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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 4th 2020

starting something

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 4th 2020
• (edited Aug 4th 2020)

I have a basic question here:

Is the composite

$S^7 \longrightarrow \mathbb{C}P^3 \longrightarrow S^4$

of

1) the canonical $S^1$-fibration $S^7 \to \mathbb{C}P^3$

followed by

2) the twistor fibration $\mathbb{C}P^3 \to S^4$

equivalent to

3) the quaternionic Hopf fibration $S^7 \to S^4$

?

I expect it is, and probably it follows readily once the definitions are unwound suitably. But I don’t see it in detail yet.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeAug 4th 2020

Oh, I see it: It’s immediate from the reformulation of the Calanbi-Penrose fibrations given on this page.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeAug 5th 2020

added statement of and references for the definition via sending complex line to quaternionic lines

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeAug 5th 2020
• (edited Aug 5th 2020)

Does an analogue occur for other Hopf fibrations? E.g., a composite

$S^15 \longrightarrow \mathbb{H}P^3 \longrightarrow S^8$

as a composite of the $S^3$-fibration $S^15 \to \mathbb{H}P^3$ and an $S^4 = \mathbb{H}P^1$-fibration $\mathbb{H}P^3 \to S^8$.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeAug 5th 2020

added pointer to this item (thanks to David C. For pointing it out):

• B. Loo and Alberto Verjovsky, On quotients of Hopf fibrations, in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 26 (1994), pp. 103-108 (hdl:10077/4637, pdf)
• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeAug 5th 2020

• Simon Brain, Giovanni Landi, Differential and Twistor Geometry of the Quantum Hopf Fibration, Commun. Math. Phys. 315 (2012):489-530 (arXiv:1103.0419)

Anonymous

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeAug 6th 2020

• Bonaventure Loo, The space of harmonic maps of $S^2$ into $S^4$, Transactions of the American Mathematical Society Vol. 313, No. 1 (1989) (jstor:2001066)
• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeAug 6th 2020

I don’t understand yet why many authors call it the “Calabi-Penrose fibration”. It seems that the first one to consider it was Atiyah 79, Sec III.1 (who credits Penrose for general inspiration, but otherwise seems to be conjuring the thing quite by himself).

What did Calabi do here, and what would be a relevant citation?

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeAug 6th 2020

I think this paper may explain

• Harlan P. Blaine, Jr. Lawson, Surfaces minimales et la construction de Calabi-Penrose, pdf
• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeAug 6th 2020

Thanks! I have added that pointer:

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeAug 6th 2020

A sorry, that’s another “Blaine Lawson”?! Am fixing it…

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeAug 6th 2020

No, wait, how many “H. Blaine Lawson, Jr.” can there be, on this planet?

It must be that the “H = Harlan” on SemanticScholar here is a glitch of automated reference-crawling? The article it self (here) doesn’t claim that “H = Harlan”.

• CommentRowNumber15.
• CommentAuthorDavid_Corfield
• CommentTimeAug 6th 2020

This page claims H is for Herbert. Presumably it’s all one person.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeAug 6th 2020
• (edited Aug 6th 2020)

Thanks for the sanity check. With that out of the way, on to the article:

It’s interesting, but also mysterious:

On p. 4 we hear that

In one of his important works, E. Calabi has found…

but despite (or because of?) this alleged importance, we don’t get to see a citation for this. Instead, the only reference that follows is… to Atiyah el. al. (!) after the mysterious claim that (if I am translating this correctly?) this is the “only way by which we understand the link nowadays”.

But anyways it does look like this article of Lawson’s could be the origin of the term “Calabi-Penrose fibration”, thanks again.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeAug 6th 2020

According to B. Loo in jstor:2001066 Calabi’s fibration is in

• E. Calabi, Quelques applications de l’analyse complexe aux surfaces d’aire minima, Topics in Complex Manifolds (Ed., H. Rossi), Les Presses de l’Univ. de Montreal, 1967, pp. 59-81. 5.

• E. Calabi, Minimal immersions of surfaces in euclidean spheres, J. Differential Geometry 1 (1967), 111-125.

But I haven’t found a copy of the first one yet, and I haven’t recognized the construction in the second one yet.

• CommentRowNumber18.
• CommentAuthorDavid_Corfield
• CommentTimeAug 6th 2020
• (edited Aug 6th 2020)

I think it’s rather

he provided a construction from which only now do we understand the connection to the one introduced by Penrose in GR…

or maybe in better English

he provided a construction whose connection to the one introduced by Penrose in GR only now do we understand…

• CommentRowNumber19.
• CommentAuthorDavid_Corfield
• CommentTimeAug 6th 2020
• (edited Aug 6th 2020)

AHS is relatively recent, hence the “seulement aujord’hui”.

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeAug 6th 2020
• (edited Aug 6th 2020)

Okay, thanks!

Meanwhile I have found this review

• Jonas Nordstrom, Calabi’s construction of Harmonic maps from $S^2$ to $S^n$, Lund University 2008 (pdf)

which gives a clear statement of what Calabi actually did here (around Lemma 2.31). It seems to require a bit of thought to see that this is equal (for $m = 2$ in Calabi’s notation) to what Atiyah introduced as the twistor fibration.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeAug 7th 2020

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeAug 14th 2020

• Eugenio Calabi, Minimal immersions of surfaces in euclidean spheres, J. Differential Geometry (1967), 111-125 (euclid:jdg/1214427884)

• Eugenio Calabi, Quelques applications de l’analyse complexe aux surfaces d’aire minima, Topics in Complex Manifolds (Ed. H. Rossi), Les Presses de l’Universit ́e de Montr ́eal (1968), 59-81 (naid:10006413960)

• CommentRowNumber23.
• CommentAuthorDavid_Corfield
• CommentTimeAug 19th 2020

Added generalizations to all even spheres and beyond studied by Robert Bryant.

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeAug 19th 2020
• (edited Aug 19th 2020)

Thanks, David.

What we are wondering about is whether there is a sensible generalization of the twistor fibration that would still be over the 4-sphere, but coming from attaching more than one 2-cell to it. Because it’s this 2-cell which gives the single gauge field $F_2$ in Twistorial Cohomotopy.

What we’d probably need is something that would still factor the quaternionc Hopf fibration $\mathrm{Sp}(2)$-equivariantly.

• CommentRowNumber25.
• CommentAuthorDavid_Corfield
• CommentTimeAug 19th 2020

I haven’t seen anything else mentioned other than the Penrose fibration composed with the antipodal map, as in Example 1 of this and p. 5 of this.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeAug 20th 2020
• (edited Aug 20th 2020)

Thanks for the pointers, these are interesting articles.

I should say that what I am looking for here is likely not related to twistors anymore. (From the point of view of Cohomotopy theory, the appearance of twistor space in the factorization of the quaternionic Hopf fibration is a surprise, not something we asked for.)

More concretely, from the discussion on p. 5 we are naturally wondering if the construction of the combined Hopf/twistor fibration could be somehow modified such that instead of the gauge group $S(U(1)^2)$ a gauge group $S(U(1)^n)$ would appear, for small $n$.

For starters, for $n =3$ this would mean that the new fiber over $S^4$ we are after has Sullivan model of this form:

\mathbb{R} \left[ \array{ f_2^{(1)}, \\ f_2^{(2)}, \\ h_3 } \right] \Big/ \left( \begin{aligned} d\, f_2^{(1)} & = 0 \\ d\, f_2^{(2)} & = 0 \\ d\;\, h_3\; & = - f_2^{(1)} \wedge f_2^{(1)} - f_2^{(1)} \wedge f_2^{(2)} - f_2^{(2)} \wedge f_2^{(2)} \end{aligned} \right)

That’s because the right hand side in the third line is the expression for the second Chern class on $B S(U(1)^3)$ in terms of the first Chern classes on the first two $B U(1)$-factors, and it’s these second Chern classes whose appearance is suggested by equation (6) on p .3.

So we are looking for an $Sp(2)$-equivariant factorization of the quaternionic Hopf fibration through a fibration over $S^4$, whose fiber has the above Sullivan model.

This may or may not exist. And if it exists, it may or may not be related to twistors.

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeNov 21st 2020

• Angel Cano, Juan Pablo Navarrete, José Seade, Section 10.1 in: Kleinian Groups and Twistor Theory, In: Complex Kleinian Groups, Progress in Mathematics, vol 303. Birkhäuser 2013 (doi:10.1007/978-3-0348-0481-3_10)
• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeMar 31st 2021
• (edited Mar 31st 2021)

(hat tip to Michael Murray here)

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeMar 31st 2021

am finally changing the page name from “Calabi-Penrose fibration” to “twistor fibration” (which does less unjustice to Atiyah et al while still highlighting Penrose’s contribution, clearly)