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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; 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="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> <h4 id="lie_theory"><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>-Lie theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>)</p> <p><strong>Background</strong></p> <p><em>Smooth structure</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie groupoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie algebroids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> <p><strong>Formal Lie groupoids</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a></p> </li> </ul> <p><strong>Homotopy</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> <p><strong>Related topics</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><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>-Lie groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a></p> </li> </ul> <p><em><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>-Lie groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> </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></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <p><em><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>-Lie algebroids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><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>-Lie algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></p> </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='#properties'>Properties</a></li> <ul> <li><a href='#HomologyAndCohomology'>Homology and cohomology</a></li> <li><a href='#as_part_of_the_ade_pattern'>As part of the ADE pattern</a></li> <li><a href='#relation_to_orientation_of_manifolds'>Relation to orientation of manifolds</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>The <em>special orthogonal group</em> or <em>rotation group</em>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(n)</annotation></semantics></math>, is the <a class="existingWikiWord" href="/nlab/show/group">group</a> of <a class="existingWikiWord" href="/nlab/show/rotations">rotations</a> in a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>.</p> <p>This is one of the <a class="existingWikiWord" href="/nlab/show/classical+Lie+groups">classical Lie groups</a>. It is the <a class="existingWikiWord" href="/nlab/show/connected+component">connected component</a> of the neutral element in the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math>.</p> <p>For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n=2</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(2)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>.</p> <p>It is the first step in the <a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</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>⋯</mi><mo>→</mo><mi>Fivebrane</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,, </annotation></semantics></math></div> <p>the next step of which is the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/physics">physics</a> the rotation group is related to <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>.</p> <h2 id="properties">Properties</h2> <h3 id="HomologyAndCohomology">Homology and cohomology</h3> <p>On the <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> <a class="existingWikiWord" href="/nlab/show/BO%28n%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>B</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">B O(n)</annotation> </semantics> </math></a> and <a class="existingWikiWord" href="/nlab/show/BSO%28n%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>B</mi> <mi>SO</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">B SO(n)</annotation> </semantics> </math></a></p> <ul> <li> <p>with <a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+2"><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></a> <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a>:</p> <p>generated by the <a class="existingWikiWord" href="/nlab/show/Stiefel-Whitney+classes">Stiefel-Whitney classes</a></p> <p>(e.g. <a href="#MilnorStasheff74">Milnor & Stasheff 74, Theorem 7.1. & Theorem 12.4.</a>)</p> </li> <li> <p>with <a class="existingWikiWord" href="/nlab/show/integer">integer</a> <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> (<a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a>):</p> <p>generated by the <a class="existingWikiWord" href="/nlab/show/integral+Stiefel-Whitney+classes">integral Stiefel-Whitney classes</a> and the <a class="existingWikiWord" href="/nlab/show/Pontrjagin+classes">Pontrjagin classes</a></p> <p>(e.g. <a href="#Brown82">Brown 82</a>, <a href="#Feshbach83">Feshbach 83</a>, <a href="#Pittie91">Pittie 91</a>, <a href="#RudolphSchmidt17">Rudolph-Schmidt 17, Theorem 4.2.23</a>)</p> </li> </ul> <h3 id="as_part_of_the_ade_pattern">As part of the ADE pattern</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/ADE+classification">ADE classification</a></strong> and <strong><a class="existingWikiWord" href="/nlab/show/McKay+correspondence">McKay correspondence</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/Dynkin+diagram">Dynkin diagram</a>/ <br /> <a class="existingWikiWord" href="/nlab/show/Dynkin+quiver">Dynkin quiver</a></th><th><a class="existingWikiWord" href="/nlab/show/dihedron">dihedron</a>,<br /> <a class="existingWikiWord" href="/nlab/show/Platonic+solid">Platonic solid</a></th><th><a class="existingWikiWord" href="/nlab/show/classification+of+finite+rotation+groups">finite subgroups of SO(3)</a></th><th><a class="existingWikiWord" href="/nlab/show/classification+of+finite+rotation+groups">finite subgroups of SU(2)</a></th><th><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a></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>A</mi> <mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">A_{n \geq 1}</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{n+1}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{n+1}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(n+1)</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A1">A1</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+2">cyclic group of order 2</a> <br /> <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;"><a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+2">cyclic group of order 2</a> <br /> <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;"><a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A2">A2</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+3">cyclic group of order 3</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_3</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+3">cyclic group of order 3</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_3</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SU%283%29">SU(3)</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/A3">A3</a> <br /> = <br /> <a class="existingWikiWord" href="/nlab/show/D3">D3</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+4">cyclic group of order 4</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_4</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+4">cyclic group of order 4</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>D</mi> <mn>2</mn></msub><mo>≃</mo><msub><mi>ℤ</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">2 D_2 \simeq \mathbb{Z}_4</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SU%284%29">SU(4)</a> <br /> <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> <br /> <a class="existingWikiWord" href="/nlab/show/Spin%286%29">Spin(6)</a></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/dihedron">dihedron</a> on <br /> <a class="existingWikiWord" href="/nlab/show/bigon">bigon</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Klein+four-group">Klein four-group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mn>4</mn></msub><mo>≃</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>×</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quaternion+group">quaternion group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>D</mi> <mn>4</mn></msub><mo>≃</mo></mrow><annotation encoding="application/x-tex">2 D_4 \simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Q8">Q8</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%288%29">SO(8)</a>, <a class="existingWikiWord" href="/nlab/show/Spin%288%29">Spin(8)</a></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/dihedron">dihedron</a> on <br /> <a class="existingWikiWord" href="/nlab/show/triangle">triangle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dihedral+group+of+order+6">dihedral group of order 6</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">D_6</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/binary+dihedral+group+of+order+12">binary dihedral group of order 12</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>D</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">2 D_6</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2810%29">SO(10)</a>, <a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a></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/dihedron">dihedron</a> on <br /> <a class="existingWikiWord" href="/nlab/show/square">square</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dihedral+group+of+order+8">dihedral group of order 8</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">D_8</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/binary+dihedral+group+of+order+16">binary dihedral group of order 16</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>D</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">2 D_{8}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2812%29">SO(12)</a>, <a class="existingWikiWord" href="/nlab/show/Spin%2812%29">Spin(12)</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></msub></mrow><annotation encoding="application/x-tex">D_{n \geq 4}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dihedron">dihedron</a>, <br /> <a class="existingWikiWord" href="/nlab/show/hosohedron">hosohedron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dihedral+group">dihedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mrow><mn>2</mn><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">D_{2(n-2)}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/binary+dihedral+group">binary dihedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>D</mi> <mrow><mn>2</mn><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">2 D_{2(n-2)}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a>, <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(2n)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(2n)</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><msub><mi>E</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">E_6</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tetrahedron">tetrahedron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tetrahedral+group">tetrahedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/binary+tetrahedral+group">binary tetrahedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>T</mi></mrow><annotation encoding="application/x-tex">2T</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E6">E6</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">E_7</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cube">cube</a>, <br /> <a class="existingWikiWord" href="/nlab/show/octahedron">octahedron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/octahedral+group">octahedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/binary+octahedral+group">binary octahedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>O</mi></mrow><annotation encoding="application/x-tex">2O</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E7">E7</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dodecahedron">dodecahedron</a>, <br /> <a class="existingWikiWord" href="/nlab/show/icosahedron">icosahedron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/icosahedral+group">icosahedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/binary+icosahedral+group">binary icosahedral group</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>I</mi></mrow><annotation encoding="application/x-tex">2I</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/E8">E8</a></td></tr> </tbody></table> </div> <h3 id="relation_to_orientation_of_manifolds">Relation to orientation of manifolds</h3> <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> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> a lift of <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> classifying map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>ℬ</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to \mathcal{B}O(n)</annotation></semantics></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> to which the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> is the same as a choice of <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> of <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>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>For the almost degenerate case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math> there are exceptional <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SO</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>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> SO(2) \simeq U(1) \simeq Spin(2) </annotation></semantics></math></div> <p>with the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> and <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> in dimension 2.</p> </li> <li> <p>the case of <a class="existingWikiWord" href="/nlab/show/SO%288%29">SO(8)</a> is special, since in the <a class="existingWikiWord" href="/nlab/show/ADE+classification">ADE classification</a> of <a class="existingWikiWord" href="/nlab/show/simple+Lie+groups">simple Lie groups</a> it corresponds to <a class="existingWikiWord" href="/nlab/show/D4">D4</a>, which makes its <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a> enjoy <em><a class="existingWikiWord" href="/nlab/show/triality">triality</a></em>.</p> </li> </ul> <p><br /></p> <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> <p><br /></p> <h2 id="related_concepts">Related concepts</h2> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">\cdots \to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Fivebrane+group">Fivebrane group</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">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/string+group">string group</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">\to</annotation></semantics></math> <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><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <strong>special orthogonal group</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a>.</p> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/group">group</a></th><th>symbol</th><th><a class="existingWikiWord" href="/nlab/show/universal+cover">universal cover</a></th><th>symbol</th><th>higher cover</th><th>symbol</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{O}(n)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin+group">Pin group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pin(n)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Tring+group">Tring group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Tring</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Tring(n)</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(n)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin+group">Spin group</a></td><td style="text-align: left;"><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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/String+group">String group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">String(n)</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{O}(n,1)</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n,1)</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+group">anti de Sitter group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{O}(n,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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(n,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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+group">conformal group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{O}(n+1,t+1)</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"></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/Narain+group">Narain group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n,n)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></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/Poincar%C3%A9+group">Poincaré group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ISO</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ISO(n,1)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+spin+group">Poincaré spin group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ISO</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widehat {ISO}(n,1)</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+group">super Poincaré group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sISO</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sISO(n,1)</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superconformal+group">superconformal group</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler+angles">Euler angles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+rotation+group">finite rotation group</a></p> </li> </ul> <h2 id="References">References</h2> <p>For general references see also at <em><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></em>.</p> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a>:</p> <ul> <li id="MilnorStasheff74"> <p><a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Characteristic classes</em>, Princeton Univ. Press (1974) [<a href="https://press.princeton.edu/books/paperback/9780691081229/characteristic-classes-am-76-volume-76">ISBN:9780691081229</a>, <a href="https://doi.org/10.1515/9781400881826">doi:10.1515/9781400881826</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/milnstas.pdf">pdf</a>]</p> </li> <li id="Brown82"> <p><a class="existingWikiWord" href="/nlab/show/Edgar+H.+Brown">Edgar H. Brown</a>, <em>The Cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mi>SO</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">B SO_n</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>BO</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">BO_n</annotation></semantics></math> with Integer Coefficients</em>, Proceedings of the American Mathematical Society Vol. 85, No. 2 (Jun., 1982), pp. 283-288 (<a href="https://www.jstor.org/stable/2044298">jstor:2044298</a>)</p> </li> <li id="Feshbach83"> <p>Mark Feshbach, <em>The Integral Cohomology Rings of the Classifying Spaces of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{O}(n)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{SO}(n)</annotation></semantics></math>,</em> Indiana Univ. Math. J. 32 (1983), 511-516 (<a href="https://doi.org/10.1512/iumj.1983.32.32036">doi:10.1512/iumj.1983.32.32036</a>)</p> </li> <li id="Pittie91"> <p><a class="existingWikiWord" href="/nlab/show/Harsh+Pittie">Harsh Pittie</a>, <em>The integral homology and cohomology rings of SO(n) and Spin(n)</em>, Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105–153 (<a href="https://doi.org/10.1016/0022-4049(91)90108-E">doi:10.1016/0022-4049(91)90108-E</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Howard+Georgi">Howard Georgi</a>, §21 & 22 in: <em>Lie Algebras In Particle Physics</em>, Westview Press (1999), CRC Press (2019) [<a href="https://doi.org/10.1201/9780429499210">doi:10.1201/9780429499210</a>]</p> <blockquote> <p>with an eye towards application to <a class="existingWikiWord" href="/nlab/show/spinors">spinors</a> in (the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model</a> of) <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> </blockquote> </li> <li id="RudolphSchmidt17"> <p><a class="existingWikiWord" href="/nlab/show/Gerd+Rudolph">Gerd Rudolph</a>, <a class="existingWikiWord" href="/nlab/show/Matthias+Schmidt">Matthias Schmidt</a>, around Theorem 4.2.23 of: <em>Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields</em>, Theoretical and Mathematical Physics series, Springer 2017 (<a href="https://link.springer.com/book/10.1007/978-94-024-0959-8">doi:10.1007/978-94-024-0959-8</a>)</p> </li> </ul> <p>See also</p> <ul> <li id="Stasheff13"><a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>The topology and algebra of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">SO(n-1) \to SO(n) \to S^{n-1}</annotation></semantics></math></em>, Herman’s seminar July 2013 (<a class="existingWikiWord" href="/nlab/files/StasheffSOn.pdf" title="pdf slides">pdf slides</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 4, 2024 at 22:04:10. 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