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\section*{7-sphere}

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

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

[[!include topology - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{QuaternionicHopfFibration}{Quaternionic Hopf fibration}\dotfill \pageref*{QuaternionicHopfFibration} \linebreak
\noindent\hyperlink{CosetSpaceRealization}{Coset space realizations}\dotfill \pageref*{CosetSpaceRealization} \linebreak
\noindent\hyperlink{exotic_7spheres}{Exotic 7-spheres}\dotfill \pageref*{exotic_7spheres} \linebreak
\noindent\hyperlink{G2Structure}{$G_2$-structure}\dotfill \pageref*{G2Structure} \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 [[sphere]] of [[dimension]] 7.

This is one of the [[parallelizable manifold|parallelizable]] spheres, as such corresponds to the [[octonions]] among the [[division algebras]], being the manifold of unit octonions, and is the only one of these which does not carry ([[Lie group|Lie]]) [[group]] structure but just [[Moufang loop]] structure.

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

\hypertarget{QuaternionicHopfFibration}{}\subsubsection*{{Quaternionic Hopf fibration}}\label{QuaternionicHopfFibration}

The 7-sphere participates in the [[quaternionic Hopf fibration]], the analog of the complex [[Hopf fibration]] with the field of [[complex numbers]] replaced by the division ring of [[quaternions]] or Hamiltonian numbers $\mathbb{H}$.

\begin{displaymath}
\itexarray{
    S^3 &\hookrightarrow& S^7
    \\
    && \downarrow^\mathrlap{p}
    \\
    && S^4 &\stackrel{}{\longrightarrow}& \mathbf{B} SU(2)
  }
\end{displaymath}
Here the idea is that $S^7$ can be construed as $\{(x, y) \in \mathbb{H}^2: {|x|}^2 + {|y|}^2 = 1\}$, with $p$ mapping $(x, y)$ to $x/y$ as an element in the [[projective line]] $\mathbb{P}^1(\mathbb{H}) \cong S^4$, with each [[fiber]] a [[torsor]] parametrized by quaternionic [[scalars]] $\lambda$ of unit [[norm]] (so $\lambda \in S^3$). This canonical $S^3$-bundle (or $SU(2)$-bundle) is classified by a map $S^4 \to \mathbf{B} SU(2)$.

\hypertarget{CosetSpaceRealization}{}\subsubsection*{{Coset space realizations}}\label{CosetSpaceRealization}

\begin{prop}
\label{QuotientOfSpin7ByG2IsS7}\hypertarget{QuotientOfSpin7ByG2IsS7}{}
\textbf{([[coset space]] of [[Spin(7)]] by [[G2]] is [[7-sphere]])}

Consider the canonical [[action]] of [[Spin(7)]] on the [[unit sphere]] in $\mathbb{R}^8$ (the [[7-sphere]]),

\begin{enumerate}%
\item This action is is [[transitive action|transitive]];


\item the [[stabilizer group]] of any point on $S^7$ is [[G2]];


\item all [[G2]]-subgroups of [[Spin(7)]] arise this way, and are all [[conjugate subgroup|conjugate]] to each other.



\end{enumerate}
Hence the [[coset space]] of [[Spin(7)]] by [[G2]] is the [[7-sphere]]

\begin{displaymath}
S^7
  \;\simeq_{diff}\;
  Spin(7)/G_2
  \,.
\end{displaymath}
\end{prop}
(e.g \hyperlink{Varadarajan01}{Varadarajan 01, Theorem 3})

Other coset realizations of the usual [[differentiable manifold|differentiable]] 7-sphere (\hyperlink{Choquet-Bruhat+DeWitt-Morette00}{Choquet-Bruhat, DeWitt-Morette 00, p. 288}):

\begin{itemize}%
\item $S^7 \simeq_{diff}$ [[Spin(6)]]$/SU(3) \simeq_{iso} SU(4)/SU(3)$ (by \href{sphere#OddDimSphereAsSpecialUnitaryCoset}{this Prop.});


\item $S^7 \simeq_{diff}  Spin(5)/SU(2)$ (\hyperlink{AwadaDuffPope83}{Awada-Duff-Pope 83}, \hyperlink{DuffNilssonPope83}{Duff-Nilsson-Pope 83})



\end{itemize}
These three coset realizations of `squashed' 7-spheres together with a fourth

\begin{itemize}%
\item $S^7 \simeq_{diff}  Spin(8)/Spin(7)$,

\end{itemize}
the realization of the `round' 7-sphere, may be seen jointly as resulting from the 8-dimensional representations of even [[Clifford algebras]] in 5, 6, 7, and 8 dimensions (see \hyperlink{Baez}{Baez}) and as such related to the four [[normed division algebras]]. See also \hyperlink{Choquet-Bruhat+DeWitt-Morette00}{Choquet-Bruhat+DeWitt-Morette00, pp. 263-274}.

The following gives an [[exotic 7-sphere]]:

\begin{itemize}%
\item $S^7 \simeq_{homeo} Sp(1)\backslash Sp(2)/Sp(1)$ ([[Gromoll-Meyer sphere]])

\end{itemize}
$\backslash$linebreak

\hypertarget{exotic_7spheres}{}\subsubsection*{{Exotic 7-spheres}}\label{exotic_7spheres}

A celebrated result of Milnor is that $S^7$ admits [[exotic smooth structures]] (see at \emph{[[exotic 7-sphere]]}), i.e., there are [[smooth manifold]] structures on the [[topological manifold]] $S^7$ that are not [[diffeomorphism|diffeomorphic]] to the standard smooth structure on $S^7$. More structurally, considering smooth structures up to [[orientation|oriented]] diffeomorphism, the different smooth structures form a [[monoid]] under a (suitable) operation of [[connected sum]], and this monoid is isomorphic to the [[cyclic group]] $\mathbb{Z}/(28)$. With the notable possible exception of $n = 4$ (where the question of existence of exotic 4-spheres is wide open), exotic spheres first occur in dimension $7$. This phenomenon is connected to the [[h-cobordism theorem]] (the monoid of smooth structures is identified with the monoid of h-cobordism classes of oriented [[homotopy spheres]]).

One explicit construction of the smooth structures is given as follows (see \hyperlink{Mil2}{Milnor 1968}). Let $W_k$ be the algebraic variety in $\mathbb{C}^5$ defined by the equation

\begin{displaymath}
z_1^{6 k - 1} + z_2^3 + z_3^2 + z_4^2 + z_5^2 = 0
\end{displaymath}
and $S_\epsilon \subset \mathbb{C}^5$ a sphere of small radius $\epsilon$ centered at the origin. Then each of the $28$ smooth structures on $S^7$ is represented by an intersection $W_k \cap S_\epsilon$, as $k$ ranges from $1$ to $28$. These manifolds sometimes go by the name \emph{Brieskorn manifolds} or \emph{[[Brieskorn spheres]]} or \emph{[[Milnor spheres]]}.

\hypertarget{G2Structure}{}\subsubsection*{{$G_2$-structure}}\label{G2Structure}

Let $\phi_0 \in \Omega^3(\mathbb{R}^7)$ be the [[associative 3-form]] and let

\begin{displaymath}
\Phi_0 \in \Omega^4(\mathbb{R} \oplus \mathbb{R}^7)
\end{displaymath}
be given by

\begin{displaymath}
\Phi_0 = d x_0 \wedge \phi_0 + \star \phi_0
\end{displaymath}
(where $x_0$ denotes the canonical coordinate on the first factor of $\mathbb{R}$ and $\phi_0$ is pulled back along the projection to $\mathbb{R}^7$) .

By construction this is its own [[Hodge dual]]

\begin{displaymath}
\Phi = \star \Phi
  \,.
\end{displaymath}
This implies that as we restrict $\Phi_0$ to

\begin{displaymath}
\mathbb{R}^8 - \{0\} \simeq \mathbb{R} \times S^7
\end{displaymath}
then there is a unique 3-form

\begin{displaymath}
\phi \in \Omega^3(S^7)
\end{displaymath}
on the 7-sphere such that

\begin{displaymath}
\Phi_0 = r^3 \wedge \phi + r^4 \star_{S^7} \phi
  \;\;\;\; (on \; \mathbb{R}^8 - \{0\})
  \,.
\end{displaymath}
This 3-form $\phi$ defines a [[G2-structure]] on $S^7$. It is \emph{nearly parallel} in that

\begin{displaymath}
d \phi = 4 \star \phi
  \,.
\end{displaymath}
(e.g. \hyperlink{Lotay12}{Lotay 12, def.2.4})

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

\begin{itemize}%
\item [[2-sphere]]


\item [[3-sphere]]


\item [[4-sphere]]


\item [[6-sphere]]


\item 7-sphere

[[exotic 7-sphere]]


\item [[n-sphere]]



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

\begin{itemize}%
\item [[Martin Cederwall]], Christian R. Preitschopf, \emph{The Seven-sphere and its Kac-Moody Algebra}, Commun. Math. Phys. 167 (1995) 373-394 (\href{http://arxiv.org/abs/hep-th/9309030}{arXiv:hep-th/9309030})


\item Takeshi \^O{}no, \emph{On the Hopf fibration $S^7 \to S^4$ over $Z$}, Nagoya Math. J. Volume 59 (1975), 59-64. (\href{http://projecteuclid.org/euclid.nmj/1118795554}{Euclid})



\end{itemize}
Relation to the [[Milnor fibration]]:

\begin{itemize}%
\item [[Kenneth Intriligator]], [[Hans Jockers]], [[Peter Mayr]], [[David Morrison]], M. Ronen Plesser, \emph{Conifold Transitions in M-theory on Calabi-Yau Fourfolds with Background Fluxes}, Adv.Theor.Math.Phys. 17 (2013) 601-699 (\href{http://arxiv.org/abs/1203.6662}{arXiv:1203.6662})

\end{itemize}
An [[ADE classification]] of finite subgroups of $SO(8)$ [[free action|acting freely]] on $S^7$ (see at \emph{[[group action on an n-sphere]]}) such that the quotient is [[spin structure|spin]] and has at least four [[Killing spinors]] (see also at [[ABJM model]]) is in

\begin{itemize}%
\item Paul de Medeiros, [[José Figueroa-O'Farrill]], Sunil Gadhia, [[Elena Méndez-Escobar]], \emph{Half-BPS quotients in M-theory: ADE with a twist}, JHEP 0910:038,2009 (\href{http://arxiv.org/abs/0909.0163}{arXiv:0909.0163}, \href{http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf}{pdf slides})


\item Paul de Medeiros, [[José Figueroa-O'Farrill]], \emph{Half-BPS M2-brane orbifolds} (\href{http://arxiv.org/abs/1007.4761}{arXiv:1007.4761})



\end{itemize}
Discussion of [[subgroups]]:

\begin{itemize}%
\item [[Veeravalli Varadarajan]], \emph{Spin(7)-subgroups of SO(8) and Spin(8)}, Expositiones Mathematicae, 19 (2001): 163-177 (\href{https://core.ac.uk/download/pdf/81114499.pdf}{pdf})

\end{itemize}
Discussion of [[exotic smooth structures]] on 7-spheres includes

\begin{itemize}%
\item Wikipedia, \emph{Exotic sphere}, \href{https://en.wikipedia.org/wiki/Exotic_sphere}{link}.

\end{itemize}
The explicit construction of exotic 7-spheres by intersecting algebraic varieties with spheres is described in

\begin{itemize}%
\item [[John Milnor]], ``Singular points of complex hypersurfaces'' , Princeton Univ. Press (1968).

\end{itemize}
Discussion of (nearly) [[G2-structures]] on $S^7$ and [[calibrated submanifolds]] includes

\begin{itemize}%
\item [[Jason Lotay]], \emph{Associative Submanifolds of the 7-Sphere}, Proc. London Math. Soc. (2012) 105 (6): 1183-1214 (\href{http://arxiv.org/abs/1006.0361}{arXiv:1006.0361}, \href{http://www.homepages.ucl.ac.uk/~ucahjdl/JDLotay_KIAS2011_slides.pdf}{talk slides})

\end{itemize}
On [[coset]]-realizations:

\begin{itemize}%
\item Linus Kramer, \emph{Octonion Hermitian quadrangles}, Bull. Belg. Math. Soc. Simon Stevin Volume 5, Number 2/3 (1998), 353-362 (\href{https://projecteuclid.org/euclid.bbms/1103409015}{euclid:1103409015})


\item Y. Choquet-Bruhat, C. DeWitt-Morette, \emph{Analysis, Manifolds and Physics Part II}, North Holland 2000


\item M. A. Awada, [[Mike Duff]], [[Christopher Pope]], \emph{$N=8$ Supergravity Breaks Down to $N=1$}, Phys. Rev. Lett. 50, 294 – Published 31 January 1983 (\href{https://doi.org/10.1103/PhysRevLett.50.294}{doi:10.1103/PhysRevLett.50.294})


\item [[Mike Duff]], [[Bengt Nilsson]], [[Christopher Pope]], \emph{Spontaneous Supersymmetry Breaking by the Squashed Seven-Sphere}, Phys. Rev. Lett. 50, 2043 – Published 27 June 1983; Erratum Phys. Rev. Lett. 51, 846 (\href{https://doi.org/10.1103/PhysRevLett.50.2043}{doi:10.1103/PhysRevLett.50.2043})


\item [[John Baez]], \emph{Rotations in the 7th Dimension}, (\href{https://golem.ph.utexas.edu/category/2007/09/rotations_in_the_7th_dimension.html}{blog post}), and \emph{TWF 195}, (\href{http://math.ucr.edu/home/baez/week195.html}{webpage})



\end{itemize}
[[!redirects 7-spheres]]

[[!redirects seven sphere]] [[!redirects seven spheres]]



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