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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> 7-sphere in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } </style> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script type="text/javascript"> <![CDATA[//><!]]> </script> </head> <body> <div id="Container"> <textarea id='content' readonly=' readonly' rows='24' cols='60' > +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Spheres +--{: .hide} [[!include spheres -- contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## 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. ## Properties ### Quaternionic Hopf fibration {#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 $\mathbb{H}$. $$ \array{ S^3 &\hookrightarrow& S^7 \\ && \downarrow^\mathrlap{p} \\ && S^4 &\stackrel{}{\longrightarrow}& \mathbf{B} SU(2) } $$ 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 $S^4 \to \mathbf{B} SU(2)$. ### Coset space realizations {#CosetSpaceRealization} +-- {: .num_prop #QuotientOfSpin7ByG2IsS7} ###### Proposition **([[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]]), 1. This action is is [[transitive action|transitive]]; 1. the [[stabilizer group]] of any point on $S^7$ is [[G₂]]; 1. all [[G₂]]-subgroups of [[Spin(7)]] arise this way, and are all [[conjugate subgroup|conjugate]] to each other. Hence the [[coset space]] of [[Spin(7)]] by [[G₂]] is the [[7-sphere]] $$ S^7 \;\simeq_{diff}\; Spin(7)/G_2 \,. $$ =-- (e.g [Varadarajan 01, Theorem 3](#Varadarajan01)) Other coset realizations of the usual [[differentiable manifold|differentiable]] 7-sphere ([Choquet-Bruhat, DeWitt-Morette 00, p. 288](#Choquet-Bruhat+DeWitt-Morette00)): * $S^7 \simeq_{diff} $ [[Spin(6)]]$/SU(3) \simeq_{iso} SU(4)/SU(3)$ (by [this Prop.](sphere#OddDimSphereAsSpecialUnitaryCoset)); * $S^7 \simeq_{diff} Spin(5)/SU(2)$ ([Awada-Duff-Pope 83](#AwadaDuffPope83), [Duff-Nilsson-Pope 83](#DuffNilssonPope83)) These three coset realizations of 'squashed' 7-spheres together with a fourth * $S^7 \simeq_{diff} Spin(8)/Spin(7)$, 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 [Baez](#Baez)) and as such related to the four [[normed division algebras]]. See also [Choquet-Bruhat+DeWitt-Morette00, pp. 263-274](#Choquet-Bruhat+DeWitt-Morette00). [[!include coset space structure on n-spheres -- table]] The following gives an [[exotic 7-sphere]]: * $S^7 \simeq_{homeo} Sp(1)\backslash Sp(2)/Sp(1)$ ([[Gromoll-Meyer sphere]]) \linebreak ### Exotic 7-spheres A celebrated result of Milnor is that $S^7$ admits [[exotic smooth structures]] (see at _[[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 [Milnor 1968](#Mil2)). Let $W_k$ be the algebraic variety in $\mathbb{C}^5$ defined by the equation $$z_1^{6 k - 1} + z_2^3 + z_3^2 + z_4^2 + z_5^2 = 0$$ 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 _Brieskorn manifolds_ or _[[Brieskorn spheres]]_ or _[[Milnor spheres]]_. ### $G_2$-structure {#G2Structure} Let $\phi_0 \in \Omega^3(\mathbb{R}^7)$ be the [[associative 3-form]] and let $$ \Phi_0 \in \Omega^4(\mathbb{R} \oplus \mathbb{R}^7) $$ be given by $$ \Phi_0 = d x_0 \wedge \phi_0 + \star \phi_0 $$ (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]] $$ \Phi = \star \Phi \,. $$ This implies that as we restrict $\Phi_0$ to $$ \mathbb{R}^8 - \{0\} \simeq \mathbb{R} \times S^7 $$ then there is a unique 3-form $$ \phi \in \Omega^3(S^7) $$ on the 7-sphere such that $$ \Phi_0 = r^3 \wedge \phi + r^4 \star_{S^7} \phi \;\;\;\; (on \; \mathbb{R}^8 - \{0\}) \,. $$ This 3-form $\phi$ defines a [[G₂-structure]] on $S^7$. It is _nearly parallel_ in that $$ d \phi = 4 \star \phi \,. $$ (e.g. [Lotay 12, def.2.4](#Lotay12)) ## Related concepts [[!include spheres -- contents]] ## References * [[Martin Cederwall]], Christian R. Preitschopf, _The Seven-sphere and its Kac-Moody Algebra_, Commun. Math. Phys. 167 (1995) 373-394 ([arXiv:hep-th/9309030](http://arxiv.org/abs/hep-th/9309030)) * Takeshi &#212;no, _On the Hopf fibration $S^7 \to S^4$ over $Z$_, Nagoya Math. J. Volume 59 (1975), 59-64. ([Euclid](http://projecteuclid.org/euclid.nmj/1118795554)) Relation to the [[Milnor fibration]]: * [[Kenneth Intriligator]], [[Hans Jockers]], [[Peter Mayr]], [[David Morrison]], M. Ronen Plesser, _Conifold Transitions in M-theory on Calabi-Yau Fourfolds with Background Fluxes_, Adv.Theor.Math.Phys. 17 (2013) 601-699 ([arXiv:1203.6662](http://arxiv.org/abs/1203.6662)) An [[ADE classification]] of finite subgroups of $SO(8)$ [[free action|acting freely]] on $S^7$ (see at _[[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 * [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], Sunil Gadhia, [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf)) * [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], _Half-BPS M2-brane orbifolds_ ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761)) Discussion of [[subgroups]]: * {#Varadarajan01} [[Veeravalli Varadarajan]], _Spin(7)-subgroups of SO(8) and Spin(8)_, Expositiones Mathematicae, 19 (2001): 163-177 ([pdf](https://core.ac.uk/download/pdf/81114499.pdf)) Discussion of [[exotic smooth structures]] on 7-spheres includes * Wikipedia, _Exotic sphere_, [link](https://en.wikipedia.org/wiki/Exotic_sphere). The explicit construction of exotic 7-spheres by intersecting algebraic varieties with spheres is described in * {#Mil2} [[John Milnor]], "Singular points of complex hypersurfaces" , Princeton Univ. Press (1968). Discussion of (nearly) [[G2-structures]] on $S^7$ and [[calibrated submanifolds]] includes * {#Lotay12} [[Jason Lotay]], _Associative Submanifolds of the 7-Sphere_, Proc. London Math. Soc. (2012) 105 (6): 1183-1214 ([arXiv:1006.0361](http://arxiv.org/abs/1006.0361), [talk slides](http://www.homepages.ucl.ac.uk/~ucahjdl/JDLotay_KIAS2011_slides.pdf)) On [[coset]]-realizations: * {#Kramer98} Linus Kramer, _Octonion Hermitian quadrangles_, Bull. Belg. Math. Soc. Simon Stevin Volume 5, Number 2/3 (1998), 353-362 ([euclid:1103409015](https://projecteuclid.org/euclid.bbms/1103409015)) * {#Choquet-Bruhat+DeWitt-Morette00} [[Yvonne Choquet-Bruhat]], [[Cécile DeWitt-Morette]], *Analysis, manifolds and physics*, Part II, North Holland (1982, 2001) $[$[ISBN:9780444860170](https://www.elsevier.com/books/analysis-manifolds-and-physics-revised-edition/choquet-bruhat/978-0-444-86017-0)$]$ * {#AwadaDuffPope83} [[Moustafa A. Awada]], [[Mike Duff]], [[Christopher Pope]], *$N=8$ Supergravity Breaks Down to $N=1$*, Phys. Rev. Lett. **50** 5 (1983) 294-297 &lbrack;[doi:10.1103/PhysRevLett.50.294](https://doi.org/10.1103/PhysRevLett.50.294)&rbrack; * {#DuffNilssonPope83} [[Mike Duff]], [[Bengt Nilsson]], [[Christopher Pope]], _Spontaneous Supersymmetry Breaking by the Squashed Seven-Sphere_, Phys. Rev. Lett. 50, 2043 – Published 27 June 1983; Erratum Phys. Rev. Lett. 51, 846 ([doi:10.1103/PhysRevLett.50.2043](https://doi.org/10.1103/PhysRevLett.50.2043)) * {#Baez} [[John Baez]], _Rotations in the 7th Dimension_, ([blog post](https://golem.ph.utexas.edu/category/2007/09/rotations_in_the_7th_dimension.html)), and _TWF 195_, ([webpage](http://math.ucr.edu/home/baez/week195.html)) [[!redirects 7-spheres]] [[!redirects seven sphere]] [[!redirects seven spheres]] </textarea> </div>  </body> </html>