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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

    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)

    diff, v11, current

    • 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)

    diff, v11, current

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 26th 2019

    If at the beginning the requirement is n>1n \gt 1 in ϕ:S 2n1S n\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>0n \gt 0.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2019

    added pointer to

    • Dale Husemöller, chapter 15 of Fibre Bundles, Graduate Texts in Mathematics 20, Springer New York (1966)

    diff, v12, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2019

    added publication data for this here:

    diff, v13, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2019

    added pointer to

    diff, v14, current

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

    added the claim (here) that the Hopf invariant of a map ϕ\phi may be read off as the unique free coefficient of the Sullivan model of ϕ\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.

    diff, v16, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2019

    added also pointer to

    diff, v18, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2019

    added also pointer to

    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)

    diff, v19, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 12th 2019

    added pointer to

    diff, v22, current

    • 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 EE-theory in a natural way.

    diff, v30, current

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 27th 2020

    You have E 8E_8 in the diagram, where you want E 2nE_{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.

    diff, v30, current

    • 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 Σ 8κ\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 𝕆P 1and𝕆P 2\mathbb{O}P^1 and \mathbb{O}P^2 (e.g. “Why octonions are bad” here).

    Which made me wonder:

    The diagrams for the 𝕂\mathbb{K}-Hopf fibrations which I was showing involve exactly and only these two cases 𝕂P 1\mathbb{K}P^1, 𝕂P 2\mathbb{K}P^2.

    So while 𝕂\mathbb{K}-orientation in EE-cohomology in the sense of lifts though

    E˜(𝕂P )E˜(𝕂P 1) \widetilde E( \mathbb{K}P^\infty ) \longrightarrow \widetilde E( \mathbb{K}P^1 )

    does not make sense for 𝕂=𝕆\mathbb{K} = \mathbb{O}, what does make sense are “orientations to stage 2”, being lifts through

    E˜(𝕂P 2)E˜(𝕂P 1). \widetilde E( \mathbb{K}P^2 ) \longrightarrow \widetilde E( \mathbb{K}P^1 ) \,.

    But these finite-stage orientations have received little attention, even for 𝕂=\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 kS 0f:S^k \to S^0 is a stable map which vanishes in EE-(co)homology. Then we get an extension E *(S k+1)E *(Cf)E *(S 0)E^*(S^{k+1}) \to E^*(Cf) \to E^*(S^0) in Ext 1(E *,E *)\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 ff (but you could think of it as an extension in any abelian category where E *(S k)E^*(S^k) lives, e.g. just as modules over E *E^* if you want). When EE 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

    Thanks, Dylan. That’s a great hint. I’ll think about this.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2020

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

    diff, v35, current

    • 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 EE satisfying the usual assumptions for the EE-Adams spectral sequence, the e-invariants all “agree”, under some pertinent isos between their Ext 1Ext^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 nS^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)

    diff, v40, current

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