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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='#Differential'>Basic differential geometry</a></li> <li><a href='#CosetSpaceStructure'>Coset space structure</a></li> <li><a href='#HomotopyGroups'>Homotopy groups</a></li> <li><a href='#bundles_over_the_4sphere'>Bundles over the 4-sphere</a></li> <ul> <li><a href='#the_quaternionic_hopf_fibration'>The quaternionic Hopf fibration</a></li> <li><a href='#the_calabipenrose_fibration'>The Calabi-Penrose fibration</a></li> <li><a href='#AsAQuotientOfTheComplexProjectivePlane'>The complex projective plane</a></li> </ul> <li><a href='#exotic_smooth_structures'>Exotic smooth structures</a></li> <li><a href='#QuaternionAction'><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> action</a></li> <ul> <li><a href='#CircleAction'>Circle action</a></li> <li><a href='#m5brane_orbifolds'>M5-brane orbifolds</a></li> </ul> <li><a href='#FreeLoopSpace'>Free and cyclic loop space</a></li> <li><a href='#diffeomorphism_group'>Diffeomorphism group</a></li> </ul> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#branched_covers'>Branched covers</a></li> </ul> </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> 4.</p> <h2 id="properties">Properties</h2> <h3 id="Differential">Basic differential geometry</h3> <p>An embedded radius-<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">R</annotation></semantics></math> 4-sphere inherits a <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> (a degree-4 <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a> from the ambient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>5</mn></mrow><annotation encoding="application/x-tex">5</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a>, namely</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><mfrac><mn>1</mn><mi>R</mi></mfrac><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow> <mn>5</mn></munderover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mi>x</mi> <mi>j</mi></msub><mi>d</mi><msub><mi>x</mi> <mn>1</mn></msub><mo>∧</mo><mi>…</mi><mo>∧</mo><mover><mrow><mi>d</mi><msub><mi>x</mi> <mi>j</mi></msub></mrow><mo>^</mo></mover><mo>∧</mo><mi>…</mi><mo>∧</mo><mi>d</mi><msub><mi>x</mi> <mn>5</mn></msub></mrow><annotation encoding="application/x-tex"> \omega = \frac{1}{R} \sum_{j=1}^{5} (-1)^{j-1} x_j d x_1 \wedge \ldots \wedge \widehat{d x_j} \wedge \ldots \wedge d x_{5} </annotation></semantics></math></div> <p>where the hat means omit that factor. This is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mi>d</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">\ast d r</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Hodge+star+operator">Hodge star operator</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^5</annotation></semantics></math> for the Euclidean <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">metric</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">d r</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/exterior+derivative">exterior derivative</a> of the radius function.</p> <p>The volume of the manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">S^4</annotation></semantics></math> with this volume form is then given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>8</mn><msup><mi>π</mi> <mn>2</mn></msup><msup><mi>R</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">8\pi^2R^4/3</annotation></semantics></math>.</p> <h3 id="CosetSpaceStructure">Coset space structure</h3> <p>As any <a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, the <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a> has the <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> <a class="existingWikiWord" href="/nlab/show/structure">structure</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>4</mn></msup><mo>≃</mo><mi>O</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>Pin</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Pin</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^4 \simeq O(5)/O(4) \simeq SO(5)/SO(4) \simeq Spin(5)/Spin(4)\simeq Pin(5)/Pin(4). </annotation></semantics></math></div> <p>There is also this:</p> <div class="num_example" id="Sp2Sp1BySp1Sp1Sp1IsS4"> <h6 id="example">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> of <a class="existingWikiWord" href="/nlab/show/Sp%282%29.Sp%281%29">Sp(2).Sp(1)</a> (<a href="SpnSp1#SpnSp1">this Def.</a>) by <a class="existingWikiWord" href="/nlab/show/Sp%281%29Sp%281%29Sp%281%29">Sp(1)Sp(1)Sp(1)</a> (<a href="SpnSp1#Spin4Spin3">this Def.</a>) is the <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mi>S</mi> <mn>4</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{ Sp(2)\cdot Sp(1) } { Sp(1)Sp(1)Sp(1) } \;\simeq\; S^4 \,. </annotation></semantics></math></div> <p>This follows essentially from the <a class="existingWikiWord" href="/nlab/show/quaternionic+Hopf+fibration">quaternionic Hopf fibration</a> and its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(2)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+function">equivariance</a>…</p> </div> <p>(e.g. <a href="#BettiolMendes15">Bettiol-Mendes 15, (3.1), (3.2), (3.3)</a>)</p> <h3 id="HomotopyGroups">Homotopy groups</h3> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of the 4-sphere in low degree are:</p> <table><thead><tr><th><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></th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</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><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_k(S^4)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;"><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">\mathbb{Z}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>×</mo><msub><mi>ℤ</mi> <mn>12</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z} \times \mathbb{Z}_{12}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ℤ</mi> <mn>2</mn> <mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2^2</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ℤ</mi> <mn>2</mn> <mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2^2</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>24</mn></msub><mo>×</mo><msub><mi>ℤ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{24} \times \mathbb{Z}_3</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>15</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{15}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td></tr> </tbody></table> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/Serre+finiteness+theorem">Serre finiteness theorem</a></em> and at <em><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+spheres">homotopy groups of spheres</a></em>.</p> <h3 id="bundles_over_the_4sphere">Bundles over the 4-sphere</h3> <h4 id="the_quaternionic_hopf_fibration">The quaternionic Hopf fibration</h4> <p>The 4-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> may be construed as</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>7</mn></msup></mtd> <mtd><mo>≃</mo><mi>S</mi><mo stretchy="false">(</mo><msup><mi>ℍ</mi> <mn>4</mn></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><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><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^7 &\simeq S(\mathbb{H}^4) \\ & \simeq \{(x, y) \in \mathbb{H}^2: {|x|}^2 + {|y|}^2 = 1\}, } </annotation></semantics></math></div> <p>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> parameterized 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> <p id="HopfParameterization"> There are other useful ways to parameterize the quaternionic Hopf fibration, such as the original <em><a class="existingWikiWord" href="/nlab/show/Hopf+construction">Hopf construction</a></em>, see there the section <em><a href="Hopf+construction#HopfFibrations">Hopf fibrations</a></em>. By this parameterization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">S^4</annotation></semantics></math> is identified as <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><mi>S</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo>⊕</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})</annotation></semantics></math>.</p> <h4 id="the_calabipenrose_fibration">The Calabi-Penrose fibration</h4> <p>See at <em><a class="existingWikiWord" href="/nlab/show/Calabi-Penrose+fibration">Calabi-Penrose fibration</a></em>.</p> <h4 id="AsAQuotientOfTheComplexProjectivePlane">The complex projective plane</h4> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Arnold-Kuiper-Massey+theorem">Arnold-Kuiper-Massey theorem</a>)</strong></p> <p>The 4-sphere is the <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a> of the <a class="existingWikiWord" href="/nlab/show/complex+projective+plane">complex projective plane</a> by the <a class="existingWikiWord" href="/nlab/show/action">action</a> of <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a> (on homogeneous coordinates):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mn>2</mn></msup><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>≃</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex"> \mathbb{C}P^2 / (-)^* \simeq S^4 </annotation></semantics></math></div></div> <h3 id="exotic_smooth_structures">Exotic smooth structures</h3> <p>It is open whether the 4-sphere admits an <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a>. See <a href="#FreedmanGompfMorrisonWalker09">Freedman, Gompf, Morrison & Walker 2009</a> for review.</p> <h3 id="QuaternionAction"><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> action</h3> <p>If we identify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>5</mn></msup><msub><mo>≃</mo> <mi>ℚ</mi></msub><mi>ℝ</mi><mo>⊕</mo><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^5 \simeq_{\mathbb{Q}} \mathbb{R} \oplus \mathbb{H}</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> with the <a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a> underlying the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a>, so that</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>4</mn></msup><mo>≃</mo><mi>S</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo>⊕</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S^4 \simeq S(\mathbb{R} \oplus \mathbb{H}) </annotation></semantics></math></div> <p>as in the discussion of the quaternionic Hopf fibration <a href="#HopfParameterization">above</a>, then there is induced an <a class="existingWikiWord" href="/nlab/show/action">action</a> of the group <a class="existingWikiWord" href="/nlab/show/special+unitary+group">SU(2)</a> on the 4-sphere, by identifying</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>S</mi><mo stretchy="false">(</mo><mi>ℚ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> SU(2) \simeq S(\mathbb{Q}) </annotation></semantics></math></div> <p>and then acting by left multiplication.</p> <h4 id="CircleAction">Circle action</h4> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Given an continuous <a class="existingWikiWord" href="/nlab/show/action">action</a> of the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> on the <a class="existingWikiWord" href="/nlab/show/topological+space">topological</a> <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a>, its <a class="existingWikiWord" href="/nlab/show/fixed+point">fixed point</a> space is of one of two types:</p> <ol> <li> <p>either it is the <a class="existingWikiWord" href="/nlab/show/0-sphere">0-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup><mo>↪</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">S^0 \hookrightarrow S^4</annotation></semantics></math></p> </li> <li> <p>or it has the <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy type</a> of an even-dimensional sphere.</p> </li> </ol> </div> <p>(<a href="#FelixOpreaTanre08">Félix-Oprea-Tanré 08, Example 7.39</a>)</p> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/group+actions+on+spheres">group actions on spheres</a></em>.</p> <p>As a special case of the <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>-action from <a href="#QuaternionAction">above</a>, we discuss the induced circle action via the embedding</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>1</mn></msup><mo>≃</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^1 \simeq U(1) \hookrightarrow SU(2) \,. </annotation></semantics></math></div> <p>Consider the following <a class="existingWikiWord" href="/nlab/show/circle+group">circle</a> <a class="existingWikiWord" href="/nlab/show/group+action+on+an+n-sphere">group action on the 4-sphere</a>:</p> <div class="num_defn" id="CircleActionOn4Sphere"> <h6 id="definition">Definition</h6> <p><strong>(<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>-action on 4-sphere)</strong></p> <p>Regard</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>4</mn></msup><mo>≃</mo><mi>S</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo>⊕</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S^4 \simeq S(\mathbb{R} \oplus \mathbb{H}) </annotation></semantics></math></div> <p>as the <a class="existingWikiWord" href="/nlab/show/unit+sphere">unit sphere</a> inside the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> (as <a class="existingWikiWord" href="/nlab/show/real+vector+spaces">real vector spaces</a>) of the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> with the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a>, and regard the <a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a> <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> as the group of unit-norm quaternions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>S</mi><mo stretchy="false">(</mo><mi>ℍ</mi><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> SU(2) \simeq S(\mathbb{H},\cdot) </annotation></semantics></math></div> <p>In particular this restricts to an <a class="existingWikiWord" href="/nlab/show/action">action</a> of the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</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>1</mn></msup><mo>≃</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S^1 \simeq U(1) \hookrightarrow SU(2) </annotation></semantics></math></div> <p>(as the <a class="existingWikiWord" href="/nlab/show/diagonal+matrices">diagonal matrices</a> inside <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>) on the 4-sphere.</p> </div> <p>The resulting ordinary <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>4</mn></msup><msub><mo stretchy="false">/</mo> <mi>ord</mi></msub><msup><mi>S</mi> <mn>1</mn></msup><mo>≃</mo><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^4/_{ord} S^1 \simeq S^3</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/projection">projection</a> <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><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^4 \to S^3</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/suspension">suspension</a> of the <a class="existingWikiWord" href="/nlab/show/complex+Hopf+fibration">complex Hopf fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^3 \to S^2</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/fixed+point">fixed point</a> set of the action is the two poles</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>0</mn></msup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mo>±</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>ℝ</mi><mo>⊕</mo><mi>ℍ</mi></mrow><annotation encoding="application/x-tex"> S^0 \;=\; \{(\pm 1, 0,0,0,0)\} \;\in\; \mathbb{R} \oplus \mathbb{H} </annotation></semantics></math></div> <p>introduced by the suspension, hence forms the <a class="existingWikiWord" href="/nlab/show/0-sphere">0-sphere</a> space. Since this is not the <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a>, the <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> <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 stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^4 // S^1</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/circle+action">circle action</a> differs from <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>, but there is still the canonical <a class="existingWikiWord" href="/nlab/show/projection">projection</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>4</mn></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>⟶</mo><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>≃</mo><msup><mi>S</mi> <mn>3</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^4//S^1 \longrightarrow S^4 / S^1 \simeq S^3 \,. </annotation></semantics></math></div> <p>Hence both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">S^4</annotation></semantics></math> and <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 stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^4 // S^1</annotation></semantics></math> are canonically <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> over <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>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/minimal+dg-module">minimal dg-module</a> presentation in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> for these projections is given in <a href="#RoigSaralegiAranguren00">Roig & Saralegi-Aranguren 00, second page</a>:</p> <div class="num_prop" id="FourSphereOverThreeSphereMinimalDgModel"> <h6 id="proposition_3">Proposition</h6> <p><strong>(<a href="#RoigSaralegiAranguren00">Roig & Saralegi-Aranguren 00, p. 2</a>)</strong></p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔩</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>3</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msup><mi>Sym</mi> <mo>•</mo></msup><mo stretchy="false">⟨</mo><munder><munder><mrow><msub><mi>h</mi> <mn>3</mn></msub></mrow><mo>⏟</mo></munder><mtext>deg 3</mtext></munder><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> CE(\mathfrak{l}(S^3))) = Sym^\bullet \langle \underset{\text{deg 3}}{\underbrace{h_3}} \rangle </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/minimal+Sullivan+model">minimal Sullivan model</a> of the <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a>. Then <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a> <a class="existingWikiWord" href="/nlab/show/minimal+dg-modules">minimal dg-modules</a> for the maps (via Def. <a class="maruku-ref" href="#CircleActionOn4Sphere"></a>)</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>4</mn></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>S</mi> <mn>3</mn></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo><mphantom><mi>AA</mi></mphantom><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>S</mi> <mn>3</mn></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo><mphantom><mi>AA</mi></mphantom><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mn>0</mn></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>S</mi> <mn>3</mn></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^4 \\ \downarrow \\ S^3 } \,,\phantom{AA} \array{ S^4//S^1 \\ \downarrow \\ S^3 } \,,\phantom{AA} \array{ S^0 \\ \downarrow \\ S^3 } </annotation></semantics></math></div> <p>as <a class="existingWikiWord" href="/nlab/show/dg-modules">dg-modules</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔩</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>3</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{l}(S^3))</annotation></semantics></math> are given as follows, respectively:</p> <div class="maruku-equation" id="eq:FourSphereAndRelatedOverThreeSphereMinimalDGModels"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mtext>fibration</mtext></mtd> <mtd><mrow><mtable><mtr><mtd><mtext>vector space underlying</mtext></mtd></mtr> <mtr><mtd><mtext>minimal dg-model</mtext></mtd></mtr></mtable></mrow></mtd> <mtd><mrow><mtable><mtr><mtd><mtext>differential on</mtext></mtd></mtr> <mtr><mtd><mtext>minimal dg-model</mtext></mtd></mtr></mtable></mrow></mtd></mtr> <mtr><mtd><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mn>4</mn></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>S</mi> <mn>3</mn></msup></mtd></mtr></mtable></mrow></mtd> <mtd><msup><mi>Sym</mi> <mo>•</mo></msup><mo stretchy="false">⟨</mo><munder><munder><mrow><msub><mi>h</mi> <mn>3</mn></msub></mrow><mo>⏟</mo></munder><mrow><mi>deg</mi><mo>=</mo><mn>3</mn></mrow></munder><mo stretchy="false">⟩</mo><mo>⊗</mo><mrow><mo>⟨</mo><munder><munder><mrow><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mrow><mn>2</mn><mi>p</mi></mrow></msub></mrow><mo>⏟</mo></munder><mrow><mi>deg</mi><mo>=</mo><mn>2</mn><mi>p</mi></mrow></munder><mo>,</mo><munder><munder><mrow><msub><mi>ω</mi> <mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn></mrow></msub></mrow><mo>⏟</mo></munder><mrow><mi>deg</mi><mo>=</mo><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn></mrow></munder><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mi>p</mi><mo>∈</mo><mi>ℕ</mi><mo>⟩</mo></mrow></mtd> <mtd><mi>d</mi><mo lspace="verythinmathspace">:</mo><mrow><mo>{</mo><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mn>0</mn></msub></mtd> <mtd><mo>↦</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msub></mtd> <mtd><mo>↦</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mrow><mn>2</mn><mi>p</mi></mrow></msub></mtd></mtr> <mtr><mtd><msub><mi>ω</mi> <mn>4</mn></msub></mtd> <mtd><mo>↦</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>ω</mi> <mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>6</mn></mrow></msub></mtd> <mtd><mo>↦</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn></mrow></msub></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mn>0</mn></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>S</mi> <mn>3</mn></msup></mtd></mtr></mtable></mrow></mtd> <mtd><msup><mi>Sym</mi> <mo>•</mo></msup><mo stretchy="false">⟨</mo><munder><munder><mrow><msub><mi>h</mi> <mn>3</mn></msub></mrow><mo>⏟</mo></munder><mrow><mi>deg</mi><mo>=</mo><mn>3</mn></mrow></munder><mo stretchy="false">⟩</mo><mo>⊗</mo><mrow><mo>⟨</mo><munder><munder><mrow><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mrow><mn>2</mn><mi>p</mi></mrow></msub></mrow><mo>⏟</mo></munder><mrow><mi>deg</mi><mo>=</mo><mn>2</mn><mi>p</mi></mrow></munder><mo>,</mo><munder><munder><mrow><msub><mi>ω</mi> <mrow><mn>2</mn><mi>p</mi></mrow></msub></mrow><mo>⏟</mo></munder><mrow><mi>deg</mi><mo>=</mo><mn>2</mn><mi>p</mi></mrow></munder><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mi>p</mi><mo>∈</mo><mi>ℕ</mi><mo>⟩</mo></mrow></mtd> <mtd><mi>d</mi><mo lspace="verythinmathspace">:</mo><mrow><mo>{</mo><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mn>0</mn></msub></mtd> <mtd><mo>↦</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msub></mtd> <mtd><mo>↦</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mrow><mn>2</mn><mi>p</mi></mrow></msub></mtd></mtr> <mtr><mtd><msub><mi>ω</mi> <mn>0</mn></msub></mtd> <mtd><mo>↦</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>ω</mi> <mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msub></mtd> <mtd><mo>↦</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mrow><mn>2</mn><mi>p</mi></mrow></msub></mtd></mtr></mtable></mrow></mrow></mtd></mtr> <mtr><mtd><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>S</mi> <mn>3</mn></msup></mtd></mtr></mtable></mrow></mtd> <mtd><msup><mi>Sym</mi> <mo>•</mo></msup><mo stretchy="false">⟨</mo><munder><munder><mrow><msub><mi>h</mi> <mn>3</mn></msub></mrow><mo>⏟</mo></munder><mrow><mi>deg</mi><mo>=</mo><mn>3</mn></mrow></munder><mo>,</mo><munder><munder><mrow><msub><mi>ω</mi> <mn>2</mn></msub></mrow><mo>⏟</mo></munder><mrow><mi>deg</mi><mo>=</mo><mn>2</mn></mrow></munder><mo stretchy="false">⟩</mo><mo>⊗</mo><mrow><mo>⟨</mo><munder><munder><mrow><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mrow><mn>2</mn><mi>p</mi></mrow></msub></mrow><mo>⏟</mo></munder><mrow><mi>deg</mi><mo>=</mo><mn>2</mn><mi>p</mi></mrow></munder><mo>,</mo><munder><munder><mrow><msub><mi>ω</mi> <mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn></mrow></msub></mrow><mo>⏟</mo></munder><mrow><mi>deg</mi><mo>=</mo><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn></mrow></munder><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mi>p</mi><mo>∈</mo><mi>ℕ</mi><mo>⟩</mo></mrow></mtd> <mtd><mi>d</mi><mo lspace="verythinmathspace">:</mo><mrow><mo>{</mo><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mn>0</mn></msub></mtd> <mtd><mo>↦</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msub></mtd> <mtd><mo>↦</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mrow><mn>2</mn><mi>p</mi></mrow></msub></mtd></mtr> <mtr><mtd><msub><mi>ω</mi> <mn>2</mn></msub></mtd> <mtd><mo>↦</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>ω</mi> <mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>4</mn></mrow></msub></mtd> <mtd><mo>↦</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mrow><mn>2</mn><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msub></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \text{fibration} & \array{\text{vector space underlying} \\ \text{minimal dg-model}} & \array{ \text{differential on} \\ \text{minimal dg-model} } \\ \array{ S^4 \\ \downarrow \\ S^3 } & Sym^\bullet \langle \underset{ deg = 3 }{ \underbrace{ h_3 }} \rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \tilde\omega_{2p} }}, \underset{deg = 2p + 4}{ \underbrace{ \omega_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \tilde\omega_0 & \mapsto 0 \\ \tilde\omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p} \\ \omega_4 & \mapsto 0 \\ \omega_{2p+6} & \mapsto h_3 \wedge \omega_{2p + 4} \end{aligned} \right. \\ \array{ S^0 \\ \downarrow \\ S^3 } & Sym^\bullet \langle \underset{ deg = 3 }{ \underbrace{ h_3 }} \rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \tilde \omega_{2p} }}, \underset{ deg = 2p }{ \underbrace{ \omega_{2p} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \tilde \omega_0 & \mapsto 0 \\ \tilde \omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p} \\ \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \end{aligned} \right. \\ \array{ S^4//S^1 \\ \downarrow \\ S^3 } & Sym^\bullet \langle \underset{ deg = 3 }{ \underbrace{ h_3 }} , \underset{ deg = 2 }{ \underbrace{ \omega_2 }} \rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \tilde \omega_{2p} }}, \underset{ deg =2p + 4 }{ \underbrace{ \omega_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \tilde \omega_0 & \mapsto 0 \\ \tilde \omega_{2p+2} &\mapsto h_3 \wedge \tilde \omega_{2p} \\ \omega_2 & \mapsto 0 \\ \omega_{2p+4} & \mapsto h_3 \wedge \omega_{2p + 2} \end{aligned} \right. } </annotation></semantics></math></div></div> <p>Beware that in the model for <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 stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^4//S^2</annotation></semantics></math> the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\omega_2</annotation></semantics></math> induces its entire polynomial algebra as generator of the dg-module.</p> <p>Notice that we changed the notation of the generators compared to <a href="#RoigSaralegiAranguren00">Roig & Saralegi-Aranguren 00, second page</a>, to bring out the pattern:</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>Roig<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>here<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></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><mphantom><mi>A</mi></mphantom><mi>a</mi><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}a\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><msub><mi>h</mi> <mn>3</mn></msub><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}h_3\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mn>1</mn><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}1\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mn>0</mn></msub><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\tilde\omega_0\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><msub><mi>c</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}c_{2n}\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\tilde\omega_{2n+2}\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><msub><mi>c</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}c_{2n+1}\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><msub><mi>ω</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>4</mn></mrow></msub><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\omega_{2n+4}\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mi>e</mi><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}e\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><msub><mi>ω</mi> <mn>2</mn></msub><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\omega_2\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><msub><mi>γ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\gamma_{2n}\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><msub><mover><mi>ω</mi><mo stretchy="false">˜</mo></mover> <mrow><mn>2</mn><mi>n</mi></mrow></msub><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\tilde\omega_{2n}\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><msub><mi>γ</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\gamma_{2n+1}\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><msub><mi>ω</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\omega_{2n}\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> <h4 id="m5brane_orbifolds">M5-brane orbifolds</h4> <p>The <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetric</a> <a class="existingWikiWord" href="/nlab/show/Freund-Rubin+compactifications">Freund-Rubin compactifications</a> of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> which are <a class="existingWikiWord" href="/nlab/show/Cartesian+products">Cartesian products</a> of 7-dimensional <a class="existingWikiWord" href="/nlab/show/anti-de+Sitter+spacetime">anti-de Sitter spacetime</a> with a compact 4-dimensional <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>7</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex"> AdS_7 \times X_4 </annotation></semantics></math></div> <p>(the <a class="existingWikiWord" href="/nlab/show/near+horizon+geometry">near horizon geometry</a> of a <a class="existingWikiWord" href="/nlab/show/black+brane">black</a> <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a>) are all of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>4</mn></msub><mo>≃</mo><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> X_4 \simeq S^4//G </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>⊂</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \subset SU(2)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/finite+group">finite</a> <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of <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> (i.e. an <a class="existingWikiWord" href="/nlab/show/ADE+classification">ADE group</a>), <a class="existingWikiWord" href="/nlab/show/action">acting</a> via the identification <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><mi>S</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo>⊕</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})</annotation></semantics></math> as <a href="#QuaternionAction">above</a>, and where the double slash denotes the <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> (<a class="existingWikiWord" href="/nlab/show/orbifold+quotient">orbifold quotient</a>).</p> <p>See (<a href="#AFHS98">AFHS 98, section 5.2</a>, <a href="#MF12">MF 12, section 8.3</a>).</p> <h3 id="FreeLoopSpace">Free and cyclic loop space</h3> <p>We discuss the <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> of the <a class="existingWikiWord" href="/nlab/show/free+loop+space">free loop space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{L}(S^4)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">S^4</annotation></semantics></math>, as well as the <a class="existingWikiWord" href="/nlab/show/cyclic+loop+space">cyclic loop space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathcal{L}(S^4)/S^1</annotation></semantics></math> using the results from <em><a class="existingWikiWord" href="/nlab/show/Sullivan+models+of+free+loop+spaces">Sullivan models of free loop spaces</a></em>:</p> <div class="num_example"> <h6 id="example_2">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">X = S^4</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a>. The corresponding <a class="existingWikiWord" href="/nlab/show/rational+n-sphere">rational n-sphere</a> has minimal Sullivan model</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">⟨</mo><msub><mi>g</mi> <mn>4</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>7</mn></msub><mo stretchy="false">⟩</mo><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\wedge^\bullet \langle g_4, g_7 \rangle, d) </annotation></semantics></math></div> <p>with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>g</mi> <mn>4</mn></msub><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>d</mi><msub><mi>g</mi> <mn>7</mn></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>g</mi> <mn>4</mn></msub><mo>∧</mo><msub><mi>g</mi> <mn>4</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d g_4 = 0\,,\;\;\;\; d g_7 = -\tfrac{1}{2} g_4 \wedge g_4 \,. </annotation></semantics></math></div> <p>Hence <a href="Sullivan+model+of+free+loop+space#SullivanModelForTheFreeLoopSpace">this prop.</a> gives for the rationalization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathcal{L}S^4</annotation></semantics></math> the model</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">⟨</mo><msub><mi>ω</mi> <mn>4</mn></msub><mo>,</mo><msub><mi>ω</mi> <mn>6</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>7</mn></msub><mo stretchy="false">⟩</mo><mo>,</mo><msub><mi>d</mi> <mrow><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ( \wedge^\bullet \langle \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4} ) </annotation></semantics></math></div> <p>with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>d</mi> <mrow><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup></mrow></msub><msub><mi>h</mi> <mn>3</mn></msub></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mrow><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup></mrow></msub><msub><mi>ω</mi> <mn>4</mn></msub></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mrow><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup></mrow></msub><msub><mi>ω</mi> <mn>6</mn></msub></mtd> <mtd><mo>=</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>4</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mrow><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup></mrow></msub><msub><mi>h</mi> <mn>7</mn></msub></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>ω</mi> <mn>4</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>4</mn></msub></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} d_{\mathcal{L}S^4} h_3 & = 0 \\ d_{\mathcal{L}S^4} \omega_4 & = 0 \\ d_{\mathcal{L}S^4} \omega_6 & = h_3 \wedge \omega_4 \\ d_{\mathcal{L}S^4} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 \\ \end{aligned} </annotation></semantics></math></div> <p>and <a href="Sullivan+model+of+free+loop+space#ModelForS1quotient">this prop</a> gives for the rationalization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathcal{L}S^4 / / S^1</annotation></semantics></math> the model</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">⟨</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>ω</mi> <mn>4</mn></msub><mo>,</mo><msub><mi>ω</mi> <mn>6</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>7</mn></msub><mo stretchy="false">⟩</mo><mo>,</mo><msub><mi>d</mi> <mrow><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ( \wedge^\bullet \langle \omega_2, \omega_4, \omega_6, h_3, h_7 \rangle , d_{\mathcal{L}S^4 / / S^1} ) </annotation></semantics></math></div> <p>with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>d</mi> <mrow><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><msub><mi>h</mi> <mn>3</mn></msub></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mrow><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><msub><mi>ω</mi> <mn>2</mn></msub></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mrow><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><msub><mi>ω</mi> <mn>4</mn></msub></mtd> <mtd><mo>=</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mrow><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><msub><mi>ω</mi> <mn>6</mn></msub></mtd> <mtd><mo>=</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>4</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>d</mi> <mrow><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><msub><mi>h</mi> <mn>7</mn></msub></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>ω</mi> <mn>4</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>4</mn></msub><mo>+</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>6</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} d_{\mathcal{L}S^4 / / S^1} h_3 & = 0 \\ d_{\mathcal{L}S^4 / / S^1} \omega_2 & = 0 \\ d_{\mathcal{L}S^4 / / S^1} \omega_4 & = h_3 \wedge \omega_2 \\ d_{\mathcal{L}S^4 / / S^1} \omega_6 & = h_3 \wedge \omega_4 \\ d_{\mathcal{L}S^4 / / S^1} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 + \omega_2 \wedge \omega_6 \end{aligned} \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>𝔤</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\hat \mathfrak{g} \to \mathfrak{g}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/central+extension">central</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extension">Lie algebra extension</a> by <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> of a finite dimensional Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>⟶</mo><mi>b</mi><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g} \longrightarrow b \mathbb{R}</annotation></semantics></math> be the corresponding <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra+cohomology">L-∞ 2-cocycle</a> with coefficients in the <a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie 2-algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">b \mathbb{R}</annotation></semantics></math>, hence (<a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">FSS 13, prop. 3.5</a>) so that there is a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>𝔤</mi><mo stretchy="false">^</mo></mover><mo>⟶</mo><mi>𝔤</mi><mover><mo>⟶</mo><mrow><msub><mi>ω</mi> <mn>2</mn></msub></mrow></mover><mi>b</mi><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> \hat \mathfrak{g} \longrightarrow \mathfrak{g} \overset{\omega_2}{\longrightarrow} b \mathbb{R} </annotation></semantics></math></div> <p>which is dually modeled by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mover><mi>𝔤</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>⊕</mo><mo stretchy="false">⟨</mo><mi>e</mi><mo stretchy="false">⟩</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>d</mi> <mover><mi>𝔤</mi><mo stretchy="false">^</mo></mover></msub><msub><mo stretchy="false">|</mo> <mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow></msub><mo>=</mo><msub><mi>d</mi> <mi>𝔤</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>d</mi> <mover><mi>𝔤</mi><mo stretchy="false">^</mo></mover></msub><mi>e</mi><mo>=</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> CE(\hat \mathfrak{g}) = ( \wedge^\bullet ( \mathfrak{g}^\ast \oplus \langle e \rangle ), d_{\hat \mathfrak{g}}|_{\mathfrak{g}^\ast} = d_{\mathfrak{g}},\; d_{\hat \mathfrak{g}} e = \omega_2) \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a space with <a class="existingWikiWord" href="/nlab/show/Sullivan+model">Sullivan model</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A_X,d_X)</annotation></semantics></math> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔩</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{l}(X)</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>, i.e. for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra whose <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A_X,d_X)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔩</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> CE(\mathfrak{l}X) = (A_X,d_X) \,. </annotation></semantics></math></div> <p>Then there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi></mrow></msub><mo stretchy="false">(</mo><mover><mi>𝔤</mi><mo stretchy="false">^</mo></mover><mo>,</mo><mi>𝔩</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi><mo stretchy="false">/</mo><mi>b</mi><mi>ℝ</mi></mrow></msub><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>𝔩</mi><mo stretchy="false">(</mo><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Hom_{L_\infty Alg}( \hat \mathfrak{g}, \mathfrak{l}(S^4) ) \;\simeq\; Hom_{L_\infty Alg/b \mathbb{R}}( \mathfrak{g}, \mathfrak{l}( \mathcal{L}S^4 / S^1 ) ) \,, </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔩</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{l}(S^4)</annotation></semantics></math> from <a href="Sullivan+model+of+free+loop+space#SullivanModelForTheFreeLoopSpace">this prop.</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔩</mi><mo stretchy="false">(</mo><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{l}(\mathcal{L}S^4 //S^1)</annotation></semantics></math> from <a href="Sullivan+model+of+free+loop+space#ModelForS1quotient">this prop.</a>, where on the right we have homs in the <a class="existingWikiWord" href="/nlab/show/slice+category">slice</a> over the <a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie 2-algebra</a>, via <a href="Sullivan+model+of+free+loop+space#ModelForS1quotient">this prop.</a></p> <p>Moreover, this isomorphism takes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>𝔤</mi><mo stretchy="false">^</mo></mover><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>4</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>7</mn></msub><mo stretchy="false">)</mo></mrow></mover><mi>𝔩</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \hat \mathfrak{g} \overset{(g_4, g_7)}{\longrightarrow} \mathfrak{l}(S^4) </annotation></semantics></math></div> <p>to</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><mi>𝔤</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>ω</mi> <mn>4</mn></msub><mo>,</mo><msub><mi>ω</mi> <mn>6</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>7</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mi>𝔩</mi><mo stretchy="false">(</mo><mi>ℒ</mi><mi>X</mi><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ω</mi> <mn>2</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>ω</mi> <mn>2</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>b</mi><mi>ℝ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathfrak{g} && \overset{(\omega_2,\omega_4, \omega_6, h_3,h_7)}{\longrightarrow} && \mathfrak{l}( \mathcal{L}X / S^1 ) \\ & {}_{\mathllap{\omega_2}}\searrow && \swarrow_{\mathrlap{\omega_2}} \\ && b \mathbb{R} } \,, </annotation></semantics></math></div> <p>where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>4</mn></msub><mo>=</mo><msub><mi>g</mi> <mn>4</mn></msub><mo>−</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><mi>e</mi><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>h</mi> <mn>7</mn></msub><mo>=</mo><msub><mi>g</mi> <mn>7</mn></msub><mo>+</mo><msub><mi>ω</mi> <mn>6</mn></msub><mo>∧</mo><mi>e</mi></mrow><annotation encoding="application/x-tex"> \omega_4 = g_4 - h_3 \wedge e \;\,, \;\;\; h_7 = g_7 + \omega_6 \wedge e </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> being the central generator in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mover><mi>𝔤</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\hat \mathfrak{g})</annotation></semantics></math> from above, and where the equations take place in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mover><mi>𝔤</mi><mo stretchy="false">^</mo></mover> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\wedge^\bullet \hat \mathfrak{g}^\ast</annotation></semantics></math> with the defining inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>↪</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\wedge^\bullet \mathfrak{g}^\ast \hookrightarrow \wedge^\bullet \mathfrak{g}^\ast</annotation></semantics></math> understood.</p> </div> <p>This is observed in (<a href="Sullivan+model+of+free+loop+space#FiorenzaSatiSchreiber16">FSS 16</a>, <a href="#FSS16b">FSS 16b</a>), where it serves to formalize, on the level of <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a>, the <a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a> of <a class="existingWikiWord" href="/nlab/show/M-branes">M-branes</a> in <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> to <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> in <a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a> (for the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is type IIA <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>9</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mn>16</mn></mstyle><mo>+</mo><mover><mstyle mathvariant="bold"><mn>16</mn></mstyle><mo>¯</mo></mover></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>𝔤</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \mathfrak{g}</annotation></semantics></math> is 11d <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mn>32</mn></mstyle></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{10,1\vert \mathbf{32}}</annotation></semantics></math>, and the cocycles are those of <a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">The brane bouquet</a>).</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>By the fact that the underlying graded algebras are free, and since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> is a generator of odd degree, the given decomposition for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\omega_4</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">h_7</annotation></semantics></math> is unique.</p> <p>Hence it is sufficient to observe that under this decomposition the defining equations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>g</mi> <mn>4</mn></msub><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>d</mi><msub><mi>g</mi> <mn>7</mn></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>g</mi> <mn>4</mn></msub><mo>∧</mo><msub><mi>g</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex"> d g_4 = 0 \,,\;\;\; d g_{7} = -\tfrac{1}{2} g_4 \wedge g_4 </annotation></semantics></math></div> <p>for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔩</mi><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathfrak{l}S^4</annotation></semantics></math>-valued cocycle on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>𝔤</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \mathfrak{g}</annotation></semantics></math> turn into the equations for a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔩</mi><mo stretchy="false">(</mo><mi>ℒ</mi><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{l} ( \mathcal{L}S^4 / S^1 )</annotation></semantics></math>-valued cocycle on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>. This is straightforward:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><msub><mi>d</mi> <mover><mi>𝔤</mi><mo stretchy="false">^</mo></mover></msub><mo stretchy="false">(</mo><msub><mi>ω</mi> <mn>4</mn></msub><mo>+</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mtd> <mtd><msub><mi>d</mi> <mi>𝔤</mi></msub><mo stretchy="false">(</mo><msub><mi>ω</mi> <mn>4</mn></msub><mo>−</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>d</mi> <mi>𝔤</mi></msub><msub><mi>h</mi> <mn>3</mn></msub><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mtd> <mtd><msub><mi>d</mi> <mi>𝔤</mi></msub><msub><mi>ω</mi> <mn>4</mn></msub><mo>=</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>d</mi> <mi>𝔤</mi></msub><msub><mi>h</mi> <mn>3</mn></msub><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & d_{\hat \mathfrak{g}} ( \omega_4 + h_3 \wedge e ) = 0 \\ \Leftrightarrow \;\;\;\; & d_{\mathfrak{g}} (\omega_4 - h_3 \wedge \omega_2) = 0 \;\;\; and \;\;\; d_{\mathfrak{g}} h_3 = 0 \\ \Leftrightarrow \;\;\;\; & d_{\mathfrak{g}} \omega_4 = h_3 \wedge \omega_2 \;\;\; and \;\;\; d_{\mathfrak{g}} h_3 = 0 \end{aligned} </annotation></semantics></math></div> <p>as well as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><msub><mi>d</mi> <mover><mi>𝔤</mi><mo stretchy="false">^</mo></mover></msub><mo stretchy="false">(</mo><msub><mi>h</mi> <mn>7</mn></msub><mo>−</mo><msub><mi>ω</mi> <mn>6</mn></msub><mo>∧</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi>ω</mi> <mn>4</mn></msub><mo>+</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><mi>e</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><msub><mi>ω</mi> <mn>4</mn></msub><mo>+</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><mi>e</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mtd> <mtd><msub><mi>d</mi> <mi>𝔤</mi></msub><msub><mi>h</mi> <mn>7</mn></msub><mo>−</mo><msub><mi>ω</mi> <mn>6</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>ω</mi> <mn>4</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>4</mn></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>−</mo><msub><mi>d</mi> <mi>𝔤</mi></msub><msub><mi>ω</mi> <mn>6</mn></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>4</mn></msub></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mtd> <mtd><msub><mi>d</mi> <mi>𝔤</mi></msub><msub><mi>h</mi> <mn>7</mn></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>ω</mi> <mn>4</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>4</mn></msub><mo>+</mo><msub><mi>ω</mi> <mn>6</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>d</mi> <mi>𝔤</mi></msub><msub><mi>h</mi> <mn>6</mn></msub><mo>=</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>4</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & d_{\hat \mathfrak{g}} ( h_7 - \omega_6 \wedge e ) = -\tfrac{1}{2}( \omega_4 + h_3 \wedge e ) \wedge (\omega_4 + h_3\wedge e) \\ \Leftrightarrow \;\;\;\; & d_\mathfrak{g} h_7 - \omega_6 \wedge \omega_2 = -\tfrac{1}{2}\omega_4 \wedge \omega_4 \;\;\; and \;\;\; - d_\mathfrak{g} \omega_6 = - h_3 \wedge \omega_4 \\ \Leftrightarrow \;\;\;\; & d_\mathfrak{g} h_7 = -\tfrac{1}{2}\omega_4 \wedge \omega_4 + \omega_6 \wedge \omega_2 \;\;\; and \;\;\; d_\mathfrak{g} h_6 = h_3 \wedge \omega_4 \end{aligned} </annotation></semantics></math></div></div> <p>The <a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit</a> of the <a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a>-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mi>ℒ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow></munder><mover><mo>⟵</mo><mi>hofib</mi></mover></munderover><mn>∞</mn><msub><mi>Grpd</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub></mrow><annotation encoding="application/x-tex"> \infty Grpd \underoverset {\underset{\mathcal{L}(-)/S^1}{\longrightarrow}} {\overset{hofib}{\longleftarrow}} {\bot} \infty Grpd_{/B S^1} </annotation></semantics></math></div> <p>(<a href="double+dimensional+reduction#GeneralReduction">this prop.</a>) applied to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+infinity-bundle">principal infinity-bundle</a></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>4</mn></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>hofib</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mtd> <mtd><munder><mo>⟶</mo><mi>c</mi></munder></mtd> <mtd><mi>B</mi><msup><mi>S</mi> <mn>1</mn></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^4 \\ {}^{\mathllap{hofib(c)}}\downarrow \\ S^4 / S^1 &\underset{c}{\longrightarrow}& B S^1 } </annotation></semantics></math></div> <p>is a natural map</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>4</mn></msup><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>⟶</mo><mi>ℒ</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex"> S^4/S^1 \longrightarrow \mathcal{L}(S^4)/S^1 </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> by the <a class="existingWikiWord" href="/nlab/show/circle+action">circle action</a> (def. <a class="maruku-ref" href="#CircleActionOn4Sphere"></a>), to the <a class="existingWikiWord" href="/nlab/show/cyclic+loop+space">cyclic loop space</a> of the 4-sphere.</p> <h3 id="diffeomorphism_group">Diffeomorphism group</h3> <p>Counterexamples (via <a class="existingWikiWord" href="/nlab/show/graph+complexes">graph complexes</a>) to the analogue of the <a class="existingWikiWord" href="/nlab/show/Smale+conjecture">Smale conjecture</a> for the 4-sphere are claimed in <a href="diffeomorphism+group#Watanabe18">Watanabe 18</a>, reviewed in <a href="diffeomorphism+group#Watanabe21">Watanabe 21</a>.</p> <h2 id="related_entries">Related entries</h2> <p><a class="existingWikiWord" href="/nlab/show/spheres+--+contents">spheres – contents</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fuzzy+4-sphere">fuzzy 4-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a></p> </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> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <ul> <li id="FreedmanGompfMorrisonWalker09"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Freedman">Michael Freedman</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Gompf">Robert Gompf</a>, <a class="existingWikiWord" href="/nlab/show/Scott+Morrison">Scott Morrison</a>, <a class="existingWikiWord" href="/nlab/show/Kevin+Walker">Kevin Walker</a>, <em>Man and machine thinking about the smooth 4-dimensional Poincaré conjecture</em>, Quantum Topology, Volume 1, Issue 2 (2010), pp. 171-208 (<a href="http://arxiv.org/abs/0906.5177">arXiv:0906.5177</a>)</p> </li> <li id="RoigSaralegiAranguren00"> <p><a class="existingWikiWord" href="/nlab/show/Agust%C3%AD+Roig">Agustí Roig</a>, <a class="existingWikiWord" href="/nlab/show/Martintxo+Saralegi-Aranguren">Martintxo Saralegi-Aranguren</a>, <em>Minimal Models for Non-Free Circle Actions</em>, Illinois Journal of Mathematics, volume 44, number 4 (2000) (<a href="https://arxiv.org/abs/math/0004141">arXiv:math/0004141</a>)</p> </li> <li id="AFHS98"> <p><a class="existingWikiWord" href="/nlab/show/Bobby+Acharya">Bobby Acharya</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Figueroa-O%27Farrill">José Figueroa-O'Farrill</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Hull">Chris Hull</a>, B. Spence, <em>Branes at conical singularities and holography</em> , Adv. Theor. Math. Phys. 2 (1998) 1249–1286</p> </li> <li id="FelixOpreaTanre08"> <p><a class="existingWikiWord" href="/nlab/show/Yves+F%C3%A9lix">Yves Félix</a>, John Oprea, <a class="existingWikiWord" href="/nlab/show/Daniel+Tanr%C3%A9">Daniel Tanré</a>, <em>Algebraic Models in Geometry</em>, Oxford University Press 2008</p> </li> <li id="MF12"> <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>, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (<a href="http://arxiv.org/abs/1007.4761">arXiv:1007.4761</a>, <a href="https://projecteuclid.org/euclid.atmp/1408561553">Euclid</a>)</p> </li> <li id="BettiolMendes15"> <p>Renato G. Bettiol, Ricardo A. E. Mendes, <em>Flag manifolds with strongly positive curvature</em>, Math. Z. 280 (2015), no. 3-4, 1031-1046 (<a href="https://arxiv.org/abs/1412.0039">arXiv:1412.0039</a>)</p> </li> <li> <p>Selman Akbulut, <em>Homotopy 4-spheres associated to an infinite order loose cork</em> (<a href="https://arxiv.org/abs/1901.08299">arXiv:1901.08299</a>)</p> </li> <li> <p>Akio Kawauchi, <em>Smooth homotopy 4-sphere</em> (<a href="https://arxiv.org/abs/1911.11904">arXiv:1911.11904</a>)</p> </li> <li> <p>David T. Gay, <em>Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins</em> (<a href="https://arxiv.org/abs/2102.12890">arXiv:2102.12890</a>)</p> </li> </ul> <h3 id="branched_covers">Branched covers</h3> <p>All <a class="existingWikiWord" href="/nlab/show/PL+manifold">PL</a> <a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a> are <em>simple</em> <a class="existingWikiWord" href="/nlab/show/branched+covers">branched covers</a> of the <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a>:</p> <ul> <li id="Piergallini95"> <p><a class="existingWikiWord" href="/nlab/show/Riccardo+Piergallini">Riccardo Piergallini</a>, <em>Four-manifolds as 4-fold branched covers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">S^4</annotation></semantics></math></em>, Topology Volume 34, Issue 3, July 1995 (<a href="https://doi.org/10.1016/0040-9383(94)00034-I">doi:10.1016/0040-9383(94)00034-I</a>, <a href="https://core.ac.uk/download/pdf/82379618.pdf">pdf</a>)</p> </li> <li id="IoriPiergallini02"> <p>Massimiliano Iori, <a class="existingWikiWord" href="/nlab/show/Riccardo+Piergallini">Riccardo Piergallini</a>, <em>4-manifolds as covers of the 4-sphere branched over non-singular surfaces</em>, Geom. Topol. 6 (2002) 393-401 (<a href="https://arxiv.org/abs/math/0203087">arXiv:math/0203087</a>)</p> </li> </ul> <p id="ConnesOnCohomotopy"> Speculative remarks on the possible role of maps from <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> to the <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a> in some kind of <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a> via <a href="spectral+triple">spectral geometry</a> (related to the <a class="existingWikiWord" href="/nlab/show/Connes-Lott-Chamseddine-Barrett+model">Connes-Lott-Chamseddine-Barrett model</a>) are in</p> <ul> <li id="ChamseddineConnesMukhanov14"> <p><a class="existingWikiWord" href="/nlab/show/Ali+Chamseddine">Ali Chamseddine</a>, <a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, Viatcheslav Mukhanov, <em>Quanta of Geometry: Noncommutative Aspects</em>, Phys. Rev. Lett. 114 (2015) 9, 091302 (<a href="https://arxiv.org/abs/1409.2471">arXiv:1409.2471</a>)</p> </li> <li id="ChamseddineConnesMukhanov14"> <p><a class="existingWikiWord" href="/nlab/show/Ali+Chamseddine">Ali Chamseddine</a>, <a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, Viatcheslav Mukhanov, <em>Geometry and the Quantum: Basics</em>, JHEP 12 (2014) 098 (<a href="https://arxiv.org/abs/1411.0977">arXiv:1411.0977</a>)</p> </li> <li id="Connes17"> <p><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, section 4 of <em>Geometry and the Quantum</em>, in <em>Foundations of Mathematics and Physics One Century After Hilbert</em>, Springer 2018. 159-196 (<a href="https://arxiv.org/abs/1703.02470">arXiv:1703.02470</a>, <a href="https://www.springer.com/gp/book/9783319648125">doi:10.1007/978-3-319-64813-2</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, from 58:00 to 1:25:00 in <em>Why Four Dimensions and the Standard Model Coupled to Gravity - A Tentative Explanation From the New Geometric Paradigm of NCG</em>, talk at IHES, 2017 (<a href="https://www.youtube.com/watch?v=qVqqftQ92kA">video recording</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 27, 2024 at 07:08:59. 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