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)

]]>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)?

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.

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

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

]]>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 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.

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

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

]]>Thanks! Fixed now.

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

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

]]>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.

]]>added pointer to

- Robert Mosher, Martin Tangora, p. 33 of
*Cohomology operations and applications in homotopy theory*, Harper \& Row 1986

added also pointer to

- Hassler Whitney, Section 31 in
*Geometric Integration Theory*, 1957 (pup:3151)

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)

added also pointer to

- Phillip Griffiths, John Morgan, Section 14.5 of
*Rational Homotopy Theory and Differential Forms*, Progress in Mathematics Volume 16, Birkhauser (2013) (doi:10.1007/978-1-4614-8468-4)

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.

]]>added pointer to

- Raoul Bott, Loring Tu, Prop. 17.22 in
*Differential Forms in Algebraic Topology*Graduate Texts in Mathematics 82, Springer 1982 (doi:10.1007/BFb0063500)

added publication data for this here:

- Dev Sinha, Ben Walter,
*Lie coalgebras and rational homotopy theory II: Hopf invariants*, Trans. Amer. Math. Soc. 365 (2013), 861-883 (arXiv:0809.5084, doi:10.1090/S0002-9947-2012-05654-6)

added pointer to

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

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

]]>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.

]]>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)

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)

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*.