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\begin{document}

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\section*{quaternionic Hopf fibration}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{EquivariantStructure}{$SO(3)$- and $Spin(5)$-Equivariant structure}\dotfill \pageref*{EquivariantStructure} \linebreak
\noindent\hyperlink{ClassInTheHomotopyGroupsOfSpheres}{Class in the homotopy groups of spheres}\dotfill \pageref*{ClassInTheHomotopyGroupsOfSpheres} \linebreak
\noindent\hyperlink{ClassInEquivariantStableHomotopyTheory}{Class in equivariant stable homotopy theory}\dotfill \pageref*{ClassInEquivariantStableHomotopyTheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{quaternionic Hopf fibration} is the [[Hopf fibration]] induced by the [[quaternions]], hence it is the [[fibration]]

\begin{displaymath}
\itexarray{
     S^3 &\hookrightarrow& S^7
     \\
     && \downarrow^{\mathrlap{p_{\mathbb{H}}}}
     \\
     && S^4
  }
\end{displaymath}
of the [[7-sphere]] over the [[4-sphere]] with [[fiber]] the [[3-sphere]], which is induced via the [[Hopf construction]] from the product operation

\begin{displaymath}
\mathbb{H} \times \mathbb{H} \stackrel{(-)\cdot (-)}{\longrightarrow}
  \mathbb{H}
\end{displaymath}
on the [[quaternions]], or else from

\begin{displaymath}
\mathbb{H} \times \mathbb{H}^{\times} \stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}
  \mathbb{H}
\end{displaymath}
to match standard conventions.

This means that if $S^7$ is regarded as the [[unit sphere]] $\{(x,y)  | {\vert x\vert}^2 + {\vert y\vert}^2 = 1\}$ in $\mathbb{H}\times \mathbb{H}$ and $S^4$ is regarded as the quaternionic [[projective space]], then $p$ is given (on points $(x,y)$ with $y \neq 0$) simply by

\begin{displaymath}
p_{\mathbb{H}} \colon (x,y) \mapsto [x;y] = [x/y; 1]
  \,,
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{EquivariantStructure}{}\subsubsection*{{$SO(3)$- and $Spin(5)$-Equivariant structure}}\label{EquivariantStructure}

Since the [[automorphism group]] of the [[quaternions]], as an $\mathbb{R}$-[[associative algebra|algebra]], is the [[special orthogonal group]] $SO(3)$

\begin{displaymath}
\mathrm{Aut}_{\mathbb{R}}(\mathbb{H}) \simeq SO(3)
\end{displaymath}
acting by [[rotation]] of the imaginary quaternions, via the [[Hopf construction]] it follows that the 7-sphere and 4-sphere inherit $SO(3)$-[[actions]] under which the quaternionic Hopf map is equivariant.

Notice that this means that $SO(3)$ acts on $S^7$ here diagonally on the \emph{two} copies of the imaginary octonions in $S^7 \hookrightarrow \mathbb{H} \oplus \mathbb{H}$ (as opposed to, say, via any one of the embeddings $SO(3) \hookrightarrow SO(8)$ and the following canonical action of $SO(8)$ on $S^7 \hookrightarrow \mathbb{R}^8$).

(see also \hyperlink{CookCrabb93}{Cook-Crabb 93})

But in fact more is true:

\begin{prop}
\label{Spin5EquivarianceOfQuaternionicHopfFibration}\hypertarget{Spin5EquivarianceOfQuaternionicHopfFibration}{}
\textbf{([[Spin(5)]]-[[action|equivariance]] of [[quaternionic Hopf fibration]])}

Consider

\begin{enumerate}%
\item the [[Spin(5)]]-[[action]] on the [[4-sphere]] $S^4$ which is induced by the defining action on $\mathbb{R}^5$ under the identification $S^4 \simeq S(\mathbb{R}^5)$;


\item the [[Spin(5)]]-action on the [[7-sphere]] $S^7$ which is induced under the exceptional [[isomorphism]] $Spin(5) \simeq Sp(2) = U(2,\mathbb{H})$ (this Prop. $\backslash$ref\{Spin(5)\#ExceptionalIsoToSp2\}) by the canonical left action of $U(2,\mathbb{H})$ on $\mathbb{H}^2$ via $S^7 \simeq S(\mathbb{H}^2)$.



\end{enumerate}
Then the [[quaternionic Hopf fibration]] $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$ is [[equivariant]] with respect to these [[actions]].

\end{prop}
This appears as (\hyperlink{GluckWarnerZiller86}{Gluck-Warner-Ziller 86, Prop. 4.1}).

The statement is also almost explicit in \hyperlink{Porteous95}{Porteous 95, p. 263}

$\backslash$begin\{center\} $\backslash$begin\{imagefromfile\} ``file\_name'': ``Spin5OnHopfH.jpg'', ``width'': 280 $\backslash$end\{imagefromfile\} $\backslash$end\{center\}

A way to make the $Spin(5)$-equivariance of the quaternionic Hopf fibration fully explicit is to observe that the quaternionic Hopf fibration is equivalently the following map of [[coset spaces]]:

\begin{displaymath}
\itexarray{
    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)}  
  }
\end{displaymath}
(\hyperlink{HasudaTomizawa09}{Hasuda-Tomizawa 09, table 1})

\begin{remark}
\label{}\hypertarget{}{}
Of the resulting [[action]] of [[Sp(2)]]$\times$[[Sp(1)]] on the [[7-sphere]] (from Prop. \ref{Spin5EquivarianceOfQuaternionicHopfFibration}), only the [[quotient group]] [[Sp(2).Sp(1)]] acts [[effective action|effectively]].

\end{remark}
\hypertarget{ClassInTheHomotopyGroupsOfSpheres}{}\subsubsection*{{Class in the homotopy groups of spheres}}\label{ClassInTheHomotopyGroupsOfSpheres}

The quaternionic Hopf fibration gives an element in the 7th [[homotopy groups of spheres|homotopy group of the 4-sphere]]

\begin{displaymath}
[p_{\mathbb{H}}] \in \pi_7(S^4) \simeq \mathbb{Z} \times (\mathbb{Z}/12)
\end{displaymath}
and in fact it is a generator of the non-torsion factor in this group.

Stably, i.e. as a generator for the [[stable homotopy groups of spheres]] in degree $7-4 = 3$, the quaternionic Hopf map becomes a [[torsion subgroup|torsion]] generator

\begin{displaymath}
[p_{\mathbb{H}}] \in \pi_3^S \simeq \mathbb{Z}/24
  \,,.
\end{displaymath}
\hypertarget{ClassInEquivariantStableHomotopyTheory}{}\subsubsection*{{Class in equivariant stable homotopy theory}}\label{ClassInEquivariantStableHomotopyTheory}

Fix a [[finite group|finite]] [[subgroup]] $G \hookrightarrow SO(3)$ which does not come from $SO(2) \hookrightarrow SO(3)$ -- i.e. not a [[cyclic group]], but one of the [[dihedral groups]] or else the [[tetrahedral group]] or [[octahedral group]] or [[icosahedral group]] (by the [[ADE classification]]).

Regard both $S^7$ and $S^4$ as pointed [[topological G-spaces]] via the $SO(3)$-action induced via automorphisms of the quaternions, as \hyperlink{EquivariantStructure}{above}. Write

\begin{displaymath}
\Sigma^\infty_G S^7, \Sigma^\infty_G S^4 \in G Spectra
\end{displaymath}
for the corresponding [[equivariant suspension spectra]].

Notice that if we took trivial $G$, then in the [[stable homotopy category]]

\begin{displaymath}
[\Sigma^\infty S^7, \Sigma^\infty S^4] \simeq \mathbb{Z}/24
\end{displaymath}
by the \hyperlink{ClassInTheHomotopyGroupsOfSpheres}{above}. In contrast:\footnote{The proof of prop. \ref{QuaternionicHopfFibrationIsDEEquivariantlyStablyNonTorsion} profited from [[Charles Rezk]], who suggested \href{http://mathoverflow.net/a/224185/381}{here} that the reduction to fixed points will make the real Hopf fibration give a non-torsion contribution, and from [[David Barnes]] who amplified the use of the Greenless-May splitting theorem.} 

\begin{prop}
\label{QuaternionicHopfFibrationIsDEEquivariantlyStablyNonTorsion}\hypertarget{QuaternionicHopfFibrationIsDEEquivariantlyStablyNonTorsion}{}
In $G$-[[equivariant homotopy theory]] this becomes a non-[[torsion subgroup|torsion group]], i.e.

\begin{displaymath}
[\Sigma^\infty_G S^7, \Sigma^\infty_G S^4]_G \simeq \mathbb{Z} \oplus \cdots
\end{displaymath}
with the quaternionic Hopf fibration, regarded as a $G$-equivariant map, representing a non-torsion element.

\end{prop}
\begin{proof}
First use the \href{rational+equivariant+stable+homotopy+theory#SplittingIntoMackeyFunctors}{Greenlees-May decomposition} which says that for any two $G$-[[equivariant spectra]] $X,Y$ and writing $\pi_\bullet(X), \pi_\bullet(Y)$ for their [[equivariant homotopy groups]], organized as [[Mackey functors]] $H \mapsto \pi_n^H(X)$ for all subgroups $H \subset G$, then the canonical map

\begin{displaymath}
[X,Y]_G
  \longrightarrow
   \underset{n}{\oplus} 
  Hom_{\mathcal{M}[G]}(\pi_n(X), \pi_n(Y))
\end{displaymath}
is [[rational equivariant stable homotopy theory|rationally]] an [[isomorphism]].

With this we are reduced to showing that there exists $n \in \mathbb{Z}$ and a morphism of [[Mackey functors]] of [[equivariant homotopy groups]] $\pi_n(\Sigma^\infty_G S^7) \to \pi_n(\Sigma^\infty_G S^4)$ which is not a torsion element in the abelian [[hom-object|hom-group]] of Mackey functors.

To analyse this, we use the [[tom Dieck splitting]] which says that the [[equivariant homotopy groups]] of [[equivariant suspension spectra]] $\Sigma^\infty_G X$ contain a [[direct sum|direct summand]] which is simply the ordinary stable homotopy groups of the naive [[fixed point]] space $X^H$:

\begin{displaymath}
\pi_n^H(\Sigma^\infty_G X) \simeq \pi_n(\Sigma^\infty (X^H)) \oplus \cdots
  \,.
\end{displaymath}
Now observe that the [[fixed points]] of the $SO(3)$-action on the quaternionic Hopf fibration that we are considering is just the [[real Hopf fibration]]:

\begin{displaymath}
(p_{\mathbb{H}})^{SO(3)} = p_{\mathbb{R}} \;\colon\; S^1 \longrightarrow S^1
\end{displaymath}
since $SO(3)$ acts transitively on the quaternionic quaternions and fixes the real quaternions. By our assumption that $G \subset SO(3)$ does not come through $SO(2) \hookrightarrow SO(3)$ it follows that this statment is still true for $G$:

\begin{displaymath}
(p_{\mathbb{H}})^{G} = p_{\mathbb{R}} \;\colon\; S^1 \longrightarrow S^1
  \,.
\end{displaymath}
But the [[real Hopf fibration]] defines a non-torsion element in $\pi_0^S \simeq \mathbb{Z}$.

In conclusion then, at $n = 1$ and $H = G$ we find that the $G$-equivariant quaternionic Hopf fibration contributes a non-torsion element in

\begin{displaymath}
Hom_{Ab}(\pi_1^G(\Sigma^\infty_G S^7), \pi_1^G(\Sigma^\infty_G S^4))
\end{displaymath}
which appears as a non-torsion element in

\begin{displaymath}
Hom_{\mathcal{M}[G]}( \pi_1(\Sigma^\infty_G S^7), \pi_1(\Sigma^\infty_G S^4) )
\end{displaymath}
and hence in $[\Sigma^\infty_G S^7, \Sigma^\infty_G S^4]_G$.

\end{proof}
See also at \emph{[[equivariant stable cohomotopy]]}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[real Hopf fibration]]


\item [[complex Hopf fibration]]


\item [[octonionic Hopf fibration]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Herman Gluck, Frank Warner, Wolfgang Ziller, \emph{The geometry of the Hopf fibrations}, L'Enseignement Math\'e{}matique, t.32 (1986), p. 173-198 (\href{https://www.researchgate.net/publication/266548925_The_geometry_of_the_Hopf_fibrations}{ResearchGate})


\item [[Reiko Miyaoka]], \emph{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. (\href{http://projecteuclid.org/euclid.ojm/1200784357}{Euclid})


\item [[Ian Porteous]], \emph{Clifford Algebras and the Classical Groups}, Cambridge Studies in Advanced Mathematics, Cambridge University Press (1995)


\item Machiko Hatsuda, Shinya Tomizawa, \emph{Coset for Hopf fibration and Squashing}, Class.Quant.Grav.26:225007, 2009 (\href{https://arxiv.org/abs/0906.1025}{arXiv:0906.1025})



\end{itemize}
Discussion in [[parameterized homotopy theory]] includes

\begin{itemize}%
\item A. L. Cook, M.C. Crabb, \emph{Fiberwise Hopf structures on sphere bundles}, J. London Math. Soc. (2) 48 (1993) 365-384 (\href{http://www.maths.ed.ac.uk/~aar/papers/crabbcook.pdf}{pdf})


\item Kouyemon Iriye, \emph{Equivariant Hopf structures on a sphere}, J. Math. Kyoto Univ. Volume 35, Number 3 (1995), 403-412 (\href{http://projecteuclid.org/euclid.kjm/1250518704}{Euclid})



\end{itemize}
Discussion in [[homotopy type theory]] is in

\begin{itemize}%
\item [[Ulrik Buchholtz]], [[Egbert Rijke]], \emph{The Cayley-Dickson Construction in Homotopy Type Theory} (\href{https://arxiv.org/abs/1610.01134}{arXiv:1610.01134})

\end{itemize}
Noteworthy [[fiber products]] with the quaternionic Hopf fibration, notably [[exotic 7-spheres]], are discussed in

\begin{itemize}%
\item Llohann D. Sperança, \emph{Explicit Constructions over the Exotic 8-sphere} (\href{https://www.ime.unicamp.br/~rigas/sigma8EncontroTopol.pdf}{pdf}, [[SperancaExoticSpheres.pdf:file]])

\end{itemize}


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