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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 20th 2016
• (edited Jan 20th 2016)

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.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 26th 2019

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)

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 26th 2019

and this one:

• André Haefliger, p. 3 of Whitehead products and differential forms, In: Schweitzer P.A. (eds) Differential Topology, Foliations and Gelfand-Fuks Cohomology. Lecture Notes in Mathematics, vol 652. Springer, Berlin, Heidelberg (doi:10.1007/BFb0063500)
• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeApr 26th 2019

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.

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeApr 26th 2019

Mind you, Adams and Atiyah allow that case, so presumably change to $n \gt 0$.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeApr 29th 2019

• Dale Husemöller, chapter 15 of Fibre Bundles, Graduate Texts in Mathematics 20, Springer New York (1966)
• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMay 21st 2019

added publication data for this here:

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMay 30th 2019

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJun 3rd 2019
• (edited Jun 3rd 2019)

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.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJun 4th 2019

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJun 5th 2019

and

• Lee Rudolph, Whitehead’s Integral Formula, Isolated Critical Points, and the Enhancement of the Milnor Number, Pure and Applied Mathematics Quarterly Volume 6, Number 2, 2010 (arXiv:0912.4974)
• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJun 12th 2019

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeNov 27th 2020

starting a section on the Hopf invariant in generalized cohomology, here.

So far I have added a homotopy pasting diagram which exhibits the Hopf invariant in any $E$-theory in a natural way.

• CommentRowNumber14.
• CommentAuthorDavid_Corfield
• CommentTimeNov 27th 2020

You have $E_8$ in the diagram, where you want $E_{2n}$.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeNov 27th 2020
• (edited Nov 27th 2020)

Thanks! Fixed now.

Also added one more diagram, showing the case of the classical Hopf fibrations.

• CommentRowNumber16.
• CommentAuthorDavid_Corfield
• CommentTimeNov 27th 2020

Does anything stop the octonionic Hopf fibration appearing in a similar diagram? “Octonionic orientation” receives precisely 0 hits.

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeNov 27th 2020

By the way, bottom right of your new diagram you should have $\Sigma^8 \kappa$.

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeNov 27th 2020
• (edited Nov 27th 2020)

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).

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.

• CommentRowNumber19.
• CommentAuthorDylan Wilson
• CommentTimeNov 27th 2020

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)?

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeNov 27th 2020

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeNov 27th 2020

added the diagrammatic proof of the homotopy Whitehead ingegral/functional cup product-formula for the Hopf invariant (here)

• CommentRowNumber22.
• CommentAuthorDavid_Corfield
• CommentTimeNov 27th 2020

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.

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeNov 29th 2020
• (edited Nov 29th 2020)

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)?

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeJan 20th 2021

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)