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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="spheres">Spheres</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unit+sphere">unit sphere</a>, <a class="existingWikiWord" href="/nlab/show/polar+coordinates">polar coordinates</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/round+sphere">round sphere</a>, <a class="existingWikiWord" href="/nlab/show/squashed+sphere">squashed sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hemisphere">hemisphere</a>, <a class="existingWikiWord" href="/nlab/show/equator">equator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stereographic+projection">stereographic projection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+spheres">homotopy groups of spheres</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+sphere">homotopy sphere</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+sphere">rational homotopy sphere</a>, <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+fibration">spherical fibration</a>, <a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a>, <a class="existingWikiWord" href="/nlab/show/stable+Cohomotopy+theory">stable Cohomotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology+sphere">homology sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+sphere">rational homotopy sphere</a>, <a class="existingWikiWord" href="/nlab/show/rational+n-sphere">rational n-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+sphere">motivic sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+actions+on+spheres">group actions on spheres</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+sphere">representation sphere</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+Cohomotopy">equivariant Cohomotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeb+sphere+theorem">Reeb sphere theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+packing">sphere packing</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Music+of+the+Spheres">Music of the Spheres</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/low+dimensional+topology">low dimensional</a> <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+projective+space">real projective space</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℝ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{R}P^1</annotation></semantics></math></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+projective+line">complex projective line</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{C}P^1</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+the+2-sphere">geometric quantization of the 2-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fuzzy+2-sphere">fuzzy 2-sphere</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fuzzy+3-sphere">fuzzy 3-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quaternionic+projective+line">quaternionic projective line</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℍ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{H}P^1</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fuzzy+4-sphere">fuzzy 4-sphere</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/5-sphere">5-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6-sphere">6-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/exotic+7-sphere">exotic 7-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/8-sphere">8-sphere</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/octonionic+projective+line">octonionic projective line</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>𝕆</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{O}P^1</annotation></semantics></math></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/13-sphere">13-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/15-sphere">15-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite-dimensional+sphere">infinite-dimensional sphere</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#QuaternionicHopfFibration'>Quaternionic Hopf fibration</a></li> <li><a href='#CosetSpaceRealization'>Coset space realizations</a></li> <li><a href='#exotic_7spheres'>Exotic 7-spheres</a></li> <li><a href='#G2Structure'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-structure</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <a class="existingWikiWord" href="/nlab/show/sphere">sphere</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 7.</p> <p>This is one of the <a class="existingWikiWord" href="/nlab/show/parallelizable+manifold">parallelizable</a> spheres, as such corresponds to the <a class="existingWikiWord" href="/nlab/show/octonions">octonions</a> among the <a class="existingWikiWord" href="/nlab/show/division+algebras">division algebras</a>, being the manifold of unit octonions, and is the only one of these which does not carry (<a class="existingWikiWord" href="/nlab/show/Lie+group">Lie</a>) <a class="existingWikiWord" href="/nlab/show/group">group</a> structure but just <a class="existingWikiWord" href="/nlab/show/Moufang+loop">Moufang loop</a> structure.</p> <h2 id="properties">Properties</h2> <h3 id="QuaternionicHopfFibration">Quaternionic Hopf fibration</h3> <p>The 7-sphere participates in the <a class="existingWikiWord" href="/nlab/show/quaternionic+Hopf+fibration">quaternionic Hopf fibration</a>, the analog of the complex <a class="existingWikiWord" href="/nlab/show/Hopf+fibration">Hopf fibration</a> with the field of <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> replaced by the division ring of <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a> or Hamiltonian numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">\mathbb{H}</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mn>3</mn></msup></mtd> <mtd><mo>↪</mo></mtd> <mtd><msup><mi>S</mi> <mn>7</mn></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>S</mi> <mn>4</mn></msup></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^3 &\hookrightarrow& S^7 \\ && \downarrow^\mathrlap{p} \\ && S^4 &\stackrel{}{\longrightarrow}& \mathbf{B} SU(2) } </annotation></semantics></math></div> <p>Here the idea is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">S^7</annotation></semantics></math> can be construed as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>ℍ</mi> <mn>2</mn></msup><mo>:</mo><msup><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">|</mo><mi>y</mi><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>=</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{(x, y) \in \mathbb{H}^2: {|x|}^2 + {|y|}^2 = 1\}</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> mapping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, y)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo stretchy="false">/</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x/y</annotation></semantics></math> as an element in the <a class="existingWikiWord" href="/nlab/show/projective+line">projective line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℙ</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>ℍ</mi><mo stretchy="false">)</mo><mo>≅</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{P}^1(\mathbb{H}) \cong S^4</annotation></semantics></math>, with each <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> a <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a> parametrized by quaternionic <a class="existingWikiWord" href="/nlab/show/scalars">scalars</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> of unit <a class="existingWikiWord" href="/nlab/show/norm">norm</a> (so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>∈</mo><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\lambda \in S^3</annotation></semantics></math>). This canonical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^3</annotation></semantics></math>-bundle (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(2)</annotation></semantics></math>-bundle) is classified by a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>4</mn></msup><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^4 \to \mathbf{B} SU(2)</annotation></semantics></math>.</p> <h3 id="CosetSpaceRealization">Coset space realizations</h3> <div class="num_prop" id="QuotientOfSpin7ByG2IsS7"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> of <a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a> by <a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a> is <a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a>)</strong></p> <p>Consider the canonical <a class="existingWikiWord" href="/nlab/show/action">action</a> of <a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a> on the <a class="existingWikiWord" href="/nlab/show/unit+sphere">unit sphere</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>8</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^8</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a>),</p> <ol> <li> <p>This action is is <a class="existingWikiWord" href="/nlab/show/transitive+action">transitive</a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/stabilizer+group">stabilizer group</a> of any point on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">S^7</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a>;</p> </li> <li> <p>all <a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a>-subgroups of <a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a> arise this way, and are all <a class="existingWikiWord" href="/nlab/show/conjugate+subgroup">conjugate</a> to each other.</p> </li> </ol> <p>Hence the <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> of <a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a> by <a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a> is the <a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><mspace width="thickmathspace"></mspace><msub><mo>≃</mo> <mi>diff</mi></msub><mspace width="thickmathspace"></mspace><mi>Spin</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^7 \;\simeq_{diff}\; Spin(7)/G_2 \,. </annotation></semantics></math></div></div> <p>(e.g <a href="#Varadarajan01">Varadarajan 01, Theorem 3</a>)</p> <p>Other coset realizations of the usual <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable</a> 7-sphere (<a href="#Choquet-Bruhat+DeWitt-Morette00">Choquet-Bruhat, DeWitt-Morette 00, p. 288</a>):</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub></mrow><annotation encoding="application/x-tex">S^7 \simeq_{diff} </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Spin%286%29">Spin(6)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><msub><mo>≃</mo> <mi>iso</mi></msub><mi>SU</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">/SU(3) \simeq_{iso} SU(4)/SU(3)</annotation></semantics></math> (by <a href="sphere#OddDimSphereAsSpecialUnitaryCoset">this Prop.</a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^7 \simeq_{diff} Spin(5)/SU(2)</annotation></semantics></math> (<a href="#AwadaDuffPope83">Awada-Duff-Pope 83</a>, <a href="#DuffNilssonPope83">Duff-Nilsson-Pope 83</a>)</p> </li> </ul> <p>These three coset realizations of ‘squashed’ 7-spheres together with a fourth</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Spin</mi><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^7 \simeq_{diff} Spin(8)/Spin(7)</annotation></semantics></math>,</li> </ul> <p>the realization of the ‘round’ 7-sphere, may be seen jointly as resulting from the 8-dimensional representations of even <a class="existingWikiWord" href="/nlab/show/Clifford+algebras">Clifford algebras</a> in 5, 6, 7, and 8 dimensions (see <a href="#Baez">Baez</a>) and as such related to the four <a class="existingWikiWord" href="/nlab/show/normed+division+algebras">normed division algebras</a>. See also <a href="#Choquet-Bruhat+DeWitt-Morette00">Choquet-Bruhat+DeWitt-Morette00, pp. 263-274</a>.</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a>-<a class="existingWikiWord" href="/nlab/show/structures">structures</a> on <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a>:</strong></p> <table><thead><tr><th><strong>standard:</strong></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^{n-1} \simeq_{diff} SO(n)/SO(n-1)</annotation></semantics></math></td><td style="text-align: left;"><a href="sphere#nSphereAsCosetSpace">this Prop.</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)</annotation></semantics></math></td><td style="text-align: left;"><a href="sphere#OddDimSphereAsSpecialUnitaryCoset">this Prop.</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Sp</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Sp</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)</annotation></semantics></math></td><td style="text-align: left;"><a href="sphere#SphereAsSymplecticUnitaryCoset">this Prop.</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/exceptional+geometry">exceptional</a>:</strong></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Spin</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">S^7 \simeq_{diff} Spin(7)/G_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%287%29%2FG%E2%82%82+is+the+7-sphere">Spin(7)/G₂ is the 7-sphere</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Spin</mi><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^7 \simeq_{diff} Spin(6)/SU(3)</annotation></semantics></math></td><td style="text-align: left;">since <a class="existingWikiWord" href="/nlab/show/Spin%286%29">Spin(6)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SU%284%29">SU(4)</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^7 \simeq_{diff} Spin(5)/SU(2)</annotation></semantics></math></td><td style="text-align: left;">since <a class="existingWikiWord" href="/nlab/show/Sp%282%29">Sp(2)</a> is <a class="existingWikiWord" href="/nlab/show/Spin%285%29">Spin(5)</a> and <a class="existingWikiWord" href="/nlab/show/Sp%281%29">Sp(1)</a> is <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a>, see <a class="existingWikiWord" href="/nlab/show/Spin%285%29%2FSU%282%29+is+the+7-sphere">Spin(5)/SU(2) is the 7-sphere</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>6</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><msub><mi>G</mi> <mn>2</mn></msub><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^6 \simeq_{diff} G_2/SU(3)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/G%E2%82%82%2FSU%283%29+is+the+6-sphere">G₂/SU(3) is the 6-sphere</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>15</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Spin</mi><mo stretchy="false">(</mo><mn>9</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^15 \simeq_{diff} Spin(9)/Spin(7)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%289%29%2FSpin%287%29+is+the+15-sphere">Spin(9)/Spin(7) is the 15-sphere</a></td></tr> </tbody></table> <p>see also <em><a class="existingWikiWord" href="/nlab/show/Spin%288%29-subgroups+and+reductions+--+table">Spin(8)-subgroups and reductions</a></em></p> <p id="HomotopyTheoretic"> <a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a> of <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a>:</p> <p><img src="/nlab/files/ExceptionalSpheres.jpg" width="730px" /></p> <p>(from <a class="existingWikiWord" href="/schreiber/show/Twisted+Cohomotopy+implies+M-theory+anomaly+cancellation">FSS 19, 3.4</a>)</p> </div> <p>The following gives an <a class="existingWikiWord" href="/nlab/show/exotic+7-sphere">exotic 7-sphere</a>:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><msub><mo>≃</mo> <mi>homeo</mi></msub><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>\</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^7 \simeq_{homeo} Sp(1)\backslash Sp(2)/Sp(1)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/Gromoll-Meyer+sphere">Gromoll-Meyer sphere</a>)</li> </ul> <p><br /></p> <h3 id="exotic_7spheres">Exotic 7-spheres</h3> <p>A celebrated result of Milnor is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">S^7</annotation></semantics></math> admits <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structures">exotic smooth structures</a> (see at <em><a class="existingWikiWord" href="/nlab/show/exotic+7-sphere">exotic 7-sphere</a></em>), i.e., there are <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> structures on the <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">S^7</annotation></semantics></math> that are not <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphic</a> to the standard smooth structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">S^7</annotation></semantics></math>. More structurally, considering smooth structures up to <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> diffeomorphism, the different smooth structures form a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> under a (suitable) operation of <a class="existingWikiWord" href="/nlab/show/connected+sum">connected sum</a>, and this monoid is isomorphic to the <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>28</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}/(28)</annotation></semantics></math>. With the notable possible exception of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">n = 4</annotation></semantics></math> (where the question of existence of exotic 4-spheres is wide open), exotic spheres first occur in dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>7</mn></mrow><annotation encoding="application/x-tex">7</annotation></semantics></math>. This phenomenon is connected to the <a class="existingWikiWord" href="/nlab/show/h-cobordism+theorem">h-cobordism theorem</a> (the monoid of smooth structures is identified with the monoid of h-cobordism classes of oriented <a class="existingWikiWord" href="/nlab/show/homotopy+spheres">homotopy spheres</a>).</p> <p>One explicit construction of the smooth structures is given as follows (see <a href="#Mil2">Milnor 1968</a>). Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">W_k</annotation></semantics></math> be the algebraic variety in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}^5</annotation></semantics></math> defined by the equation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>z</mi> <mn>1</mn> <mrow><mn>6</mn><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>+</mo><msubsup><mi>z</mi> <mn>2</mn> <mn>3</mn></msubsup><mo>+</mo><msubsup><mi>z</mi> <mn>3</mn> <mn>2</mn></msubsup><mo>+</mo><msubsup><mi>z</mi> <mn>4</mn> <mn>2</mn></msubsup><mo>+</mo><msubsup><mi>z</mi> <mn>5</mn> <mn>2</mn></msubsup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">z_1^{6 k - 1} + z_2^3 + z_3^2 + z_4^2 + z_5^2 = 0</annotation></semantics></math></div> <p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>ϵ</mi></msub><mo>⊂</mo><msup><mi>ℂ</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">S_\epsilon \subset \mathbb{C}^5</annotation></semantics></math> a sphere of small radius <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> centered at the origin. Then each of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>28</mn></mrow><annotation encoding="application/x-tex">28</annotation></semantics></math> smooth structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">S^7</annotation></semantics></math> is represented by an intersection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>k</mi></msub><mo>∩</mo><msub><mi>S</mi> <mi>ϵ</mi></msub></mrow><annotation encoding="application/x-tex">W_k \cap S_\epsilon</annotation></semantics></math>, as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> ranges from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>28</mn></mrow><annotation encoding="application/x-tex">28</annotation></semantics></math>. These manifolds sometimes go by the name <em>Brieskorn manifolds</em> or <em><a class="existingWikiWord" href="/nlab/show/Brieskorn+spheres">Brieskorn spheres</a></em> or <em><a class="existingWikiWord" href="/nlab/show/Milnor+spheres">Milnor spheres</a></em>.</p> <h3 id="G2Structure"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_2</annotation></semantics></math>-structure</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>0</mn></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>7</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi_0 \in \Omega^3(\mathbb{R}^7)</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/associative+3-form">associative 3-form</a> and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mn>0</mn></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>ℝ</mi><mo>⊕</mo><msup><mi>ℝ</mi> <mn>7</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Phi_0 \in \Omega^4(\mathbb{R} \oplus \mathbb{R}^7) </annotation></semantics></math></div> <p>be given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mn>0</mn></msub><mo>=</mo><mi>d</mi><msub><mi>x</mi> <mn>0</mn></msub><mo>∧</mo><msub><mi>ϕ</mi> <mn>0</mn></msub><mo>+</mo><mo>⋆</mo><msub><mi>ϕ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> \Phi_0 = d x_0 \wedge \phi_0 + \star \phi_0 </annotation></semantics></math></div> <p>(where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math> denotes the canonical coordinate on the first factor of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\phi_0</annotation></semantics></math> is pulled back along the projection to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^7</annotation></semantics></math>) .</p> <p>By construction this is its own <a class="existingWikiWord" href="/nlab/show/Hodge+dual">Hodge dual</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mo>=</mo><mo>⋆</mo><mi>Φ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Phi = \star \Phi \,. </annotation></semantics></math></div> <p>This implies that as we restrict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\Phi_0</annotation></semantics></math> to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>8</mn></msup><mo>−</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>≃</mo><mi>ℝ</mi><mo>×</mo><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex"> \mathbb{R}^8 - \{0\} \simeq \mathbb{R} \times S^7 </annotation></semantics></math></div> <p>then there is a unique 3-form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>7</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \phi \in \Omega^3(S^7) </annotation></semantics></math></div> <p>on the 7-sphere such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mn>0</mn></msub><mo>=</mo><msup><mi>r</mi> <mn>3</mn></msup><mo>∧</mo><mi>ϕ</mi><mo>+</mo><msup><mi>r</mi> <mn>4</mn></msup><msub><mo>⋆</mo> <mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow></msub><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>on</mi><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mn>8</mn></msup><mo>−</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Phi_0 = r^3 \wedge \phi + r^4 \star_{S^7} \phi \;\;\;\; (on \; \mathbb{R}^8 - \{0\}) \,. </annotation></semantics></math></div> <p>This 3-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> defines a <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-structure">G₂-structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">S^7</annotation></semantics></math>. It is <em>nearly parallel</em> in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>ϕ</mi><mo>=</mo><mn>4</mn><mo>⋆</mo><mi>ϕ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d \phi = 4 \star \phi \,. </annotation></semantics></math></div> <p>(e.g. <a href="#Lotay12">Lotay 12, def.2.4</a>)</p> <h2 id="related_concepts">Related concepts</h2> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unit+sphere">unit sphere</a>, <a class="existingWikiWord" href="/nlab/show/polar+coordinates">polar coordinates</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/round+sphere">round sphere</a>, <a class="existingWikiWord" href="/nlab/show/squashed+sphere">squashed sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hemisphere">hemisphere</a>, <a class="existingWikiWord" href="/nlab/show/equator">equator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stereographic+projection">stereographic projection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+spheres">homotopy groups of spheres</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+sphere">homotopy sphere</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+sphere">rational homotopy sphere</a>, <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+fibration">spherical fibration</a>, <a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a>, <a class="existingWikiWord" href="/nlab/show/stable+Cohomotopy+theory">stable Cohomotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology+sphere">homology sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+sphere">rational homotopy sphere</a>, <a class="existingWikiWord" href="/nlab/show/rational+n-sphere">rational n-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+sphere">motivic sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+actions+on+spheres">group actions on spheres</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+sphere">representation sphere</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+Cohomotopy">equivariant Cohomotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeb+sphere+theorem">Reeb sphere theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+packing">sphere packing</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Music+of+the+Spheres">Music of the Spheres</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/low+dimensional+topology">low dimensional</a> <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+projective+space">real projective space</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℝ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{R}P^1</annotation></semantics></math></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+projective+line">complex projective line</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{C}P^1</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+the+2-sphere">geometric quantization of the 2-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fuzzy+2-sphere">fuzzy 2-sphere</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fuzzy+3-sphere">fuzzy 3-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quaternionic+projective+line">quaternionic projective line</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℍ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{H}P^1</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fuzzy+4-sphere">fuzzy 4-sphere</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/5-sphere">5-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6-sphere">6-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/exotic+7-sphere">exotic 7-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/8-sphere">8-sphere</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/octonionic+projective+line">octonionic projective line</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>𝕆</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{O}P^1</annotation></semantics></math></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/13-sphere">13-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/15-sphere">15-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite-dimensional+sphere">infinite-dimensional sphere</a></p> </li> </ul> </div> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Martin+Cederwall">Martin Cederwall</a>, Christian R. Preitschopf, <em>The Seven-sphere and its Kac-Moody Algebra</em>, Commun. Math. Phys. 167 (1995) 373-394 (<a href="http://arxiv.org/abs/hep-th/9309030">arXiv:hep-th/9309030</a>)</p> </li> <li> <p>Takeshi Ôno, <em>On the Hopf fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">S^7 \to S^4</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></em>, Nagoya Math. J. Volume 59 (1975), 59-64. (<a href="http://projecteuclid.org/euclid.nmj/1118795554">Euclid</a>)</p> </li> </ul> <p>Relation to the <a class="existingWikiWord" href="/nlab/show/Milnor+fibration">Milnor fibration</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kenneth+Intriligator">Kenneth Intriligator</a>, <a class="existingWikiWord" href="/nlab/show/Hans+Jockers">Hans Jockers</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Mayr">Peter Mayr</a>, <a class="existingWikiWord" href="/nlab/show/David+Morrison">David Morrison</a>, M. Ronen Plesser, <em>Conifold Transitions in M-theory on Calabi-Yau Fourfolds with Background Fluxes</em>, Adv.Theor.Math.Phys. 17 (2013) 601-699 (<a href="http://arxiv.org/abs/1203.6662">arXiv:1203.6662</a>)</li> </ul> <p>An <a class="existingWikiWord" href="/nlab/show/ADE+classification">ADE classification</a> of finite subgroups of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(8)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/free+action">acting freely</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">S^7</annotation></semantics></math> (see at <em><a class="existingWikiWord" href="/nlab/show/group+action+on+an+n-sphere">group action on an n-sphere</a></em>) such that the quotient is <a class="existingWikiWord" href="/nlab/show/spin+structure">spin</a> and has at least four <a class="existingWikiWord" href="/nlab/show/Killing+spinors">Killing spinors</a> (see also at <a class="existingWikiWord" href="/nlab/show/ABJM+model">ABJM model</a>) is in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+de+Medeiros">Paul de Medeiros</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Figueroa-O%27Farrill">José Figueroa-O'Farrill</a>, Sunil Gadhia, <a class="existingWikiWord" href="/nlab/show/Elena+M%C3%A9ndez-Escobar">Elena Méndez-Escobar</a>, <em>Half-BPS quotients in M-theory: ADE with a twist</em>, JHEP 0910:038,2009 (<a href="http://arxiv.org/abs/0909.0163">arXiv:0909.0163</a>, <a href="http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf">pdf slides</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+de+Medeiros">Paul de Medeiros</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Figueroa-O%27Farrill">José Figueroa-O'Farrill</a>, <em>Half-BPS M2-brane orbifolds</em> (<a href="http://arxiv.org/abs/1007.4761">arXiv:1007.4761</a>)</p> </li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</a>:</p> <ul> <li id="Varadarajan01"><a class="existingWikiWord" href="/nlab/show/Veeravalli+Varadarajan">Veeravalli Varadarajan</a>, <em>Spin(7)-subgroups of SO(8) and Spin(8)</em>, Expositiones Mathematicae, 19 (2001): 163-177 (<a href="https://core.ac.uk/download/pdf/81114499.pdf">pdf</a>)</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structures">exotic smooth structures</a> on 7-spheres includes</p> <ul> <li>Wikipedia, <em>Exotic sphere</em>, <a href="https://en.wikipedia.org/wiki/Exotic_sphere">link</a>.</li> </ul> <p>The explicit construction of exotic 7-spheres by intersecting algebraic varieties with spheres is described in</p> <ul> <li id="Mil2"><a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, “Singular points of complex hypersurfaces” , Princeton Univ. Press (1968).</li> </ul> <p>Discussion of (nearly) <a class="existingWikiWord" href="/nlab/show/G2-structures">G2-structures</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">S^7</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/calibrated+submanifolds">calibrated submanifolds</a> includes</p> <ul> <li id="Lotay12"><a class="existingWikiWord" href="/nlab/show/Jason+Lotay">Jason Lotay</a>, <em>Associative Submanifolds of the 7-Sphere</em>, Proc. London Math. Soc. (2012) 105 (6): 1183-1214 (<a href="http://arxiv.org/abs/1006.0361">arXiv:1006.0361</a>, <a href="http://www.homepages.ucl.ac.uk/~ucahjdl/JDLotay_KIAS2011_slides.pdf">talk slides</a>)</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/coset">coset</a>-realizations:</p> <ul> <li id="Kramer98"> <p>Linus Kramer, <em>Octonion Hermitian quadrangles</em>, Bull. Belg. Math. Soc. Simon Stevin Volume 5, Number 2/3 (1998), 353-362 (<a href="https://projecteuclid.org/euclid.bbms/1103409015">euclid:1103409015</a>)</p> </li> <li id="Choquet-Bruhat+DeWitt-Morette00"> <p><a class="existingWikiWord" href="/nlab/show/Yvonne+Choquet-Bruhat">Yvonne Choquet-Bruhat</a>, <a class="existingWikiWord" href="/nlab/show/C%C3%A9cile+DeWitt-Morette">Cécile DeWitt-Morette</a>, <em>Analysis, manifolds and physics</em>, Part II, North Holland (1982, 2001) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.elsevier.com/books/analysis-manifolds-and-physics-revised-edition/choquet-bruhat/978-0-444-86017-0">ISBN:9780444860170</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="AwadaDuffPope83"> <p><a class="existingWikiWord" href="/nlab/show/Moustafa+A.+Awada">Moustafa A. Awada</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">N=8</annotation></semantics></math> Supergravity Breaks Down to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math></em>, Phys. Rev. Lett. <strong>50</strong> 5 (1983) 294-297 [<a href="https://doi.org/10.1103/PhysRevLett.50.294">doi:10.1103/PhysRevLett.50.294</a>]</p> </li> <li id="DuffNilssonPope83"> <p><a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, <a class="existingWikiWord" href="/nlab/show/Bengt+Nilsson">Bengt Nilsson</a>, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, <em>Spontaneous Supersymmetry Breaking by the Squashed Seven-Sphere</em>, Phys. Rev. Lett. 50, 2043 – Published 27 June 1983; Erratum Phys. Rev. Lett. 51, 846 (<a href="https://doi.org/10.1103/PhysRevLett.50.2043">doi:10.1103/PhysRevLett.50.2043</a>)</p> </li> <li id="Baez"> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <em>Rotations in the 7th Dimension</em>, (<a href="https://golem.ph.utexas.edu/category/2007/09/rotations_in_the_7th_dimension.html">blog post</a>), and <em>TWF 195</em>, (<a href="http://math.ucr.edu/home/baez/week195.html">webpage</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 18, 2024 at 10:44:26. 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