$-s. Am I reading that right? (The definition on p. 5 needs some unravelling…)
By the way, does anyone discuss the e-invariant in equivariant cohomology (K-theory or otherwise)?
I have now spelled out (here) the argument for the essentially unique existence of the trivialization of the cup square on $S^n$, using appeal to connective covers of ring spectra
(previously the paragraph just had a pointer to the idea of this argument in Lurie, Lec. 4. Exmpl. 8, which however I have kept)
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