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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="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='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#internal_to_a_general_category'>Internal to a general category</a></li> <li><a href='#internal_to_'>Internal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math></a></li> <li><a href='#for_lie_groups_and_klein_geometry'>For Lie groups and Klein geometry</a></li> <li><a href='#ForInfinityGroups'>For <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>-groups</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#for_normal_subgroups'>For normal subgroups</a></li> <li><a href='#QuotientMaps'>Topology of the quotient map</a></li> <li><a href='#as_a_homotopy_fiber'>As a homotopy fiber</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#spheres'><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>-Spheres</a></li> <li><a href='#QuotientMapsOfCosetSpaces'>Sequences of coset spaces</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Given a <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>, then their <em>coset object</em> is the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math>, hence the set of <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> where two are regarded as equivalent if they differ by right multiplication with an element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>, then the quotient is a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and usually called the <em>coset space</em>. This is in particular a <a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a>, see there for more.</p> <h2 id="definition">Definition</h2> <h3 id="internal_to_a_general_category">Internal to a general category</h3> <p>In a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup object</a>, the left/right <em>object of cosets</em> is the <a class="existingWikiWord" href="/nlab/show/orbit">object of orbits</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> under left/right multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>.</p> <p>Explicitly, the left coset space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/coequalizes">coequalizes</a> the parallel morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>×</mo><mi>G</mi><munderover><mo>⇉</mo><mi>μ</mi><mrow><msub><mi>proj</mi> <mi>G</mi></msub></mrow></munderover><mi>G</mi></mrow><annotation encoding="application/x-tex"> H \times G \underoverset{\mu}{proj_G}\rightrightarrows G </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> is (the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>×</mo><mi>G</mi><mo>↪</mo><mi>G</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H\times G \hookrightarrow G\times G</annotation></semantics></math> composed with) the group multiplication.</p> <p>Simiarly, the right coset space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>\</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H\backslash G</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/coequalizes">coequalizes</a> the parallel morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>×</mo><mi>H</mi><munderover><mo>⇉</mo><mrow><msub><mi>proj</mi> <mi>G</mi></msub></mrow><mi>μ</mi></munderover><mi>G</mi></mrow><annotation encoding="application/x-tex"> G \times H \underoverset{proj_G}{\mu}\rightrightarrows G </annotation></semantics></math></div> <h3 id="internal_to_">Internal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math></h3> <p>Specializing the above definition to the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is the well-pointed topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>, given an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, its orbit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mi>H</mi></mrow><annotation encoding="application/x-tex">g H</annotation></semantics></math> is an element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math> and is called a <em>left coset</em>.</p> <p>Using <a class="existingWikiWord" href="/nlab/show/comprehension">comprehension</a>, we can write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo>=</mo><mo stretchy="false">{</mo><mi>g</mi><mi>H</mi><mo stretchy="false">|</mo><mi>g</mi><mo>∈</mo><mi>G</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> G/H = \{g H | g \in G\} </annotation></semantics></math></div> <p>Similarly there is a coset on the right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>\</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \backslash G</annotation></semantics></math>.</p> <h3 id="for_lie_groups_and_klein_geometry">For Lie groups and Klein geometry</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G</annotation></semantics></math> is an inclusion of <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a> then the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math> is also called a <em><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a></em>.</p> <h3 id="ForInfinityGroups">For <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>-groups</h3> <p>More generally, given an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \to G</annotation></semantics></math>, hence equivalently a morphism of their <a class="existingWikiWord" href="/nlab/show/deloopings">deloopings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}H \to \mathbf{B}G</annotation></semantics></math>, then the <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math> is given by the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of this map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ G/H &\longrightarrow& \mathbf{B}H \\ && \downarrow \\ && \mathbf{B}G } \,. </annotation></semantics></math></div> <p>See at <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></em> for more on this definition. See at <em><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></em> and <em><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></em> for the corresponding concepts of <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>.</p> <h2 id="properties">Properties</h2> <h3 id="for_normal_subgroups">For normal subgroups</h3> <p>The coset inherits the structure of a group if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a>.</p> <p>Unless <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is abelian, considering both left and right coset spaces provide different information.</p> <h3 id="QuotientMaps">Topology of the quotient map</h3> <div class="num_prop" id="QuotientProjectionForCompactLieGroupActingFreelyOnManifoldIsPrincipa"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> equipped with a <a class="existingWikiWord" href="/nlab/show/free+action">free</a> smooth <a class="existingWikiWord" href="/nlab/show/action">action</a> on <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>, then the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <a class="existingWikiWord" href="/nlab/show/projection">projection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⟶</mo><mi>X</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> X \longrightarrow X/G </annotation></semantics></math></div> <p>is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> (hence in particular a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>).</p> </div> <p>This is originally due to (<a href="#Gleason50">Gleason 50</a>). See e.g. (<a href="#Cohen">Cohen, theorem 1.3</a>)</p> <div class="num_cor" id="QuotientProjectionForCompactLieSubgroupIsPrincipal"> <h6 id="corollary">Corollary</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \subset G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact</a> <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a>, then the <a class="existingWikiWord" href="/nlab/show/coset">coset</a> <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <a class="existingWikiWord" href="/nlab/show/projection">projection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>⟶</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex"> G \longrightarrow G/H </annotation></semantics></math></div> <p>is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> (hence in particular a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>).</p> </div> <p>This is originally due to (<a href="#Samelson41">Samelson 41</a>).</p> <div class="num_prop" id="ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>⊂</mo><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">K \subset H \subset G</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed</a> <a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</a>, then the <a class="existingWikiWord" href="/nlab/show/projection">projection</a> map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><mo stretchy="false">/</mo><mi>K</mi><mo>⟶</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex"> p \;\colon\; G/K \longrightarrow G/H </annotation></semantics></math></div> <p>is a locally trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">/</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">H/K</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> (hence in particular a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>).</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Observe that the projection map in question is equivalently</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><msub><mo>×</mo> <mi>H</mi></msub><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">/</mo><mi>K</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> G \times_H (H/K) \longrightarrow G/H \,, </annotation></semantics></math></div> <p>(where on the left we form the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> and then divide out the <a class="existingWikiWord" href="/nlab/show/diagonal+action">diagonal action</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>). This exhibits it as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">/</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">H/K</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> of corollary <a class="maruku-ref" href="#QuotientProjectionForCompactLieSubgroupIsPrincipal"></a>.</p> </div> <p> <div class='num_prop' id='CosetSpaceCoprojectionsWithLocalSections'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/coset+space+coprojections+with+local+sections">coset space coprojections with local sections</a>)</strong><br /> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \subset G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a>.</p> <p>Sufficient conditions for the <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> <a class="existingWikiWord" href="/nlab/show/coprojection">coprojection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mover><mo>→</mo><mi>q</mi></mover><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G \overset{q}{\to} G/H</annotation></semantics></math> to admit <a class="existingWikiWord" href="/nlab/show/local+sections">local sections</a>, in that there is an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">\underset{i \in I}{\sqcup}U_i \to G/H</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/continuous+section">continuous section</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>𝒰</mi></msub></mrow><annotation encoding="application/x-tex">\sigma_{\mathcal{U}}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of <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> to the cover,</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></mtd> <mtd></mtd> <mtd><msub><mi>G</mi> <mrow><mo stretchy="false">|</mo><mi>𝒰</mi></mrow></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∃</mo><mi>σ</mi></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msup></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><mo>∃</mo><mspace width="thickmathspace"></mspace></mrow></mpadded><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub></mtd> <mtd><mo>=</mo></mtd> <mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && G_{\vert \mathcal{U}} &\longrightarrow& G \\ & {}^{\mathllap{ \exists \sigma }} \nearrow & \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ \mathllap{ \exists \; } \underset{i \in I}{\sqcup} U_i &=& \underset{i \in I}{\sqcup} U_i &\longrightarrow& G/H \mathrlap{\,,} } </annotation></semantics></math></div> <p>include the following:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is any <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> <p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a></p> <p>(in particular for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/closed+subgroup">closed subgroup</a> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> itself is a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a>, since <a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a>).</p> <p>(<a href="#Gleason50">Gleason 50, Thm. 4.1</a>)</p> </li> </ul> <p>or:</p> <ul> <li> <p>The underlying <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separable+metric+space">separable metric space</a>;</p> </li> <li> <p>of <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension+of+a+separable+metric+space">dimension of a separable metric space</a></p> </li> </ol> <p>(e.g. if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>)</p> <p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>⊂</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \subset G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/closed+subgroup">closed subgroup</a>.</p> <p>(<a href="#Mostert53">Mostert 53, Theorem 3</a>)</p> </li> </ul> <p></p> </div> </p> <h3 id="as_a_homotopy_fiber">As a homotopy fiber</h3> <p> <div class='num_remark' id='InGeometricHomotopyTheory'> <h6>Remark</h6> <p>In <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric homotopy theory</a> (in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>), for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>⟶</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \longrightarrow G</annotation></semantics></math> any homomorphisms of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> objects, then the natural projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>⟶</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G \longrightarrow G/H</annotation></semantics></math>, generally realizes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math>. This is exhibited by a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>G</mi></mtd> <mtd><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><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ G & \longrightarrow &* \\ \downarrow && \downarrow \\ G/H &\longrightarrow& \mathbf{B}H } \,. </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}H</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>. This also equivalently exhibits the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> (see there for more).</p> <p>By the reverse <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> for <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a> (<a href="pasting+law+for+pullbacks#ReversePastingLawForInfinityGroupoids">here</a>, using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex">\ast \to \mathbf{B}H</annotation></semantics></math> is an <a href="effective+epimorphism+in+an+infinity1-category">effective epimorphism</a> by definition of <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>) then we get the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</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><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ G/H & \longrightarrow &\mathbf{B}H \\ \downarrow && \downarrow \\ * & \longrightarrow & \mathbf{B}G } </annotation></semantics></math></div> <p>which exhibits the coset as 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><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}H \to \mathbf{B}G</annotation></semantics></math>.</p> <p>See also <a href="#SatiSchreiber21">SS21, Ex. 3.2.35</a>.</p> </div> </p> <h2 id="examples">Examples</h2> <h3 id="spheres"><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>-Spheres</h3> <div class="num_example" id="nSphereAsCosetSpace"> <h6 id="example">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a> are coset spaces of <a class="existingWikiWord" href="/nlab/show/orthogonal+groups">orthogonal groups</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>≃</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>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"> S^n \simeq O(n+1)/O(n) \,. </annotation></semantics></math></div> <p>The odd-dimensional spheres are also coset spaces of <a class="existingWikiWord" href="/nlab/show/unitary+groups">unitary groups</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>≃</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S^{2n+1} \simeq U(n+1)/U(n) </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Regarding the first statement:</p> <p>Fix a <a class="existingWikiWord" href="/nlab/show/unit+vector">unit vector</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n+1}</annotation></semantics></math>. Then its <a class="existingWikiWord" href="/nlab/show/orbit">orbit</a> under the defining <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><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/action">action</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n+1}</annotation></semantics></math> is clearly the canonical embedding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>↪</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^n \hookrightarrow \mathbb{R}^{n+1}</annotation></semantics></math>. But precisely the subgroup 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>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+1)</annotation></semantics></math> that consists of rotations around the axis formed by that unit vector <a class="existingWikiWord" href="/nlab/show/stabilizer+group">stabilizes</a> it, and that subgroup is isomorphic to <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>, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>≃</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^n \simeq O(n+1)/O(n)</annotation></semantics></math>.</p> <p>The second statement follows by the same kind of reasoning:</p> <p>Clearly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n+1)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/transitive+action">acts transitively</a> on the <a class="existingWikiWord" href="/nlab/show/unit+sphere">unit sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^{2n+1}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}^{n+1}</annotation></semantics></math>. It remains to see that its <a class="existingWikiWord" href="/nlab/show/stabilizer+subgroup">stabilizer subgroup</a> of any point on this sphere is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n)</annotation></semantics></math>. If we take the point with <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,0, 0, \cdots,0)</annotation></semantics></math> and regard elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n+1)</annotation></semantics></math> as <a class="existingWikiWord" href="/nlab/show/matrices">matrices</a>, then the stabilizer subgroup consists of matrices of the block diagonal form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd> <mtd><mover><mn>0</mn><mo stretchy="false">→</mo></mover></mtd></mtr> <mtr><mtd><mover><mn>0</mn><mo stretchy="false">→</mo></mover></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( \array{ 1 & \vec 0 \\ \vec 0 & A } \right) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in U(n)</annotation></semantics></math>.</p> </div> <p>There are also various exceptional realizations of spheres as coset spaces. For instance:</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a>-<a class="existingWikiWord" href="/nlab/show/structures">structures</a> on <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a>:</strong></p> <table><thead><tr><th><strong>standard:</strong></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^{n-1} \simeq_{diff} SO(n)/SO(n-1)</annotation></semantics></math></td><td style="text-align: left;"><a href="sphere#nSphereAsCosetSpace">this Prop.</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)</annotation></semantics></math></td><td style="text-align: left;"><a href="sphere#OddDimSphereAsSpecialUnitaryCoset">this Prop.</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Sp</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Sp</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)</annotation></semantics></math></td><td style="text-align: left;"><a href="sphere#SphereAsSymplecticUnitaryCoset">this Prop.</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/exceptional+geometry">exceptional</a>:</strong></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Spin</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>G</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">S^7 \simeq_{diff} Spin(7)/G_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%287%29%2FG%E2%82%82+is+the+7-sphere">Spin(7)/G₂ is the 7-sphere</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Spin</mi><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^7 \simeq_{diff} Spin(6)/SU(3)</annotation></semantics></math></td><td style="text-align: left;">since <a class="existingWikiWord" href="/nlab/show/Spin%286%29">Spin(6)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SU%284%29">SU(4)</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^7 \simeq_{diff} Spin(5)/SU(2)</annotation></semantics></math></td><td style="text-align: left;">since <a class="existingWikiWord" href="/nlab/show/Sp%282%29">Sp(2)</a> is <a class="existingWikiWord" href="/nlab/show/Spin%285%29">Spin(5)</a> and <a class="existingWikiWord" href="/nlab/show/Sp%281%29">Sp(1)</a> is <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a>, see <a class="existingWikiWord" href="/nlab/show/Spin%285%29%2FSU%282%29+is+the+7-sphere">Spin(5)/SU(2) is the 7-sphere</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>6</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><msub><mi>G</mi> <mn>2</mn></msub><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^6 \simeq_{diff} G_2/SU(3)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/G%E2%82%82%2FSU%283%29+is+the+6-sphere">G₂/SU(3) is the 6-sphere</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>15</mn></msup><msub><mo>≃</mo> <mi>diff</mi></msub><mi>Spin</mi><mo stretchy="false">(</mo><mn>9</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^15 \simeq_{diff} Spin(9)/Spin(7)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%289%29%2FSpin%287%29+is+the+15-sphere">Spin(9)/Spin(7) is the 15-sphere</a></td></tr> </tbody></table> <p>see also <em><a class="existingWikiWord" href="/nlab/show/Spin%288%29-subgroups+and+reductions+--+table">Spin(8)-subgroups and reductions</a></em></p> <p id="HomotopyTheoretic"> <a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a> of <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a>:</p> <p><img src="/nlab/files/ExceptionalSpheres.jpg" width="730px" /></p> <p>(from <a class="existingWikiWord" href="/schreiber/show/Twisted+Cohomotopy+implies+M-theory+anomaly+cancellation">FSS 19, 3.4</a>)</p> </div> <p><br /></p> <h3 id="QuotientMapsOfCosetSpaces">Sequences of coset spaces</h3> <p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>↪</mo><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">K \hookrightarrow H \hookrightarrow G</annotation></semantics></math> two consecutive group inclusions with their induced coset <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <a class="existingWikiWord" href="/nlab/show/projections">projections</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><mi>H</mi><mo stretchy="false">/</mo><mi>K</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>G</mi><mo stretchy="false">/</mo><mi>K</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ H/K & \longrightarrow& G/K \\ && \downarrow \\ && G/H } \,. </annotation></semantics></math></div> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>K</mi><mo>→</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/K \to G/H</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>, for instance in the situation of prop. <a class="maruku-ref" href="#ProjectionOfCosetsIsFiberBundleForClosedSubgroupsOfCompactLieGroup"></a> (so that this is indeed a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> with respect to the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>) then it induces the corresponding <a class="existingWikiWord" href="/nlab/show/long+exact+sequence+of+homotopy+groups">long exact sequence of homotopy groups</a></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><msub><mi>π</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">/</mo><mi>K</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">/</mo><mi>K</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">/</mo><mi>K</mi><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \to \pi_{n+1}(G/H) \longrightarrow \pi_n(H/K) \longrightarrow \pi_n(G/K) \longrightarrow \pi_n(G/H) \longrightarrow \pi_{n-1}(H/K) \to \cdots \,. </annotation></semantics></math></div> <div class="num_example" id="CofiberSequencesOfCosetsOfOrthogonalGroups"> <h6 id="example_2">Example</h6> <p>Consider a sequence of inclusions of <a class="existingWikiWord" href="/nlab/show/orthogonal+groups">orthogonal groups</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> O(n) \hookrightarrow O(n+1) \hookrightarrow O(n+k) \,. </annotation></semantics></math></div> <p>Then by example <a class="maruku-ref" href="#nSphereAsCosetSpace"></a> we have that <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><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">O(n+1)/O(n) \simeq S^n</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a> and by corollary <a class="maruku-ref" href="#QuotientProjectionForCompactLieSubgroupIsPrincipal"></a> the quotient map is a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a>. Hence there is a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence+of+homotopy+groups">long exact sequence of homotopy groups</a> of the form</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><msub><mi>π</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \to \pi_q(S^n) \longrightarrow \pi_q(O(n+k)/O(n)) \longrightarrow \pi_q(O(n+k)/O(n+1)) \longrightarrow \pi_{q-1}(S^n) \to \cdots \,. </annotation></semantics></math></div> <p>Now for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">q \lt n</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\pi_q(S^n) = 0</annotation></semantics></math> and hence in this range we have <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mrow><mo>•</mo><mo><</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>π</mi> <mrow><mo>•</mo><mo><</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_{\bullet \lt n}(O(n+k)/O(n)) \stackrel{\simeq}{\longrightarrow} \pi_{\bullet \lt n}(O(n+k)/O(n+1)) \,. </annotation></semantics></math></div></div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coadjoint+orbit">coadjoint orbit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/index+of+a+subgroup">index of a subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/class+equation">class equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flag+variety">flag variety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/double+coset">double coset</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="Samelson41"> <p><a class="existingWikiWord" href="/nlab/show/Hans+Samelson">Hans Samelson</a>, <em>Beiträge zur Topologie der Gruppenmannigfaltigkeiten</em>, Ann. of Math. 2, 42, (1941), 1091 - 1137. (<a href="https://www.jstor.org/stable/1970463">jstor:1970463</a>, <a href="https://doi.org/10.2307/1970463">doi:10.2307/1970463</a>)</p> </li> <li id="Gleason50"> <p><a class="existingWikiWord" href="/nlab/show/Andrew+Gleason">Andrew Gleason</a>, <em>Spaces with a compact Lie group of transformations</em>, Proc. of A.M.S 1, (1950), 35 - 43 (<a href="https://www.jstor.org/stable/2032430">jstor:2032430</a>, <a href="https://doi.org/10.2307/2032430">doi:10.2307/2032430</a>)</p> </li> <li id="Steenrod51"> <p><a class="existingWikiWord" href="/nlab/show/Norman+Steenrod">Norman Steenrod</a>, Section I.7 of: <em>The topology of fibre bundles</em>, Princeton Mathematical Series 14, Princeton Univ. Press, 1951.</p> </li> <li id="Mostert53"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Mostert">Paul Mostert</a>, <em>Local Cross Sections in Locally Compact Groups</em>, Proceedings of the American Mathematical Society, Vol. 4, No. 4 (Aug., 1953), pp.645-649 (<a href="https://www.jstor.org/stable/2032540">jstor:2032540</a>, <a href="https://doi.org/10.2307/2032540">doi:10.2307/2032540</a>)</p> </li> <li id="Bredon72"> <p><a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a>, Section I.4 of: <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+compact+transformation+groups">Introduction to compact transformation groups</a></em>, Academic Press 1972 (<a href="https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1">ISBN 9780080873596</a>, <a href="http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf">pdf</a>)</p> </li> <li id="Cohen"> <p>R. Cohen, <em>Topology of fiber bundles</em>, Lecture notes (<a href="http://math.stanford.edu/~ralph/fiber.pdf">pdf</a>)</p> </li> </ul> <p>On coset spaces with the same <a class="existingWikiWord" href="/nlab/show/rational+cohomology">rational cohomology</a> as a <a class="existingWikiWord" href="/nlab/show/product+space">product</a> of <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Linus+Kramer">Linus Kramer</a>, <em>Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurface</em>, Memoirs of the American Mathematical Society number 752 (<a href="http://arxiv.org/abs/math/0109133">arXiv:math/0109133</a>, <a href="http://dx.doi.org/10.1090/memo/0752">doi:10.1090/memo/0752</a>, <a href="http://books.google.com/books?id=SA8O6ihrDFkC&printsec=frontcover&hl=de&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false">GoogleBooks</a>)</li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos+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>-topos theory</a>:</p> <ul> <li id="SatiSchreiber21"><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, Ex. 3.2.35 (<a href="https://arxiv.org/pdf/2112.13654.pdf#page=104">p. 104</a>) of: <em><a class="existingWikiWord" href="/schreiber/show/Equivariant+principal+infinity-bundles">Equivariant principal infinity-bundles</a></em> [<a href="https://arxiv.org/abs/2112.13654">arXiv:2112.13654</a>]</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/coset+spaces">coset spaces</a> (<a class="existingWikiWord" href="/nlab/show/homogeneous+spaces">homogeneous spaces</a>) and their <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+forms">Maurer-Cartan forms</a> in application to <a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation</a> of (<a class="existingWikiWord" href="/nlab/show/supergravity">super</a>-)<a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <a class="existingWikiWord" href="/nlab/show/L.+J.+Romans">L. J. Romans</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+P.+Warner">Nicholas P. Warner</a>, <em>Symmetries of coset spaces and Kaluza-Klein supergravity</em>, Annals of Physics <strong>157</strong> 2 (1984) 394-407 [<a href="https://doi.org/10.1016/0003-4916(84)90066-6">doi:10.1016/0003-4916(84)90066-6</a>]</p> </li> <li id="CastellaniDAuriaFre"> <p><a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, §I.6 in: <em><a class="existingWikiWord" href="/nlab/show/Supergravity+and+Superstrings+-+A+Geometric+Perspective">Supergravity and Superstrings - A Geometric Perspective</a></em>, World Scientific (1991) [<a href="https://doi.org/10.1142/0224">doi:10.1142/0224</a>, toc: <a class="existingWikiWord" href="/nlab/files/CDF91-TOC.pdf" title="pdf">pdf</a>, ch I.6: <a class="existingWikiWord" href="/nlab/files/CastellaniDAuriaFre-ChI6.pdf" title="pdf">pdf</a></p> </li> <li id="Castellani01"> <p><a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <em>On G/H geometry and its use in M-theory compactifications</em>, Annals Phys. <strong>287</strong> (2001) 1-13 [<a href="https://arxiv.org/abs/hep-th/9912277">arXiv:hep-th/9912277</a>, <a href="https://doi.org/10.1006/aphy.2000.6097">doi:10.1006/aphy.2000.6097</a>]</p> </li> </ul> <p>Further in <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a>:</p> <ul> <li>Ismaël Ahlouche Lahlali, Josh A. O’Connor: <em>Coset symmetries and coadjoint orbits</em> [<a href="https://arxiv.org/abs/2411.05918">arXiv:2411.05918</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 12, 2024 at 10:33:02. See the <a href="/nlab/history/coset" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/coset" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/12605/#Item_3">Discuss</a><span class="backintime"><a href="/nlab/revision/coset/25" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/coset" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/coset" accesskey="S" class="navlink" id="history" rel="nofollow">History (25 revisions)</a> <a href="/nlab/show/coset/cite" style="color: black">Cite</a> <a href="/nlab/print/coset" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/coset" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>