Thanks, I have copied it over also to his “Selected writings” list, here

]]>Added another comment on subfibrations, but we don’t have a definition of that concept yet. Presumably a fibration involving subspaces.

]]>Added a reference and result about lack of $S^1$ subfibrations.

- Maurizio Parton, Paolo Piccinni,
*The Role of Spin(9) in Octonionic Geometry*, (arXiv:1810.06288).

Given the starring role played by the quaternionic Hopf fibration in Urs’s recent paper, we might expect the octonionic version to have featured in physics somewhere.

I see Joe Polchinski gives it a role in Open Heterotic Strings, p. 7, discussed on p. 3 of When D-branes Break.

]]>I guess it could come in the context of a discussion of $Spin(3) \simeq SU(2) \simeq Sp(1)$-equivariance. Presumably a similar idea applies as with the quaternion case: $Sp(1)$ acts on unit quaternions, $S^3$, and $Spin(3)$ acting on 2-sphere by its action on 3-space. Then the fibration itself is equivariant.

]]>Yes, I suppose so. Would be good to add.

]]>In view of #24, perhaps we could also have

$\array{ S^1 &\overset{fib(h_{\mathbb{C}})}{\longrightarrow}& S^{3} &\overset{h_{\mathbb{C}}}{\longrightarrow}& S^2 \\ = && = && = \\ \frac{Spin(2)}{Spin(1)} &\longrightarrow& \frac{Spin(3)}{Spin(1)} &\longrightarrow& \frac{Spin(3)}{Spin(2)} }$And perhaps then in the real case if we can cope with $Spin(0)$.

]]>added pointer to

- Herman Gluck, Frank Warner, Wolfgang Ziller,
*The geometry of the Hopf fibrations*, L’Enseignement Mathématique, t.32 (1986), p. 173-198

which in its Prop. 7.1 explicitly states and proves the $Spin(9)$-equivariance of the octonionic Hopf fibration

(but fails to mention the coset representation that makes this manifest)

]]>made more explicit that statement

$\array{ S^7 &\overset{fib(h_{\mathbb{O}})}{\longrightarrow}& S^{15} &\overset{h_{\mathbb{O}}}{\longrightarrow}& S^8 \\ = && = && = \\ \frac{Spin(8)}{Spin(7)} &\longrightarrow& \frac{Spin(9)}{Spin(7)} &\longrightarrow& \frac{Spin(9)}{Spin(8)} }$which

- Liviu Ornea, Maurizio Parton, Paolo Piccinni, Victor Vuletescu,
*Spin(9) geometry of the octonionic Hopf fibration*, (arXiv:1208.0899, doi:10.1007/s00031-013-9233-x)

make but in passing, on their p. 7.

Analogous coset-presentations of the other Hopf fibrations must be well know, such as

$\array{ S^3 &\overset{fib(h_{\mathbb{H}})}{\longrightarrow}& S^{7} &\overset{h_{\mathbb{H}}}{\longrightarrow}& S^4 \\ = && = && = \\ \frac{Spin(4)}{Spin(3)} &\longrightarrow& \frac{Spin(5)}{Spin(3)} &\longrightarrow& \frac{Spin(5)}{Spin(4)} }$What would be a reference that makes this explicit?

]]>I’d like to know the fixed points of the octonionic Hopf fibration under subgroups of $G_2 = Aut_{\mathbb{R}}(\mathbb{O})$; probably I’d particularly like to know the fixed points under the standard $SO(4)$-subgroup and its further $SO(3)$-subgroup.

I still have little idea, but at least I found this article here:

- Reiko Miyaoka,
*The linear isotropy group of $G_2/SO(4)$, the Hopf fibering and isoparametric hypersurfaces*, Osaka J. Math. Volume 30, Number 2 (1993), 179-202. (Euclid)

which in its section 2 gives a detailed desription of these subgroups. I have added this pointer to the entry now.

Curiously, the article is otherwise concerned with understanding not the octonionic but the quaternionic Hopf fibration, and maybe the relation between the above subgroups and the quaternionic Hopf fibration is just what I really need. But not sure yet. Need to put this aside for the moment.

]]>I added some parts from above. There are no doubt more interesting things to say about $spin(9)$ etc.

]]>Yes, as mentioned in #10, the Hopf construction (curiously enough) gives the negative of the Hopf fibration. (And both -1 and +1 are generators of $\mathbb{Z}$…, if that’s what you were alluding to?)

]]>I’m not entirely convinced that the identification of $\Sigma S^7$ as given with $S^8$ represents the element 1 in $\pi_8(S^8)$, and not the element -1. There’s an implicit isomorphism $[0,1]\simeq[-1,1]$ in Urs’ parameterisation, reversing the orientation.

But from the formula for the degrees given, assuming inversion has degree -1, then the usual Hopf fibration using $x\cdoty^{-1}$ would have degree -1, whereas the general construction with smash and suspension would have degree +1. It’s not clear to me what’s going on, since the Hopf fibration are taken to be the generators in their respective homotopy groups.

]]>Ah yes. of course. But then what? So these are inequivalent constructions of the Hopf fibration? Different elements of $\pi_{15}(S^8)$? Is that why you say the “negative of the standard Hopf fibration”?

]]>The parameterization of $S^7 \star S^7$ that I gave in the entry manifestly identifies it with the unit sphere in $\mathbb{O}^2$, in its standard coordinates. Moreover, the parameterization I gave for $\Sigma S^7$ is manifestly so that $z$ parameterizes a hemisphere of $S^4$ and the two possible solutions for $t$ identify it as one or the other hemisphere. Each hemisphere in turn is naturally identified via stereographic projection with the chart $[z;1]$ or $[1,z]$ of $\mathbb{P}\mathbb{O}^1$.

That’s why I said that the standard parameterization of the Hopf fibrations is via $(x,y) \mapsto x y^{-1}$ (in one chart) or $(x,y ) \mapsto x^{-1} y$ (in the other), while the Hopf construction gives $(x,y) \mapsto x y$.

]]>Re #16, I saw that. I was just wondering how the Hopf construction map, i.e., product, looks under the change of coordinates.

So there’s $S^15$ as the unit sphere in $\mathbb{O}^2$ and as $S^7 * S^7$, and $S^8$ as lines in $\mathbb{O}^2$ and as $\Sigma S^7$. What happens when we pass from the unit sphere in $\mathbb{O}^2$ to $S^7 * S^7$ to $\Sigma S^7$ to lines in $\mathbb{O}^2$?

]]>David, re #11: the map in the Hopf construction is just the product operation. In the parameterization which I had chosen in the entry it is

$\array{ S^7 & \longrightarrow & S^4 \\ \left\{ (x,t,y) \,, {\vert x \vert}^2 = 2t \,,\; {\vert y\vert}^2 = 2 - 2t \right\} &\stackrel{{(x,y) \mapsto z \coloneqq x \cdot y} \atop {t \mapsto t}}{\longrightarrow}& \left\{ (z,t) \,,\; {\vert z \vert}^2 + (1 - 2t)^2 = 1 \right\} } \,.$Notice that it is the multiplicativity of the norm in division algebras which makes this parameterization work: if ${\vert x \vert}^2 = 2t$ and ${\vert y\vert}^2 = 2 - 2t$ then it follows that

$\begin{aligned} {\vert x \cdot y\vert}^2 + (1- 2t)^2 & = {\vert x \vert}^2 {\vert y \vert}^2 + (1-2t)^2 \\ & = 2t (2-2t) + (1 - 2t)^2 \\ & = 1 \end{aligned} \,,$(I have added this further clarification now also to the entry, at the end of the subsection here).

]]>Thanks, Todd!

]]>I’ve added some material to join of topological spaces, including an example which gives more details on what I was driving at in the previous comment.

Milnor construction needs to be made more precise in a couple of spots, which I intend to get to later.

]]>I’ve added some material to join of topological spaces, including an example which gives more details on what I was driving at in the previous comment.

]]>It seems that the more robust way of speaking about the Hopf fibrations is via the Hopf construction.

Oh, good point. I guess that would be a first step towards creating a classifying bundle for an $H$-space $X$, with the ability to pass to later stages mandated by satisfaction of $A_\infty$ conditions on $X$ (in the octonionic case we can’t get past step one).

]]>Maybe the different parameterizations of $S^8$ and $S^15$ are relevant. What is the map that takes the parameterization of $S^{15}$ at Hopf construction to the unit sphere in $\mathbb{O}^2$?

$S^{7}\ast S^{7} = (S^{7}\times I \times S^{7})/\sim \simeq \left\{ (x,t,y) \,, {\vert x \vert}^2 = 2t \,,\; {\vert y\vert}^2 = 2 - 2t \right\}$ ]]>Sorry for coming in late. It seems that the more robust way of speaking about the Hopf fibrations is via the *Hopf construction*. That does not mention projective spaces and it needs nothing but a binary pairing operation to exist.

The only thing is, unless I am missing something, that if you plug in the product operation of a division algebra into the Hopf construction as is, then the Hopf construction (which uses $(x,y) \mapsto x \cdot y$) spits out the negative of the standard Hopf fibration (which instead uses $(x,y)\mapsto x \cdot y^{-1}$ on one chart and $(x,y)\mapsto x^{-1} \cdot y$ on the other), by this proposition. I had tried to spell out the relation here.

]]>Thanks, Todd.

I started 15-sphere as it appears to have one or two special properties.

]]>