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\begin{document}

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\section*{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{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms}

[[!include manifolds and cobordisms - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{DefinitionFor3Sphere}{On the 3-sphere}\dotfill \pageref*{DefinitionFor3Sphere} \linebreak
\noindent\hyperlink{HomotopyTheoreticCharacterization}{Homotopy-theoretic characterization}\dotfill \pageref*{HomotopyTheoreticCharacterization} \linebreak
\noindent\hyperlink{realization_via_the_complex_numbers}{Realization via the complex numbers}\dotfill \pageref*{realization_via_the_complex_numbers} \linebreak
\noindent\hyperlink{RealizationViaQuaternions}{Realization via quaternions}\dotfill \pageref*{RealizationViaQuaternions} \linebreak
\noindent\hyperlink{realization_via_the_hopf_construction}{Realization via the Hopf construction}\dotfill \pageref*{realization_via_the_hopf_construction} \linebreak
\noindent\hyperlink{spin3actionequivariance}{[[Spin(3)]]-[[action|equivariance]]}\dotfill \pageref*{spin3actionequivariance} \linebreak
\noindent\hyperlink{OnAllFourSpheres}{On the 1-sphere, 3-sphere, 7-sphere and 15-sphere}\dotfill \pageref*{OnAllFourSpheres} \linebreak
\noindent\hyperlink{via_norms_and_projections}{Via norms and projections}\dotfill \pageref*{via_norms_and_projections} \linebreak
\noindent\hyperlink{via_the_hopf_construction}{Via the Hopf construction}\dotfill \pageref*{via_the_hopf_construction} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToStableHomotopyGroupsOfSpheres}{Relation to stable homotopy groups of spheres}\dotfill \pageref*{RelationToStableHomotopyGroupsOfSpheres} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{magnetic_monopoles}{Magnetic monopoles}\dotfill \pageref*{magnetic_monopoles} \linebreak
\noindent\hyperlink{ktheory}{K-theory}\dotfill \pageref*{ktheory} \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{complex Hopf fibration} (named after [[Heinz Hopf]]) is a canonical nontrivial [[circle principal bundle]] over the [[2-sphere]] whose total space is the [[3-sphere]].

\begin{displaymath}
S^1 \hookrightarrow S^3 \to S^2 
  \,.
\end{displaymath}
Its canonically [[associated bundle|associated]] [[complex line bundle]] is the [[basic line bundle on the 2-sphere]].

This we discuss below in

\begin{itemize}%
\item \emph{\hyperlink{DefinitionFor3Sphere}{On the 3-sphere}}

\end{itemize}
More generally, there are four Hopf fibrations, on the 1-sphere, the 3-sphere, the 7-sphere and the 15-sphere, respectively. This we discuss in

\begin{itemize}%
\item \emph{\hyperlink{OnAllFourSpheres}{On the 1-sphere, 3-sphere, 7-sphere and 15-sphere}}.

\end{itemize}
\hypertarget{DefinitionFor3Sphere}{}\subsection*{{On the 3-sphere}}\label{DefinitionFor3Sphere}

\hypertarget{HomotopyTheoreticCharacterization}{}\subsubsection*{{Homotopy-theoretic characterization}}\label{HomotopyTheoreticCharacterization}

The [[Eilenberg-MacLane space]] $K(\mathbb{Z},2) \simeq B S^1$ is the [[classifying space]] for [[circle group]] [[principal bundles]]. By its very nature, it has a single nontrivial [[homotopy group]], the second, and this is isomorphic to the group of [[integers]]

\begin{displaymath}
\pi_2(K(\mathbb{Z},2)) \simeq \mathbb{Z}
  \,.
\end{displaymath}
This means that there is, up to [[homotopy]], a canonical (up to sign), continuous map from the 2-sphere

\begin{displaymath}
\phi : S^2 \to K(\mathbb{Z},2)
  \,,
\end{displaymath}
such that $[\phi] \in \pi_2(K(\mathbb{Z},2)) = \pm 1 \in \mathbb{Z}$.

As any map into $K(\mathbb{Z},2)$ this classifies a [[circle group]] [[principal bundle]] over its [[domain]]. This is the Hopf fibration, fitting into the long [[fiber sequence]]

\begin{displaymath}
\itexarray{
    S^1 &\hookrightarrow& S^3
    \\  
    && \downarrow
    \\
    && S^2
    &\stackrel{\phi}{\to}&
    B S^1 \simeq K(\mathbb{Z},2)
  }
  \,.
\end{displaymath}
In other words, the Hopf fibration is the $U(1)$-bundle with unit first [[Chern class]] on $S^2$.

\hypertarget{realization_via_the_complex_numbers}{}\subsubsection*{{Realization via the complex numbers}}\label{realization_via_the_complex_numbers}

An explicit [[topological space]] presenting the Hopf fibration may be obtained as follows. Identify

\begin{displaymath}
S^3 \simeq \{(z_0, z_1) \in \mathbb{C}\times \mathbb{C} \,|\, {|z_0|}^2 + {|z_1|}^2 = 1\}
\end{displaymath}
and

\begin{displaymath}
S^2 \simeq \mathbb{C P}^1 \simeq \mathbb{C} \sqcup \{\infty\}
\end{displaymath}
Then the [[continuous function]] $S^3 \to S^2$ defined by

\begin{displaymath}
(z_0, z_1) \mapsto \frac{z_0}{z_1}
\end{displaymath}
gives the Hopf fibration. (Thus, the Hopf fibration is a circle bundle naturally associated with the canonical [[line bundle]].) Alternatively, if we use

\begin{displaymath}
S^2 \simeq \{(z, x) \in \mathbb{C} \times \mathbb{R} 
  \,|\,
  {|z|}^2 + x^2 = 1\}
  \,.
\end{displaymath}
and identify this presentation of the 2-sphere with the complex projective line via [[stereographic projection]], the Hopf fibration is identified with the map $S^3 \to S^2$ given by sending

\begin{displaymath}
(z_0, z_1) \mapsto (2 z_0 z_1^* , {|z_0|}^2 - {|z_1|}^2).
\end{displaymath}
\hypertarget{RealizationViaQuaternions}{}\subsubsection*{{Realization via quaternions}}\label{RealizationViaQuaternions}

Alternatively, we may regard $S^3 \simeq S(\mathbb{H})$ as the [[unit sphere]] in the [[quaternions]] and $S^2 \simeq S\left( \mathbb{H}_{\mathrm{im}}\right)$ as the unit sphere in the [[imaginary part|imaginary]] [[quaternions]]. Under this identification, the complex Hopf fibration is equivalently represented by

\begin{displaymath}
\itexarray{
    S(\mathbb{H})
    &\longrightarrow&
    S\left( \mathbb{H}_{\mathrm{im}}\right)
    \\
    q 
      &\mapsto&
    q \cdot \mathbf{i} \cdot \overline{q}
  }
\end{displaymath}
where $\mathbf{i} \in S\left( \mathbb{H}_{\mathrm{im}}\right)$ is any unit imaginary quaternion.

\hypertarget{realization_via_the_hopf_construction}{}\subsubsection*{{Realization via the Hopf construction}}\label{realization_via_the_hopf_construction}

Regard $S^1 = U(1)$ as equipped with its [[circle group]] structure. This makes $S^1$ in particular an [[H-space]]. The Hopf fibration $S^1 \to S^3 \to S^2$ is the [[Hopf construction]] applied to this H-space.

\hypertarget{spin3actionequivariance}{}\subsubsection*{{[[Spin(3)]]-[[action|equivariance]]}}\label{spin3actionequivariance}

Consider

\begin{enumerate}%
\item the [[Spin(3)]]-[[action]] on the [[2-sphere]] $S^2$ which is induced by the defining action on $\mathbb{R}^3$ under the identification $S^2 \simeq S(\mathbb{R}^3)$;


\item the [[Spin(3)]]-action on the [[3-sphere]] $S^3$ which is induced under the exceptional [[isomorphism]] $Spin(3) \simeq Sp(1) = U(1,\mathbb{H})$ by the canonical left action of $U(1,\mathbb{H})$ on $\mathbb{H}$ via $S^3 \simeq S(\mathbb{H})$.



\end{enumerate}
Then the complex Hopf fibration $S^3 \overset{h_{\mathbb{C}}}{\longrightarrow} S^2$ is [[equivariant]] with respect to these [[actions]].

A way to make the $Spin(3)$-equivariance of the complex Hopf fibration fully explicit is to observe that the it is equivalently the following map of [[coset spaces]]:

\begin{displaymath}
\itexarray{
    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)}  
  }
\end{displaymath}
\hypertarget{OnAllFourSpheres}{}\subsection*{{On the 1-sphere, 3-sphere, 7-sphere and 15-sphere}}\label{OnAllFourSpheres}

\hypertarget{via_norms_and_projections}{}\subsubsection*{{Via norms and projections}}\label{via_norms_and_projections}

For each of the [[normed division algebra|normed division algebras]] over $\mathbb{R}$, the [[real numbers]], [[complex numbers]], [[quaternions]], [[octonions]]

\begin{displaymath}
A = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O},
\end{displaymath}
there is a corresponding Hopf fibration of [[Hopf invariant one]].

The total space of the fibration is the space of pairs $(\alpha, \beta) \in A^2$ of unit norm: ${|\alpha|}^2 + {|\beta|}^2 = 1$. This gives [[spheres]] of dimension 1, [[3-sphere|3]], [[7-sphere|7]], and 15 respectively. The base space of the fibration is [[projective space|projective]] 1-space $\mathbb{P}^1(A)$, giving spheres of dimension 1, 2, 4, and 8, respectively. In each case, the Hopf fibration is the map

\begin{displaymath}
S^{2^n - 1} \to S^{2^{n-1}}
\end{displaymath}
($n = 1, 2, 3, 4$) which sends the pair $(\alpha, \beta)$ to $\alpha/\beta$.

\hypertarget{via_the_hopf_construction}{}\subsubsection*{{Via the Hopf construction}}\label{via_the_hopf_construction}

When $X$ is a [[sphere]] that is an $H$-space, namely, one of the [[groups]] $S^0 = 1$ the [[trivial group]], $S^1 = \mathbb{Z}/2$ the [[group of order 2]], the [[3-sphere]] [[special unitary group]] $S^3 = SU(2)$; or the [[7-sphere]] $S^7$ with its [[Moufang loop]] structure, then the Hopf construction produces the above four Hopf fibrations:

\begin{enumerate}%
\item $S^0 \hookrightarrow S^1 \to S^1$ -- [[real Hopf fibration]]
\item $S^1 \hookrightarrow S^3 \to S^2$ -- [[complex Hopf fibration]]
\item $S^3 \hookrightarrow S^7 \to S^4$ -- [[quaternionic Hopf fibration]]
\item $S^7 \hookrightarrow S^{15} \to S^8$ -- [[octonionic Hopf fibration]]

\end{enumerate}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToStableHomotopyGroupsOfSpheres}{}\subsubsection*{{Relation to stable homotopy groups of spheres}}\label{RelationToStableHomotopyGroupsOfSpheres}

Let

$H_{\mathbb{C}} \in \pi_3(S^2)$,

$H_{\mathbb{H}} \in \pi_7(S^4)$

$H_{\mathbb{O}} \in \pi_{15}(S^8)$

be the [[homotopy class]] of the [[complex Hopf fibration]], the [[quaternionic Hopf fibration]] and the [[octonionic Hopf fibration]], respectively. Then their [[suspensions]] are the generators of the corresponding [[stable homotopy groups of spheres]]:

\begin{displaymath}
\begin{aligned}
  \Sigma H_{\mathbb{C}}
  & =
  \pm 1
    \in 
  \mathbb{Z}_2
  \simeq
  \pi_1^{st}
  \\
  \Sigma H_{\mathbb{H}}
  & =
  \pm 1
    \in 
  \mathbb{Z}_{24} 
    \simeq
  \pi_3^{st}
  \\
  \Sigma H_{\mathbb{O}}
  & =
  \pm 1
    \in 
  \mathbb{Z}_{240} 
    \simeq
  \pi_7^{st}
  \end{aligned}
\end{displaymath}
see \href{https://mathoverflow.net/a/224082/381}{this MO comment}

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{magnetic_monopoles}{}\subsubsection*{{Magnetic monopoles}}\label{magnetic_monopoles}

When [[line bundles]] are regarded as models for the topological structure underlying the [[electromagnetic field]] the Hopf fibration is often called ``the [[magnetic monopole]]''. We may think of the $S^2$ homotopically as being the 3-dimensional [[Cartesian space]] with origin removed $\mathbb{R}^3 - \{0\}$ and think of this as being 3-dimensional physical space with a unit point [[magnetic charge]] at the origin removed. The corresponding [[electromagnetic field]] away from the origin is given by a [[connection on a bundle|connection]] on the corresponding Hopf fibration bundle.

\hypertarget{ktheory}{}\subsubsection*{{K-theory}}\label{ktheory}

In complex [[K-theory]], the Hopf fibration represents a class $H$ which generates the [[cohomology ring]] $K_U(S^2)$, and satisfying the relation $H^2 = 2 \cdot H - 1$, or $(H-1)^2 = 0$. (So in particular $H$ has an [[inverse]] $H^{-1} = 2- H$, see at [[Bott generator]].)

A succinct formulation of [[Bott periodicity]] for [[complex K-theory]] is that for a space $X$ whose [[homotopy type]] is that of a [[CW-complex]], we have

\begin{displaymath}
K(S^2 \times X) \cong K(S^2) \otimes K(X)
\end{displaymath}
(It would be interesting to see whether this can be proved by internalizing the (classically easy) calculation for $K(S^2)$ to the topos of sheaves over $X$.)

The Hopf fibrations over other [[normed division algebras]] also figure in the more complicated case of [[real K-theory]] $K_O$: they can be used to provide generators for the non-zero [[homotopy groups]] $\pi_n(B O)$ for the [[classifying space]] of the [[stable orthogonal group]], which are periodic of period 8 (not coincidentally, 8 is the dimension of the largest normed division algebra $\mathbb{O}$). To be followed up on.

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

\begin{itemize}%
\item [[Hopf construction]]


\item [[Hopf invariant one]]


\item [[quaternionic Hopf fibration]]



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

Reviews include

\begin{itemize}%
\item \emph{\href{https://chiasme.wordpress.com/2014/01/04/third-homotopy-group-of-the-sphere-and-hopf-fibration/}{Third homotopy group of the sphere and Hopf fibration}}

\end{itemize}
For more discussion in [[homotopy type theory]] see also at

\begin{itemize}%
\item \href{https://ncatlab.org/homotopytypetheory/show/HomePage}{HoTT Wiki}, \emph{[[homotopytypetheory:Hopf fibration]]}

\end{itemize}
Discussion of [[supersymmetry|supersymmetric]] Hopf fibrations:

\begin{itemize}%
\item [[A. P. Balachandran]], G. Marmo, B.-S. Skagerstam and A. Stern, section 9.3 of \emph{Gauge Symmetries and Fibre Bundles}, Lect. Notes in Physics 188, Springer-Verlag, Berlin, 1983 (\href{https://arxiv.org/abs/1702.08910}{arXiv:1702.08910})


\item Simon Davis, section 3 of \emph{Supersymmetry and the Hopf fibration} (\href{https://doi.org/10.4995/agt.2012.1623}{doi:10.4995/agt.2012.1623})



\end{itemize}
[[!redirects Hopf fibrations]]

[[!redirects complex Hopf fibration]]



\end{document}