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Since there wasn’t a page octonionic Hopf fibration, I started one, copying over the start of quaternionic Hopf fibration. The weirdness of the octonions doesn’t prevent anything there, does it?
$p_\mathbb{O}$ is not exactly given by the formula $[x/y;1]$ as one must be careful in defining $\mathbb{OP}^1$: it’s not just $(\mathbb{O}\times \mathbb{O}\setminus\{(0,0)\})/\mathbb{O}^\times$.
Also, it’s not clear how much one can think of it as being a “Moufang loop principal bundle”. Clearly there is an “action” map $S^15\times S(\mathbb{O}) \to S^15$, and I’m pretty convinced this induces a diffeomorphism $S^15\times S(\mathbb{O}) \to S^15\times_{S^8} S^{15}$, but while there is probably a sensible notion of what it means for a loop to act on a set or space, I don’t know what it is.
OK, I had a suspicion I should be wary. I’ve put ’Under construction’.
There’s an account of the fibration on p.7 of this. I wonder of I can extract that.
Does this look right?
Because of the noncommutativity of the octonions, we cannot simply imitate the construction for the quaternions in quaternionic Hopf fibration. First, we decompose $\mathbb{O}^2$ into the octonionic lines, $l_m := \{(x, m x)|x \in \mathbb{O}\}$ or $l_{\infty} := \{(0, y)|y \in \mathbb{O}\}$. In this way the fibration $\mathbb{O}^2 \setminus 0 \to S^8 = \{m \in \mathbb{O}\} \union \{\infty\}$ is obtained, with fibers $\mathbb{O} \setminus 0$, and the intersection with the unit sphere $S^{15} \subset \mathbb{O}^2$ provides the octonionic Hopf fibration.
Non-associativity, you mean?
On octonionic projective geometry, you might try reading Baez starting here.
Hmm, now I wondering whether it really was wrong as it was originally. In #2 David R. said
$p_\mathbb{O}$ is not exactly given by the formula $[x/y;1]$ as one must be careful in defining $\mathbb{OP}^1$: it’s not just $(\mathbb{O}\times \mathbb{O}\setminus\{(0,0)\})/\mathbb{O}^\times$.
But doesn’t this MO answer say that’s what $p_\mathbb{O}$ is?
David C., I think you’re safe. Baez here describes the projective line for any of the four normed division algebras $\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$ as being the space of $2 \times 2$ Hermitian idempotents of rank $1$, and for the octonionic case these are the matrices of the form $a_{11} = x^\ast x, a_{12} = x^\ast y, a_{21} = y^\ast x, a_{22} = y^\ast y$ where ${\|x\|}^2 + {\|y\|}^2 = 1$ (so $(x, y) \in S^{15}$). A little more analysis shows the manifold of such matrices is $S^8$, with the projection $p: S^{15} \to S^8$ essentially as you described it, but see that page for the complete details.
Thanks, Todd.
I started 15-sphere as it appears to have one or two special properties.
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.
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 $\mathbb{O}^2$?
$S^{7}\ast S^{7} = (S^{7}\times I \times S^{7})/\sim \simeq \left\{ (x,t,y) \,, {\vert x \vert}^2 = 2t \,,\; {\vert y\vert}^2 = 2 - 2t \right\}$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).
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.
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.
Thanks, Todd!
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).
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 $\mathbb{O}^2$?
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 $(x,y) \mapsto x y$.
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”?
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.
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 added some parts from above. There are no doubt more interesting things to say about $spin(9)$ etc.
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:
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.
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
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?
added pointer to
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)
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 $Spin(0)$.
Yes, I suppose so. Would be good to add.
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.
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.
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