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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="higher_spin_geometry">Higher spin geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/spin+geometry">spin geometry</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/string+geometry">string geometry</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/fivebrane+geometry">fivebrane geometry</a></strong> …</p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a></p> </li> </ul> <h2 id="spin_geometry">Spin geometry</h2> <p><a class="existingWikiWord" href="/nlab/show/spin+geometry">spin geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/pin+group">pin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/semi-spin+group">semi-spin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/central+product+spin+group">central product spin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/spin%5Ec+group">spin^c group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+representation">spin representation</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor">spinor</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/charge+conjugation+matrix">charge conjugation matrix</a>, <a class="existingWikiWord" href="/nlab/show/Fierz+identity">Fierz identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/real+spin+representation">real spin representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+conjugate">Dirac conjugate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+spinor">Dirac spinor</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+spinor">Weyl spinor</a>, <a class="existingWikiWord" href="/nlab/show/Majorana+spinor">Majorana spinor</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/index+theory">index theory</a>, <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+equation">Dirac equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+field">Dirac field</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/rotation+groups">rotation groups</a> in <a class="existingWikiWord" href="/nlab/show/low-dimensional+topology">low</a> <a class="existingWikiWord" href="/nlab/show/dimensions">dimensions</a></strong>:</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/classification+of+simple+Lie+groups">Dynkin label</a></th><th><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">sp. orth. group</a></th><th><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></th><th><a class="existingWikiWord" href="/nlab/show/pin+group">pin group</a></th><th><a class="existingWikiWord" href="/nlab/show/semi-spin+group">semi-spin group</a></th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%282%29">SO(2)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%282%29">Spin(2)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%282%29">Pin(2)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%283%29">SO(3)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%283%29">Spin(3)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%283%29">Pin(3)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%284%29">SO(4)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%284%29">Spin(4)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%284%29">Pin(4)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%285%29">SO(5)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%285%29">Spin(5)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%285%29">Pin(5)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D3</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%286%29">SO(6)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%286%29">Spin(6)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B3</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%287%29">SO(7)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4">D4</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%288%29">SO(8)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%288%29">Spin(8)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a href="semi-spin+group#SemiSpin8">SO(8)</a></td></tr> <tr><td style="text-align: left;">B4</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%289%29">SO(9)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%289%29">Spin(9)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D5">D5</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2810%29">SO(10)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B5</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2811%29">SO(11)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2811%29">Spin(11)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D6">D6</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2812%29">SO(12)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2812%29">Spin(12)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</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>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D8</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2816%29">SO(16)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2816%29">Spin(16)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SemiSpin%2816%29">SemiSpin(16)</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</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>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D16</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2832%29">SO(32)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2832%29">Spin(32)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29">SemiSpin(32)</a></td></tr> </tbody></table> <p>see also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Spin%285%29.Spin%283%29">Spin(5).Spin(3)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+rotation+groups">finite rotation groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ADE-classification">ADE-classification</a></p> </li> </ul> </div> <h2 id="string_geometry">String geometry</h2> <p><a class="existingWikiWord" href="/nlab/show/string+geometry">string geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/string%5Ec+2-group">string^c 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/string%5Ec+structure">string^c structure</a></p> </li> </ul> <h2 id="fivebrane_geometry">Fivebrane geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></p> </li> </ul> <h2 id="ninebrane_geometry">Ninebrane geometry</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/ninebrane+10-group">ninebrane 10-group</a></li> </ul> </div></div> <h4 id="group_theory">Group Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+object">group object</a>, <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></li> <li><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+action">group action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></li> <li><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></li> <li><a class="existingWikiWord" href="/nlab/show/progroup">progroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>. <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a>, <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classification+of+finite+simple+groups">classification of finite simple groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sporadic+finite+simple+groups">sporadic finite simple groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+variety">abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topological+group">compact topological group</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kac-Moody+group">Kac-Moody group</a></p> </li> </ul> <p><strong>Super-Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> <p><strong>Higher groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>, <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-group">n-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+groupal+n-groupoid">k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>, <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-group">fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a>, <a class="existingWikiWord" href="/nlab/show/Ext-group">Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#group_extension'>Group extension</a></li> <li><a href='#general'>General</a></li> <li><a href='#AsHomotopyFiberOfSmoothW3'>As the homotopy fiber of the smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>W</mi></mstyle> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{W}_3</annotation></semantics></math></a></li> <li><a href='#relation_to_metaplectic_group_'>Relation to metaplectic group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Mp</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Mp^c</annotation></semantics></math></a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>By definition of the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n)</annotation></semantics></math> there is a canonical inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo>↪</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z}/2\mathbb{Z}\hookrightarrow Spin(n) </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/group+of+order+2">group of order 2</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Cl</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n)\hookrightarrow GL_1(Cl(\mathbb{R}^n))</annotation></semantics></math> canonically realized by even <a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a> elements of unit norm, this is given by the inclusion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{+1,-1\}</annotation></semantics></math>.</p> </div> <blockquote> <p>We frequently write <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> as shorthand for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2\mathbb{Z}</annotation></semantics></math>.</p> </blockquote> <div class="num_defn" id="DirectDefinitionOfSpinC"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin^c(n)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/quotient+group">quotient group</a></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><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Spin^c(n) & \coloneqq Spin(n) \times_{\mathbb{Z}_2} U(1) \\ & = (Spin(n) \times U(1))/{\mathbb{Z}_2} \,, \end{aligned} </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/product">product</a> of the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> with the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> by the common <a class="existingWikiWord" href="/nlab/show/subgroup">sub</a>-<a class="existingWikiWord" href="/nlab/show/group+of+order+2">group of order 2</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>↪</mo><mi>Spin</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \hookrightarrow Spin</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>↪</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \hookrightarrow U(1)</annotation></semantics></math> (i.e.: the <a class="existingWikiWord" href="/nlab/show/central+product+group">central product group</a>).</p> </div> <p>Usually the only the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n \geq 3</annotation></semantics></math> is considered.</p> <p>Some authors (e.g. <a href="#Gompf97">Gompf 97, p. 2</a>) denote this 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><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Spin^c(n) & \coloneqq Spin(n)\cdot Spin(2) \\ & \simeq Spin(n) \cdot U(1) \end{aligned} </annotation></semantics></math></div> <p>following the notation <a class="existingWikiWord" href="/nlab/show/Sp%28n%29.Sp%281%29">Sp(n).Sp(1)</a> (see <a href="SpnSp1#SpinnSpin2IsSpinc">there</a>).</p> <h2 id="examples">Examples</h2> <ul> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n=3</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/exceptional+isomorphism">exceptional isomorphism</a> between <a class="existingWikiWord" href="/nlab/show/Spin%283%29">Spin(3)</a> and <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a> extends to an isomorphism between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin^c(3)</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(2)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Spin^c(3) \;\simeq\; U(2) </annotation></semantics></math></div> <p>over the exceptional isomorphism <a class="existingWikiWord" href="/nlab/show/SO%283%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>SO</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>≃</mo> <mi>PU</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">SO(3) \simeq PU(2)</annotation> </semantics> </math></a>, as both of these <a class="existingWikiWord" href="/nlab/show/quotient+groups">quotient groups</a> are quotients by the respective <a class="existingWikiWord" href="/nlab/show/centers">centers</a>, both identifiable with the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>) (<a href="#OzbagciStipsicz">Ozbagci–Stipsicz 2004</a> Section 6.2). This isomorphism follows from considering the surjective homomorphism <a class="existingWikiWord" href="/nlab/show/SU%282%29"><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> <mo>×</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>→</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">SU(2) \times U(1) \to U(2)</annotation> </semantics> </math></a> given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>z</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">(A,z) \mapsto z A</annotation></semantics></math>, and noticing its <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> is precisely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo>±</mo><mo stretchy="false">(</mo><mi>I</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\pm(I,1)\}</annotation></semantics></math>, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(2)</annotation></semantics></math> satisfies the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin^c(3)</annotation></semantics></math> as a quotient.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">n=4</annotation></semantics></math>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>=</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo stretchy="false">/</mo><mo stretchy="false">{</mo><mo>±</mo><mo stretchy="false">(</mo><mi>I</mi><mo>,</mo><mi>I</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>≃</mo><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> Spin^c(4) = \big(SU(2)\times SU(2)\times U(1)\big) / \{\pm(I,I,1)\} \simeq U(2)\times_{U(1)} U(2). </annotation></semantics></math></div> <p>This latter group is the <a class="existingWikiWord" href="/nlab/show/fibre+product">fibre product</a> of groups over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>, namely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mi>det</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{(A,B)\in U(2)\times U(2)\mid \det(A) = \det(B)\}</annotation></semantics></math> (<a href="#OzbagciStipsicz">Ozbagci–Stipsicz 2004</a> Section 6.3). The construction can be seen by considering the surjective homomorphism</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>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><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><mo>×</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> SU(2) \times SU(2) \times U(1) \to SU(2) \times SU(2) \times U(1) \times U(2) \to U(2) \times U(2) </annotation></semantics></math></div> <p>defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>z</mi><mi>A</mi><mo>,</mo><mi>z</mi><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,B,z) \mapsto (z A,z B)</annotation></semantics></math>, which has image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(2)\times_{U(1)} U(2)</annotation></semantics></math>. Similarly to the case of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin^c(3)</annotation></semantics></math>, the kernel consists of triples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,B,z)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mi>A</mi><mo>=</mo><mi>I</mi><mo>=</mo><mi>z</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">z A = I = z B</annotation></semantics></math>, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">/</mo><mi>z</mi><mo stretchy="false">)</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">A = B = (1/z) I</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><msup><mi>z</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\det(A) = 1/z^2 = 1</annotation></semantics></math>, we must have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">z=\pm 1</annotation></semantics></math>, and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi><mo>=</mo><mo>±</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">A=B=\pm I</annotation></semantics></math> with the same sign as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math>. Thus the kernel is precisely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo>±</mo><mo stretchy="false">(</mo><mi>I</mi><mo>,</mo><mi>I</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\pm(I,I,1)\}</annotation></semantics></math>, and so again by the universal property we get the isomorphism as stated.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Spin%284%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>Spin</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">Spin(4)</annotation> </semantics> </math></a> subgroup can be seen as the subgroup of pairs of unitary matrices with both of them having determinant 1.</p> </li> <li> <p>Using the exceptional isomorphism <a class="existingWikiWord" href="/nlab/show/Spin%286%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>Spin</mi> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>≃</mo> <mi>SU</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">Spin(6) \simeq SU(4)</annotation> </semantics> </math></a>, and the multiplication map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(4)\times U(1) \to U(4)</annotation></semantics></math> analogous to the above, it can be seen that the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin^c(6)</annotation></semantics></math> is the connected <a class="existingWikiWord" href="/nlab/show/double+cover">double cover</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(4)</annotation></semantics></math> corresponding to the (unique) index-2 subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>ℤ</mi><mo>↪</mo><mi>ℤ</mi><mo>≃</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2\mathbb{Z} \hookrightarrow \mathbb{Z} \simeq \pi_1(U(4))</annotation></semantics></math>. This is because the multiplication map is surjective and has kernel canonically isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mu_4</annotation></semantics></math>, the fourth <a class="existingWikiWord" href="/nlab/show/roots+of+unity">roots of unity</a>, via the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mn>4</mn></msub><mo>→</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu_4\to SU(4)\times U(1)</annotation></semantics></math> sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ζ</mi><mo>↦</mo><mo stretchy="false">(</mo><msup><mi>ζ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>I</mi><mo>,</mo><mi>ζ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\zeta\mapsto (\zeta^{-1}I,\zeta)</annotation></semantics></math>. Hence there are a pair of 2:1 surjective homomorphisms</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>4</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex"> SU(4) \times U(1) \to Spin^c(6) \to U(4), </annotation></semantics></math></div> <p>and hence the result.</p> </li> </ul> <p>The last two examples can also be considered as <a class="existingWikiWord" href="/nlab/show/exceptional+isomorphisms">exceptional isomorphisms</a>, even if not to one of the more <a class="existingWikiWord" href="/nlab/show/classical+Lie+groups">classical Lie groups</a>.</p> <h2 id="properties">Properties</h2> <h3 id="group_extension">Group extension</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>We have a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Spin</mi> <mi>c</mi></msup><mo>→</mo><mi>SO</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> U(1) \to Spin^c \to SO \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">U(1) \to Spin^c</annotation></semantics></math> is the canonical inclusion into the defining product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>Spin</mi><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>Spin</mi><msub><mo>×</mo> <mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1) \to Spin \times U(1) \to Spin \times_{\mathbb{Z}_2} U(1)</annotation></semantics></math>.</p> </div> <h3 id="general">General</h3> <h3 id="AsHomotopyFiberOfSmoothW3">As the homotopy fiber of the smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>W</mi></mstyle> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{W}_3</annotation></semantics></math></h3> <p>We discuss in the following that</p> <ol> <li> <p>the universal third <a class="existingWikiWord" href="/nlab/show/integral+Stiefel-Whitney+class">integral Stiefel-Whitney class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">W_3</annotation></semantics></math> has an essentially unique lift from <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> to <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>;</p> </li> <li> <p>the smooth <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup><mo>∈</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}Spin^c \in Smooth\infty Grpd</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>W</mi></mstyle> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{W}_3</annotation></semantics></math>, hence is the <a class="existingWikiWord" href="/nlab/show/circle+n-bundle">circle 2-bundle</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} SO</annotation></semantics></math> classified by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>W</mi></mstyle> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{W}_3</annotation></semantics></math>.</p> </li> </ol> <div class="num_prop" id="SpinCAsHomotopyPullbackOfW2AndC1"> <h6 id="proposition_2">Proposition</h6> <p>We have a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> diagram</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>det</mi></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>1</mn></msub><mi>mod</mi><mn>2</mn></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B} Spin^c &\stackrel{\mathbf{B}det}{\longrightarrow}& \mathbf{B}U(1) \\ \big\downarrow && \big\downarrow{}^{\mathrlap{\mathbf{c}_1 mod 2}} \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\longrightarrow}& \mathbf{B}^2 \mathbb{Z}_2 } </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>, where</p> <ul> <li> <p>the right morphism is the universal first <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a> modulo 2;</p> </li> <li> <p>the bottom morphism is the universal second <a class="existingWikiWord" href="/nlab/show/Stiefel-Whitney+class">Stiefel-Whitney class</a>.</p> </li> </ul> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>We present the sitation as usual in the projective <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> over <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> by <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-anafunctors">∞-anafunctors</a>.</p> <p>The first <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a> is given by the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-anafunctor">∞-anafunctor</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>ℤ</mi><mo>→</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>ℤ</mi><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>ℤ</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}(\mathbb{Z} \to \mathbb{R}) &\stackrel{\mathbf{c}_1}{\to}& \mathbf{B}(\mathbb{Z} \to 1) = \mathbf{B}^2 \mathbb{Z} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B} U(1) } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>G</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G_1 \to G_0)</annotation></semantics></math> denotes a presentation of a <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a> by a <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>.</p> <p>The second <a class="existingWikiWord" href="/nlab/show/Stiefel-Whitney+class">Stiefel-Whitney class</a> is given by</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><mi>Spin</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}(\mathbb{Z}_2 \to Spin) &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}(\mathbb{Z}_2 \to 1) = \mathbf{B}^2 \mathbb{Z}_2 \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B} SO } \,. </annotation></semantics></math></div> <p>Notice that the top horizontal morphism here is a <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a>.</p> <p>Therefore the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> in question is (as discussed there) given by the ordinary <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> in</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>Q</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>ℤ</mi><mo>→</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><mi>Spin</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Q &\to& \mathbf{B}(\mathbb{Z} \to \mathbb{R}) \\ \downarrow && \downarrow \\ \mathbf{B}(\mathbb{Z}_2 \to Spin) &\to& \mathbf{B}^2 \mathbb{Z}_2 } \,. </annotation></semantics></math></div> <p>This pullback is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>ℤ</mi><mover><mo>→</mo><mo>∂</mo></mover><mi>Spin</mi><mo>×</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}(\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R})</annotation></semantics></math>, where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mo lspace="verythinmathspace">:</mo><mi>n</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>n</mi><mspace width="thinmathspace"></mspace><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial\colon n \mapsto ( n \,mod\, 2 , n) \,. </annotation></semantics></math></div> <p>This is equivalent to</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><mo stretchy="false">(</mo><mi>ℤ</mi><mover><mo>→</mo><mo>∂</mo></mover><mi>Spin</mi><mo>×</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mover><mo>→</mo><mrow><mo>∂</mo><mo>′</mo></mrow></mover><mi>Spin</mi><mo>×</mo><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mover><mo>→</mo><mrow><mo>∂</mo><mo>′</mo></mrow></mover><mi>Spin</mi><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} (\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R}) & \simeq (\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times (\mathbb{R}/2\mathbb{Z})) \\ & \simeq (\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times U(1)) \end{aligned} \,, </annotation></semantics></math></div> <p>(notice the non-standard identification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1) \simeq \mathbb{R}/(2\mathbb{Z})</annotation></semantics></math> here, which is important below in prop. <a class="maruku-ref" href="#UniversalDeterminantLineBundleMap"></a> for the identification of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi></mrow><annotation encoding="application/x-tex">det</annotation></semantics></math>) where now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mo>′</mo></mrow><annotation encoding="application/x-tex">\partial'</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> embedding of the <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mo>′</mo><mo lspace="verythinmathspace">:</mo><mi>σ</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>σ</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial'\colon \sigma \mapsto (\sigma, \sigma) \,. </annotation></semantics></math></div> <p>This in turn is equivalent to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>Spin</mi><msub><mo>×</mo> <mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow></msub><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B} (Spin \times_{\mathbb{Z}_2} U(1)) \,, </annotation></semantics></math></div> <p>which is def. <a class="maruku-ref" href="#DirectDefinitionOfSpinC"></a>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Compare this with the similar but different <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> that defines the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}Spin &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } </annotation></semantics></math></div></div> <div class="num_prop" id="UniversalDeterminantLineBundleMap"> <h6 id="proposition_3">Proposition</h6> <p>Under the identificaton <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup><mo>≃</mo><mi>Spin</mi><munder><mo>×</mo><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow></munder><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin^c \simeq Spin \underset{\mathbb{Z}_2}{\times} U(1)</annotation></semantics></math> the “universal <a class="existingWikiWord" href="/nlab/show/determinant+line+bundle">determinant line bundle</a> map”</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo lspace="verythinmathspace">:</mo><msup><mi>Spin</mi> <mi>c</mi></msup><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> det \colon Spin^c \to U(1) </annotation></semantics></math></div> <p>is given in components by</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><mi>g</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>↦</mo><mn>2</mn><mi>c</mi></mrow><annotation encoding="application/x-tex"> (g,c) \mapsto 2 c </annotation></semantics></math></div> <p>(where on the right we write the group structure additively).</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By the proof of prop. <a class="maruku-ref" href="#SpinCAsHomotopyPullbackOfW2AndC1"></a> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-factor in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup><mo>≃</mo><mi>Spin</mi><munder><mo>×</mo><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow></munder><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin^c \simeq Spin \underset{\mathbb{Z}_2}{\times}U(1)</annotation></semantics></math> arises from the identification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">U(1) \simeq \mathbb{R}/2\mathbb{Z}</annotation></semantics></math>. But under the horizontal map as it appears in the homotopy pullback in that proof this corresponds to multiplication by 2.</p> </div> <div class="num_prop" id="SmoothRefinementOfBockstein"> <h6 id="proposition_4">Proposition</h6> <p>The third <em><a class="existingWikiWord" href="/nlab/show/integral+Stiefel-Whitney+class">integral Stiefel-Whitney class</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mn>3</mn></msub><mo>≔</mo><msub><mi>β</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>w</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mi>B</mi><mi>SO</mi><mover><mo>→</mo><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow></mover><msup><mi>B</mi> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mover><mo>→</mo><mrow><msub><mi>β</mi> <mn>2</mn></msub></mrow></mover><msup><mi>B</mi> <mn>3</mn></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> W_3 \coloneqq \beta_2 \circ w_2 \colon B SO \stackrel{w_2}{\to} B^2 \mathbb{Z}_2 \stackrel{\beta_2}{\to} B^3 \mathbb{Z} </annotation></semantics></math></div> <p>has an essentially unique lift through <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo lspace="verythinmathspace">:</mo></mrow><annotation encoding="application/x-tex">{\vert-\vert}\colon </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>→</mo><mi>Π</mi></mover></mrow><annotation encoding="application/x-tex">\stackrel{\Pi}{\to}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>→</mo><mo>≃</mo></mover></mrow><annotation encoding="application/x-tex">\stackrel{\simeq}{\to}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <p>given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>W</mi></mstyle> <mn>3</mn></msub><mo>=</mo><msub><mstyle mathvariant="bold"><mi>β</mi></mstyle> <mn>2</mn></msub><mo>∘</mo><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>β</mi></mstyle> <mn>2</mn></msub></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{W}_3 = \mathbf{\beta}_2 \circ \mathbf{w}_2 \colon \mathbf{B} SO(n) \stackrel{w_2}{\to} \mathbf{B}^2 \mathbb{Z}_2 \stackrel{\mathbf{\beta}_2}{\to} \mathbf{B}^2 U(1) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>β</mi></mstyle> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{\beta}_2</annotation></semantics></math> is simply given by the canonical subgroup embedding.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Once we establish that this is a lift at all, the essential uniqueness follows from the respective theorem at <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid+--+structures">smooth ∞-groupoid – structures</a>.</p> <p>The ordinary <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\beta_2</annotation></semantics></math> is presented by the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-anafunctor">∞-anafunctor</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><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mover><mo>→</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>ℤ</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z}) &\to& \mathbf{B}^2 (\mathbb{Z} \to 1) = \mathbf{B}^3 \mathbb{Z} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^2 \mathbb{Z}_2 } \,. </annotation></semantics></math></div> <p>Accordingly we need to lift the canonical presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>β</mi></mstyle> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{\beta}_2</annotation></semantics></math> to a comparable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-anafunctor. This is accomplished by</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><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mover><mo>→</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mover><mstyle mathvariant="bold"><mi>β</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mover><mo>→</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover><mi>ℝ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>β</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z}) &\stackrel{\hat \mathbf{\beta}_2}{\to}& \mathbf{B}^2 (\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{R}) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^2 \mathbb{Z}_2 &\stackrel{\mathbf{\beta}_2}{\to}& \mathbf{B}^2 U(1) } \,. </annotation></semantics></math></div> <p>Here the top horizontal morphism is induced from the morphism of <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>s that is given by the commuting diagram</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>ℤ</mi></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd><mi>ℤ</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>⋅</mo><mn>2</mn></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>⋅</mo><mn>2</mn></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ℤ</mi></mtd> <mtd><mover><mo>↪</mo><mrow></mrow></mover></mtd> <mtd><mi>ℝ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{Z} &\stackrel{id}{\to}& \mathbb{Z} \\ \downarrow^{\mathrlap{\cdot 2}} && \downarrow^{\mathrlap{\cdot 2}} \\ \mathbb{Z} &\stackrel{}{\hookrightarrow}& \mathbb{R} } \,. </annotation></semantics></math></div> <p>Since <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> is contractible, we have indeed under <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> an equivalence</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><mo stretchy="false">|</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mover><mo>→</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover><mi>ℤ</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo stretchy="false">|</mo><msub><mover><mstyle mathvariant="bold"><mi>β</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub><mo stretchy="false">|</mo></mrow></mover></mtd> <mtd><mo stretchy="false">|</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mover><mo>→</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover><mi>ℝ</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">|</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mover><mo>→</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover><mi>ℤ</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">|</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">|</mo><msup><mi>B</mi> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">|</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>β</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mo stretchy="false">|</mo><msup><mi>B</mi> <mn>3</mn></msup><mi>ℤ</mi><mo stretchy="false">|</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \vert\mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z})\vert &\stackrel{\vert \hat {\mathbf{\beta}}_2\vert}{\to}& \vert \mathbf{B}^2 (\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{R}) \vert \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \vert\mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z})\vert &\to& \vert\mathbf{B}^2(\mathbb{Z} \to 1)\vert \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \vert B^2 \mathbb{Z}_2\vert & \stackrel{\beta_2}{\to}& \vert B^3 \mathbb{Z}\vert } \,. </annotation></semantics></math></div></div> <div class="num_prop" id="HomotopyFiberOfSmoothBeta2"> <h6 id="proposition_5">Proposition</h6> <p>The sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>1</mn></msub><mi>mod</mi><mn>2</mn></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>β</mi></mstyle> <mn>2</mn></msub></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B} U(1) \stackrel{\mathbf{c}_1 mod 2}{\to} \mathbf{B}^2 \mathbb{Z}_2 \stackrel{\mathbf{\beta}_2}{\to} \mathbf{B}^2 U(1) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>β</mi></mstyle> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{\beta}_2</annotation></semantics></math> is the smoothly refined <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a> from prop. <a class="maruku-ref" href="#SmoothRefinementOfBockstein"></a>, is a <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>The homotopy fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B} \mathbb{Z}_2 \to \mathbf{B}U(1)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>≃</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)/\mathbb{Z}_2 \simeq U(1)</annotation></semantics></math>. Thinking of this is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℤ</mi><mover><mo>→</mo><mrow><mo>⋅</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></mover><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{Z} \stackrel{\cdot 1/2}{\to} \mathbb{R})</annotation></semantics></math> one sees that the inclusion of this fiber is indeed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>1</mn></msub><mi>mod</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbf{c}_1 mod 2</annotation></semantics></math>.</p> </div> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{B}Spin^c</annotation></semantics></math> of the Lie group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> is the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of the universal third smooth <a class="existingWikiWord" href="/nlab/show/integral+Stiefel-Whitney+class">integral Stiefel-Whitney class</a> from <a class="maruku-ref" href="#SmoothRefinementOfBockstein"></a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>W</mi></mstyle> <mn>3</mn></msub></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}Spin^c \to \mathbf{B} SO \stackrel{\mathbf{W}_3}{\to} \mathbf{B}^2 U(1) \,, </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>Then consider the <a class="existingWikiWord" href="/nlab/show/pasting+diagram">pasting diagram</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>1</mn></msub><mi>mod</mi><mn>2</mn></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>β</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}Spin^c &\to& \mathbf{B} U(1) &\to& {*} \\ \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}} && \downarrow \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 &\stackrel{\mathbf{\beta}_2}{\to}& \mathbf{B}^2 U(1) } \,. </annotation></semantics></math></div> <p>The right square is a homotopy pullback by prop. <a class="maruku-ref" href="#HomotopyFiberOfSmoothBeta2"></a>. The left square is a homotopy pullback by prop. <a class="maruku-ref" href="#SpinCAsHomotopyPullbackOfW2AndC1"></a>. The bottom composite is the smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>W</mi></mstyle> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{W}_3</annotation></semantics></math> by prop <a class="maruku-ref" href="#SmoothRefinementOfBockstein"></a>.</p> <p>This implies by claim by the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>.</p> </div> <h3 id="relation_to_metaplectic_group_">Relation to metaplectic group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Mp</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Mp^c</annotation></semantics></math></h3> <p>There is a direct analogy between <a class="existingWikiWord" href="/nlab/show/Spin">Spin</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec">Spin^c</a> and the <a class="existingWikiWord" href="/nlab/show/metaplectic+groups">metaplectic groups</a> <a class="existingWikiWord" href="/nlab/show/Mp">Mp</a> and <a class="existingWikiWord" href="/nlab/show/Mp%5Ec">Mp^c</a> (see there for more).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a>, <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">spin^c</annotation></semantics></math>-group</strong>, <a class="existingWikiWord" href="/nlab/show/spin%CA%B0+group">spinʰ group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSpin%E1%B6%9C">MSpinᶜ</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin%E1%B6%9C+structure">spinᶜ structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%E1%B6%9C+structure">twisted spinᶜ structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin%CA%B0+structure">spinʰ structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string%E1%B6%9C+2-group">stringᶜ 2-group</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/H.+Blaine+Lawson">H. Blaine Lawson</a>, <a class="existingWikiWord" href="/nlab/show/Marie-Louise+Michelsohn">Marie-Louise Michelsohn</a>, Appendix D in: <em><a class="existingWikiWord" href="/nlab/show/Spin+geometry">Spin geometry</a></em>, Princeton University Press (1989)</li> </ul> <p>For more see the references at <em><a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a></em>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/exceptional+isomorphisms">exceptional isomorphisms</a> in low dimension are described in</p> <ul> <li id="OzbagciStipsicz">B. Ozbagci, A. I. Stipsicz, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math> Structures on 3- and 4-Manifolds</em>, in: <em>Surgery on Contact 3-Manifolds and Stein Surfaces</em> Bolyai Society Mathematical Studies <strong>13</strong>, Springer (2004) [<a href="https://doi.org/10.1007/978-3-662-10167-4_6">doi:10.1007/978-3-662-10167-4_6</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 5, 2024 at 13:50:54. 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