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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="spheres">Spheres</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unit+sphere">unit sphere</a>, <a class="existingWikiWord" href="/nlab/show/polar+coordinates">polar coordinates</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/round+sphere">round sphere</a>, <a class="existingWikiWord" href="/nlab/show/squashed+sphere">squashed sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hemisphere">hemisphere</a>, <a class="existingWikiWord" href="/nlab/show/equator">equator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stereographic+projection">stereographic projection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+spheres">homotopy groups of spheres</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+sphere">homotopy sphere</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+sphere">rational homotopy sphere</a>, <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+fibration">spherical fibration</a>, <a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a>, <a class="existingWikiWord" href="/nlab/show/stable+Cohomotopy+theory">stable Cohomotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology+sphere">homology sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+sphere">rational homotopy sphere</a>, <a class="existingWikiWord" href="/nlab/show/rational+n-sphere">rational n-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+sphere">motivic sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+actions+on+spheres">group actions on spheres</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+sphere">representation sphere</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+Cohomotopy">equivariant Cohomotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeb+sphere+theorem">Reeb sphere theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+packing">sphere packing</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Music+of+the+Spheres">Music of the Spheres</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/low+dimensional+topology">low dimensional</a> <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+projective+space">real projective space</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℝ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{R}P^1</annotation></semantics></math></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+projective+line">complex projective line</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{C}P^1</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+the+2-sphere">geometric quantization of the 2-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fuzzy+2-sphere">fuzzy 2-sphere</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fuzzy+3-sphere">fuzzy 3-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quaternionic+projective+line">quaternionic projective line</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℍ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{H}P^1</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fuzzy+4-sphere">fuzzy 4-sphere</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/5-sphere">5-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6-sphere">6-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/exotic+7-sphere">exotic 7-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/8-sphere">8-sphere</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/octonionic+projective+line">octonionic projective line</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>𝕆</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{O}P^1</annotation></semantics></math></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/13-sphere">13-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/15-sphere">15-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite-dimensional+sphere">infinite-dimensional sphere</a></p> </li> </ul> </div></div> <h4 id="topology">Topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>)</p> <p>see also <em><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><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> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> <h4 id="manifolds_and_cobordisms">Manifolds and cobordisms</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a></strong> and <strong><a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a>, <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Cobordism+and+Complex+Oriented+Cohomology">Introduction</a></em></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+transformation">coordinate transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atlas">atlas</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, ,<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite+dimensional+manifold">infinite dimensional manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Banach+manifold">Banach manifold</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+manifold">Hilbert manifold</a>, <a class="existingWikiWord" href="/nlab/show/ILH+manifold">ILH manifold</a>, <a class="existingWikiWord" href="/nlab/show/Frechet+manifold">Frechet manifold</a>, <a class="existingWikiWord" href="/nlab/show/convenient+manifold">convenient manifold</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Genera and invariants</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/signature+genus">signature genus</a>, <a class="existingWikiWord" href="/nlab/show/Kervaire+invariant">Kervaire invariant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-hat+genus">A-hat genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-manifolds">2-manifolds</a>/<a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of a surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+transversality+theorem">Thom's transversality theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galatius-Tillmann-Madsen-Weiss+theorem">Galatius-Tillmann-Madsen-Weiss theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometrization+conjecture">geometrization conjecture</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptization+conjecture">elliptization conjecture</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#finitedimensional_spheres'>Finite-dimensional spheres</a></li> <li><a href='#InfiniteDimensionalSphere'>Infinite dimensional spheres</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#basic'>Basic</a></li> <li><a href='#CWstructures'>CW-structures</a></li> <li><a href='#LabelCosetSpaceStructure'>Coset space structure</a></li> <ul> <li><a href='#AsQuotientsOfCompactLieGroups'>As quotients of compact Lie groups</a></li> <li><a href='#AsQuotientsOfLorentzGroups'>As quotients of Lorentz groups</a></li> </ul> <li><a href='#spin_structure'>Spin structure</a></li> <li><a href='#parallelizability'>Parallelizability</a></li> <li><a href='#branched_covers'>Branched covers</a></li> <li><a href='#iterated_loop_spaces'>Iterated loop spaces</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> <li><a href='#formalization'>Formalization</a></li> <li><a href='#group_actions_on_spheres'>Group actions on spheres</a></li> <li><a href='#geometric_structures_on_spheres'>Geometric structures on spheres</a></li> <li><a href='#embeddings_of_spheres'>Embeddings of spheres</a></li> <li><a href='#iterated_loop_spaces_2'>Iterated loop spaces</a></li> <li><a href='#topological_complexity'>Topological complexity</a></li> </ul> </ul> </div> <h2 id="definition">Definition</h2> <h3 id="finitedimensional_spheres">Finite-dimensional spheres</h3> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>The <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 unit <strong>sphere</strong> , or simply <strong><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>-sphere</strong>, is the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> given by the <a class="existingWikiWord" href="/nlab/show/subset">subset</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <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> consisting of all points <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> whose distance from the origin is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></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><mo stretchy="false">{</mo><mi>x</mi><mo>:</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo><mo>=</mo><mn>1</mn><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^n = \{ x: \mathbb{R}^{n+1} \;|\; \|x\| = 1 \} \,. </annotation></semantics></math></div> <p>The <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 sphere of radius <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>S</mi> <mi>r</mi> <mi>n</mi></msubsup><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo>:</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo><mo>=</mo><mi>r</mi><mo stretchy="false">}</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^n_r = \{ x: \mathbb{R}^{n+1} \;|\; \|x\| = r \} .</annotation></semantics></math></div> <p>Topologically, this is equivalent (<a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a>) to the unit sphere for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">r \gt 0</annotation></semantics></math>, or a <a class="existingWikiWord" href="/nlab/show/point">point</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">r = 0</annotation></semantics></math>.</p> <p>This is naturally a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</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>, with the <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a> induced by the standard smooth structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>.</p> </div> <h3 id="InfiniteDimensionalSphere">Infinite dimensional spheres</h3> <p>One can also talk about the <a class="existingWikiWord" href="/nlab/show/infinite-dimensional+sphere">infinite-dimensional sphere</a> in an arbitrary (possibly infinite-dimensional) <a class="existingWikiWord" href="/nlab/show/normed+vector+space">normed vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</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>S</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo>:</mo><mi>V</mi><mspace width="thickmathspace"></mspace><mtext>such that</mtext><mspace width="thickmathspace"></mspace><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow><mo>=</mo><mn>1</mn><mo stretchy="false">}</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> S(V) = \{ x: V \;\text{such that}\; {\|x\|} = 1 \} .</annotation></semantics></math></div> <p>If a <a class="existingWikiWord" href="/nlab/show/LCTVS">locally convex topological vector space</a> admits a continuous linear injection into a <a class="existingWikiWord" href="/nlab/show/normed+vector+space">normed vector space</a>, this can be used to define its sphere. If not, one can still define the sphere as a <em>quotient</em> of the space of non-zero vectors under the scalar action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,\infty)</annotation></semantics></math>.</p> <p>Homotopy theorists (e.g. <a href="#tomDieck2008">tom Dieck 2008, example 8.3.7</a>) define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">S^\infty</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/directed+colimit">directed colimit</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^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><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>↪</mo><msup><mi>S</mi> <mn>0</mn></msup><mo>↪</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>↪</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>↪</mo><mi>⋯</mi><msup><mi>S</mi> <mn>∞</mn></msup><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^{-1} \hookrightarrow S^0 \hookrightarrow S^1 \hookrightarrow S^2 \hookrightarrow \cdots S^\infty .</annotation></semantics></math></div> <p>Note that this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">S^\infty</annotation></semantics></math> is <em>not</em> homeomorphic to the sphere of any metrizable space as defined above, since the metrizable <a class="existingWikiWord" href="/nlab/show/CW+complex">CW-complexes</a> are precisely the locally finite <a class="existingWikiWord" href="/nlab/show/CW+complex">CW-complexes</a> (<a href="#FP1990">Fritsch–Piccinini 1990: 48, prop. 1.5.17</a>), which <a href="#CWstructures"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">S^\infty</annotation></semantics></math> is not</a> (every open <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>-cell intersects all closed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math>-cells with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>≥</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m \ge n</annotation></semantics></math>)</p> <p>In themselves, infinite-dimensional spheres provide nothing new to <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, as they are at least weakly contractible and usually <a class="existingWikiWord" href="/nlab/show/contractible+space">contractible</a>. However, they are a very useful source of big contractible spaces and so are often used as a starting point for making concrete models of <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a>.</p> <p>If the vector space is a <a class="existingWikiWord" href="/nlab/show/shift+space">shift space</a>, then contractibility is straightforward to prove.</p> <div class="num_theorem" id="theorem"> <h6 id="theorem">Theorem</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/shift+space">shift space</a> of some order. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">S V</annotation></semantics></math> be its sphere (either via a norm or as the quotient of non-zero vectors). Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">S V</annotation></semantics></math> is contractible.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">T \colon V \to V</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/shift+map">shift map</a>. The idea is to homotop the sphere onto the image of <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>, and then down to a point.</p> <p>It is simplest to start with the non-zero vectors, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">V \setminus \{0\}</annotation></semantics></math>. As <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> is injective, it restricts to a map from this space to itself which commutes with the scalar action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,\infty)</annotation></semantics></math>. Define a homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><mo stretchy="false">(</mo><mi>V</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mi>V</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">H \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\}</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>t</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo><mi>v</mi><mo>+</mo><mi>t</mi><mi>T</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">H_t(v) = (1 - t)v + t T v</annotation></semantics></math>. It is clear that, assuming it is well-defined, it is a homotopy from the identity to <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>. To see that it is well-defined, we need to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>t</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_t(v)</annotation></semantics></math> is never zero. The only place where it could be zero would be on an eigenvector of <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>, but as <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> is a <a class="existingWikiWord" href="/nlab/show/shift+map">shift map</a> then it has none.</p> <p>As <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> is a <a class="existingWikiWord" href="/nlab/show/shift+map">shift map</a>, it is not surjective and so we can pick some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">v_0</annotation></semantics></math> not in its image. Then we define a homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><mo stretchy="false">(</mo><mi>V</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>→</mo><mi>V</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">G \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\}</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>t</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo><mi>T</mi><mi>v</mi><mo>+</mo><mi>t</mi><msub><mi>v</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">G_t(v) = (1 - t)T v + t v_0</annotation></semantics></math>. As <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">v_0</annotation></semantics></math> is not in the image of <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>, this is well-defined on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">V \setminus \{0\}</annotation></semantics></math>. Combining these two homotopies results in the desired contraction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">V \setminus \{0\}</annotation></semantics></math>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> admits a suitable function defining a spherical subset (such as a norm) then we can modify the above to a contraction of the spherical subset simply by dividing out by this function. If not, as the homotopies above all commute with the scalar action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,\infty)</annotation></semantics></math>, they descend to the definition of the sphere as the quotient of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∖</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">V \setminus \{0\}</annotation></semantics></math>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="basic">Basic</h3> <ul> <li> <p>The <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>-sphere is the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/ball">ball</a>.</p> </li> <li> <p>These spheres, or rather their underlying <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> or <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial sets</a>, are fundamental in (ungeneralised) <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>. In a sense, <a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a> says that these are all that you need; no further <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+homotopy+theory">generalised homotopy theory</a> (in a sense <a class="existingWikiWord" href="/nlab/show/Eckmann%E2%80%93Hilton+duality">dual</a> to <a class="existingWikiWord" href="/nlab/show/Eilenberg%E2%80%93Steenrod+cohomology+theory">Eilenberg–Steenrod cohomology theory</a>) is needed.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/positive+dimension+spheres+are+H-cogroup+objects">positive dimension spheres are H-cogroup objects</a>, and this is the origin of the <a class="existingWikiWord" href="/nlab/show/group">group</a> structure on <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>).</p> </li> </ul> <p> <div class='num_prop'> <h6>Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/topological+complexity">topological complexity</a> of the sphere is</p> <div class="maruku-equation" id="eq:TopologicalComplexityOfSpheres"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>TC</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left left"><mtr><mtd><mn>2</mn></mtd> <mtd><mo>;</mo><mi>n</mi><mspace width="1em"></mspace><mi>odd</mi></mtd></mtr> <mtr><mtd><mn>3</mn></mtd> <mtd><mo>;</mo><mi>n</mi><mspace width="1em"></mspace><mi>even</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> TC \big( S^n \big) \;=\; \left\{ \begin{array}{ll} 2 & ; n \quad odd \\ 3 & ; n \quad even \end{array} \right. </annotation></semantics></math></div> <p></p> </div> </p> <p>(<a href="#Farber01">Farber 01, Theorem 8</a>)</p> <p>This proposition can be generalized:</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/topological+complexity">topological complexity</a> of a product of spheres is</p> <div class="maruku-equation" id="eq:TopologicalComplexityOfProductOfSpheres"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>TC</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>m</mi></msup><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left left"><mtr><mtd><mi>n</mi><mo>+</mo><mn>1</mn></mtd> <mtd><mo>;</mo><mi>m</mi><mspace width="1em"></mspace><mi>odd</mi></mtd></mtr> <mtr><mtd><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mtd> <mtd><mo>;</mo><mi>m</mi><mspace width="1em"></mspace><mi>even</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> TC \big( (S^m)^n \big) \;=\; \left\{ \begin{array}{ll} n+1 & ; m \quad odd \\ 2n+1 & ; m \quad even \end{array} \right. </annotation></semantics></math></div> <p></p> </div> </p> <p>(<a href="#Farber01">Farber 01, Theorem 13</a>)</p> <p>A special case of this proposition is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TC</mi><mo stretchy="false">(</mo><msup><mi>T</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">TC(T^n)=n+1</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/topological+complexity">topological complexity</a> of the <a class="existingWikiWord" href="/nlab/show/torus">torus</a>.</p> <h3 id="CWstructures">CW-structures</h3> <p>The <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>-sphere is an <a class="existingWikiWord" href="/nlab/show/dimension+of+a+cell+complex"><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> <a class="existingWikiWord" href="/nlab/show/CW+complex">CW complex</a> in several ways:</p> <ul> <li> <p>The <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>-sphere (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n \ge 0</annotation></semantics></math>) admits, for every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>∈</mo><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">x_0 \in S^n </annotation></semantics></math>, a <a class="existingWikiWord" href="/nlab/show/CW+complex">CW-structure</a> with one <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math> and one <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>-cell <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><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>0</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">S^n \setminus \{ x_0 \}</annotation></semantics></math>, by <a class="existingWikiWord" href="/nlab/show/stereographic+projection">stereographic projection</a>. (<a href="#tomDieck2008">tom Dieck 2008, example 8.3.7</a>)</p> </li> <li> <p>The <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>-sphere (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n \ge 0</annotation></semantics></math>) can also be constructed from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-sphere by attaching <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>-cells (the north and south hemispheres) to the equator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-sphere. Iteratively applying this construction starting with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><mo>∅</mo></mrow><annotation encoding="application/x-tex">S^{-1}=\varnothing</annotation></semantics></math> yields a <a class="existingWikiWord" href="/nlab/show/CW+complex">CW complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math> with two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-cells in each dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \le n</annotation></semantics></math>, and subcomplex inclusions <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>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^n \subseteq S^{n+1}</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow><annotation encoding="application/x-tex">n \in \mathbf{N}</annotation></semantics></math>; the colimit of this sequence is (by definition) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">S^\infty</annotation></semantics></math>. (<a href="#tomDieck2008">tom Dieck 2008, example 8.3.7</a>)</p> </li> </ul> <h3 id="LabelCosetSpaceStructure">Coset space structure</h3> <h4 id="AsQuotientsOfCompactLieGroups">As quotients of compact Lie groups</h4> <div class="num_prop" id="nSphereAsCosetSpace"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a> are <a class="existingWikiWord" href="/nlab/show/coset+spaces">coset spaces</a> 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><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><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>Similarly for the corresponding <a class="existingWikiWord" href="/nlab/show/special+orthogonal+groups">special 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><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S^n \;\simeq\; SO(n+1)/SO(n) </annotation></semantics></math></div> <p>and <a class="existingWikiWord" href="/nlab/show/spin+groups">spin 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><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S^n \;\simeq\; Spin(n+1)/Spin(n) </annotation></semantics></math></div> <p>and <a class="existingWikiWord" href="/nlab/show/pin+groups">pin 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><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Pin</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Pin</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\; Pin(n+1)/Pin(n) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <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> </div> <p>Similarly, the analogous argument for <a class="existingWikiWord" href="/nlab/show/unit+spheres">unit spheres</a> inside (the <a class="existingWikiWord" href="/nlab/show/real+vector+spaces">real vector spaces</a> underlying) <a class="existingWikiWord" href="/nlab/show/complex+vector+spaces">complex vector spaces</a>, we have</p> <div class="num_prop" id="OddDimSphereAsSpecialUnitaryCoset"> <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>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/n-sphere">(2k+1)-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>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^{2k+1}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> of <a class="existingWikiWord" href="/nlab/show/special+unitary+groups">special 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>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>SU</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SU</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^{2k+1} \;\simeq\; SU(k+1)/SU(k) \,. </annotation></semantics></math></div></div> <p>And still similarly, the analogous argument for <a class="existingWikiWord" href="/nlab/show/unit+spheres">unit spheres</a> inside (the <a class="existingWikiWord" href="/nlab/show/real+vector+spaces">real vector spaces</a> underlying) <a class="existingWikiWord" href="/nlab/show/quaternionic+vector+spaces">quaternionic vector spaces</a>, we have</p> <div class="num_prop" id="SphereAsSymplecticUnitaryCoset"> <h6 id="proposition_3">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k \geq 1</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/n-sphere">(4k-1)-sphere</a> <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>k</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^{4k-1}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> of <a class="existingWikiWord" href="/nlab/show/quaternionic+unitary+groups">quaternionic 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>4</mn><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Sp</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Sp</mi><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^{4k-1} \;\simeq\; Sp(k)/Sp(k-1) \,. </annotation></semantics></math></div></div> <p>Generally:</p> <div class="num_prop" id="TransitiveEffectiveActionsOfConnectedLieGroupsOnSpheres"> <h6 id="proposition_4">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a> <a class="existingWikiWord" href="/nlab/show/compact+Lie+groups">compact Lie groups</a> with <a class="existingWikiWord" href="/nlab/show/effective+action">effective</a> <a class="existingWikiWord" href="/nlab/show/transitive+actions">transitive actions</a> on <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a> are precisely (up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>) the following:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/SO%28n%29">SO(n)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/U%28n%29">U(n)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/SU%28n%29">SU(n)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sp%28n%29">Sp(n)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sp%28n%29.Sp%281%29">Sp(n).SO(2)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sp%28n%29.Sp%281%29">Sp(n).Sp(1)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spin%289%29">Spin(9)</a></p> </li> </ul> <p>with <a class="existingWikiWord" href="/nlab/show/coset+spaces">coset spaces</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><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></mtd> <mtd><mo>≃</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msup><mi>S</mi> <mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><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></mtd> <mtd><mo>≃</mo><msup><mi>S</mi> <mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><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></mtd> <mtd><mo>≃</mo><msup><mi>S</mi> <mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><mi>Sp</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><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><mo>⋅</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msup><mi>S</mi> <mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><mi>Sp</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><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><mo>⋅</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msup><mi>S</mi> <mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><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></mtd> <mtd><mo>≃</mo><msup><mi>S</mi> <mn>6</mn></msup></mtd></mtr> <mtr><mtd><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></mtd> <mtd><mo>≃</mo><msup><mi>S</mi> <mn>7</mn></msup></mtd></mtr> <mtr><mtd><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></mtd> <mtd><mo>≃</mo><msup><mi>S</mi> <mn>15</mn></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} SO(n)/SO(n-1) & \simeq S^{n-1} \\ U(n)/U(n-1) & \simeq S^{2n-1} \\ SU(n)/SU(n-1) & \simeq S^{2n-1} \\ Sp(n)/Sp(n-1) & \simeq S^{4n-1} \\ Sp(n)\cdot SO(2)/Sp(n-1)\cdot SO(2) & \simeq S^{4n-1} \\ Sp(n)\cdot Sp(1)/Sp(n-1)\cdot Sp(1) & \simeq S^{4n-1} \\ G_2/SU(3) & \simeq S^6 \\ Spin(7)/G_2 & \simeq S^7 \\ Spin(9)/Spin(7) & \simeq S^{15} \end{aligned} </annotation></semantics></math></div></div> <p>This goes back to <a href="#MontgomerySamelson43">Montgomery & Samelson (1943)</a>, see <a href="#GrayGreen70">Gray & Green (1970), p. 1-2</a>, also <a href="#BorelSerre53">Borel & Serre (1953), 17.1</a>.</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> in Prop. <a class="maruku-ref" href="#nSphereAsCosetSpace"></a> and Prop. <a class="maruku-ref" href="#OddDimSphereAsSpecialUnitaryCoset"></a> above hold in the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> (<a class="existingWikiWord" href="/nlab/show/homeomorphisms">homeomorphisms</a>), but in fact also in the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> (<a class="existingWikiWord" href="/nlab/show/diffeomorphisms">diffeomorphisms</a>) and even in the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/Riemannian+manifolds">Riemannian manifolds</a> (<a class="existingWikiWord" href="/nlab/show/isometries">isometries</a>).</p> <p>The other <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> realizations of some <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a> in Prop. <a class="maruku-ref" href="#TransitiveEffectiveActionsOfConnectedLieGroupsOnSpheres"></a> are <a class="existingWikiWord" href="/nlab/show/homeomorphisms">homeomorphisms</a>, but not necessarily <a class="existingWikiWord" href="/nlab/show/isometries">isometries</a> (“<a class="existingWikiWord" href="/nlab/show/squashed+spheres">squashed spheres</a>”). There is also a <a class="existingWikiWord" href="/nlab/show/double+coset+space">double coset space</a> realization which is not even a <a class="existingWikiWord" href="/nlab/show/diffeomorphisms">diffeomorphisms</a> (“<a class="existingWikiWord" href="/nlab/show/exotic+sphere">exotic sphere</a>”, the <a class="existingWikiWord" href="/nlab/show/Gromoll-Meyer+sphere">Gromoll-Meyer sphere</a>).</p> <p>For more see <em><a href="7-sphere#CosetSpaceRealization">7-sphere – Coset space realization</a></em>.</p> </div> <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> <h4 id="AsQuotientsOfLorentzGroups">As quotients of Lorentz groups</h4> <p>If one drops the assumption of <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compactness</a>, then there are further coset space realizations of <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, notably as <a class="existingWikiWord" href="/nlab/show/quotient+group">quotients</a> of <a class="existingWikiWord" href="/nlab/show/Lorentz+groups">Lorentz groups</a> by <a class="existingWikiWord" href="/nlab/show/parabolic+subgroups">parabolic subgroups</a>: <em><a class="existingWikiWord" href="/nlab/show/celestial+spheres">celestial spheres</a></em>, e.g.: <a href="#Toller03">Toller (2003, p. 18)</a>, <a href="#Varlamov06">Varlamov (2006, p. 6)</a>, <a href="https://math.stackexchange.com/a/4092474/58526">Math.SE:a/4092474</a>.</p> <h3 id="spin_structure">Spin structure</h3> <p> <div class='num_remark'> <h6>Example</h6> <p>The <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>-sphere, for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, carries a canonical <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, induced from its <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a>-realization <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>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^n \simeq Spin(n+1)/Spin(n)</annotation></semantics></math> (<a href="#LabelCosetSpaceStructure">above</a>), as a special case of the canonical <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>-structure on <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 href="G-structure#CanonicalHStructureOnFModH">this example</a>).</p> </div> </p> <p>Other ways to see this:</p> <ul> <li id="Nowaczyk15"> <p>Nikolai Nowaczyk, Theorem A.6.6 in: <em>Dirac Eigenvalues of higher Multiplicity</em>, Regensburg 2015 (<a href="https://arxiv.org/abs/1501.04045">arXiv:1501.04045</a>)</p> </li> <li id="SpinorsInGeometryAndPhysics"> <p>S. Gutt, <em>Killing spinors on spheres and projective spaces</em>, p. 238-248 in: A. Trautman, G. Furlan (eds.) <em>Spinors in Geometry and Physics – Trieste 11-13 September 1986</em>, World Scientific 1988 (<a href="https://doi.org/10.1142/9789814541510">doi:10.1142/9789814541510</a>, <a href="https://books.google.ae/books?id=d14GCwAAQBAJ&lpg=PA243&ots=_tH_He8UFg&dq=%22spin%20structure%20on%20spheres%22&pg=PA243#v=onepage&q=%22spin%20structure%20on%20spheres%22&f=false">GBooks, p. 243</a>)</p> </li> </ul> <p><br /></p> <h3 id="parallelizability">Parallelizability</h3> <ul> <li> <p>Precisely four spheres are <a class="existingWikiWord" href="/nlab/show/parallelizable">parallelizable</a>, and three of these are so via <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> structure (hence are the only spheres with Lie group structure) (see at <em><a class="existingWikiWord" href="/nlab/show/Hopf+invariant+one+theorem">Hopf invariant one theorem</a></em>):</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">S^0</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/group+of+order+two">group of order two</a>, the group of units of the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>, the group of unit <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^3</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(2)</annotation></semantics></math>, the group of unit <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">S^7</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/Moufang+loop">Moufang loop</a> of unit <a class="existingWikiWord" href="/nlab/show/octonions">octonions</a>)</p> </li> </ul> </li> </ul> <h3 id="branched_covers">Branched covers</h3> <p>Every <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>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/PL+manifold">PL manifold</a> admits a <a class="existingWikiWord" href="/nlab/show/branched+covering">branched covering</a> of the <a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a> (<a href="branched+cover#Alexander20">Alexander 20</a>).</p> <p>By the <a class="existingWikiWord" href="/nlab/show/Riemann+existence+theorem">Riemann existence theorem</a>, every <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a> <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a> admits the <a class="existingWikiWord" href="/nlab/show/structure">structure</a> of a branched cover by a <a class="existingWikiWord" href="/nlab/show/holomorphic+function">holomorphic function</a> to the <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a>. See <a href="branched+cover+of+the+Riemann+sphere#RiemannSurfaces">there</a> at <em><a class="existingWikiWord" href="/nlab/show/branched+cover+of+the+Riemann+sphere">branched cover of the Riemann sphere</a></em>.</p> <div style="text-align: center"> <img src="https://ncatlab.org/nlab/files/TorusBranchedCoverOverSphere.jpg" width="500" /> </div> <blockquote> <p>graphics grabbed from <a href="branched+cover#ChamseddineConnesMukhanov14">Chamseddine-Connes-Mukhanov 14, Figure 1</a>, <a href="branched+cover#Connes17">Connes 17, Figure 11</a></p> </blockquote> <p>For <a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a> branched covering the <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a> see (<a href="branched+cover#Montesinos74">Montesinos 74</a>).</p> <p>All <a class="existingWikiWord" href="/nlab/show/PL+manifold">PL</a> <a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a> are <em>simple</em> branched covers of the <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a> (<a href="#Piergallini95">Piergallini 95</a>, <a href="branched+cover#IoriPiergallini02">Iori-Piergallini 02</a>).</p> <p>But the <a class="existingWikiWord" href="/nlab/show/n-torus">n-torus</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n \geq 3</annotation></semantics></math> is <em>not a <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic</a></em> branched over of the <a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a> (<a href="branched+cover#HirschNeumann75">Hirsch-Neumann 75</a>)</p> <h3 id="iterated_loop_spaces">Iterated loop spaces</h3> <div class="num_prop" id="RationalCohomologyOfIteratedLoopSpaceOf2kSphere"> <h6 id="proposition_5">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/rational+cohomology">rational cohomology</a> of <a class="existingWikiWord" href="/nlab/show/iterated+loop+space">iterated loop space</a> of the <a class="existingWikiWord" href="/nlab/show/n-sphere">2k-sphere</a>)</strong></p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>D</mi><mo><</mo><mi>n</mi><mo>=</mo><mn>2</mn><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex"> 1 \leq D \lt n = 2k \in \mathbb{N} </annotation></semantics></math></div> <p>(hence two <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>, one of them required to be <a class="existingWikiWord" href="/nlab/show/even+number">even</a> and the other required to be smaller than the first) and consider the <a class="existingWikiWord" href="/nlab/show/iterated+loop+space">D-fold loop space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>D</mi></msup><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Omega^D S^n</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a>.</p> <p>Its <a class="existingWikiWord" href="/nlab/show/rational+cohomology">rational</a> <a class="existingWikiWord" href="/nlab/show/cohomology+ring">cohomology ring</a> is the <a class="existingWikiWord" href="/nlab/show/free+construction">free</a> <a class="existingWikiWord" href="/nlab/show/graded-commutative+algebra">graded-commutative algebra</a> over <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">\mathbb{Q}</annotation></semantics></math> on one <a class="existingWikiWord" href="/nlab/show/generators+and+relations">generator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mrow><mi>n</mi><mo>−</mo><mi>D</mi></mrow></msub></mrow><annotation encoding="application/x-tex">e_{n-D}</annotation></semantics></math> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">n - D</annotation></semantics></math> and one generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mrow><mn>2</mn><mi>n</mi><mo>−</mo><mi>D</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">a_{2n - D - 1}</annotation></semantics></math> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mi>D</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">2n - D - 1</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Ω</mi> <mi>D</mi></msup><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mi>ℚ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℚ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><msub><mi>e</mi> <mrow><mi>n</mi><mo>−</mo><mi>D</mi></mrow></msub><mo>,</mo><msub><mi>a</mi> <mrow><mn>2</mn><mi>n</mi><mo>−</mo><mi>D</mi><mo>−</mo><mn>1</mn></mrow></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^\bullet \big( \Omega^D S^n , \mathbb{Q} \big) \;\simeq\; \mathbb{Q}\big[ e_{n - D}, a_{2n - D - 1} \big] \,. </annotation></semantics></math></div></div> <p>(<a href="#KallelSjerve99">Kallel-Sjerve 99, Prop. 4.10</a>)</p> <h2 id="Examples">Examples</h2> <ul> <li> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1)</annotation></semantics></math>-sphere is the <a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/0-sphere">0-sphere</a> is the <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> of two <a class="existingWikiWord" href="/nlab/show/points">points</a> (the classical <a class="existingWikiWord" href="/nlab/show/boolean+domain">boolean domain</a>).</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/1-sphere">1-sphere</a> is the <a class="existingWikiWord" href="/nlab/show/circle">circle</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a> is usual sphere from ordinary geometry. This canonically carries the structure of a <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> which makes it the <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a> and <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a>, <a class="existingWikiWord" href="/nlab/show/5-sphere">5-sphere</a> and <a class="existingWikiWord" href="/nlab/show/6-sphere">6-sphere</a> and <a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a> are interesting, too.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/round+sphere">round sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fuzzy+sphere">fuzzy sphere</a>, <a class="existingWikiWord" href="/nlab/show/Podle%C5%9B+sphere">Podleś sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hemisphere">hemisphere</a>, <a class="existingWikiWord" href="/nlab/show/equator">equator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unit+sphere">unit sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stereographic+projection">stereographic projection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+fiber+bundle">sphere fiber bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+type">sphere type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polar+coordinates">polar coordinates</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeb+sphere+theorem">Reeb sphere theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+spheres">homotopy groups of spheres</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+n-sphere">rational n-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+fibration">spherical fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+the+2-sphere">geometric quantization of the 2-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+sphere">representation sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+sphere">motivic sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+actions+on+spheres">group actions on spheres</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology+sphere">homology sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homology+sphere">rational homology sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+sphere">homotopy sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+sphere">rational homotopy sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+packing">sphere packing</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tate+sphere">Tate sphere</a></p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/non-abelian+cohomology">non-abelian</a> <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a> <a class="existingWikiWord" href="/nlab/show/representable+functor">represented</a> by <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a> is <a class="existingWikiWord" href="/nlab/show/Cohomotopy+cohomology+theory">Cohomotopy cohomology theory</a>.</p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="FP1990"> <p><a class="existingWikiWord" href="/nlab/show/Rudolf+Fritsch">Rudolf Fritsch</a>, <a class="existingWikiWord" href="/nlab/show/Renzo+A.+Piccinini">Renzo A. Piccinini</a>, <em>Cellular structures in topology</em>, Cambridge studies in advanced mathematics Vol. 19, Cambridge University Press (1990). (<a href="https://doi.org/10.1017/CBO9780511983948">doi:10.1017/CBO9780511983948</a>)</p> </li> <li id="tomDieck2008"> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <em>Algebraic topology</em>. European Mathematical Society, Zürich (2008) (<a href="https://doi.org/10.4171/048">doi:10.4171/048</a>)</p> </li> </ul> <h3 id="formalization">Formalization</h3> <p>Axiomatization of the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the 1-sphere (the <a class="existingWikiWord" href="/nlab/show/circle">circle</a>) and the 2-sphere, as <a class="existingWikiWord" href="/nlab/show/higher+inductive+types">higher inductive types</a>, is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>, section 6.4 of <em><a class="existingWikiWord" href="/nlab/show/Homotopy+Type+Theory+--+Univalent+Foundations+of+Mathematics">Homotopy Type Theory – Univalent Foundations of Mathematics</a></em></li> </ul> <p>Visualization of the idea of the construction for the 2-sphere is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andrej+Bauer">Andrej Bauer</a>, <em>HoTT <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math></em> (<a href="https://vimeo.com/119606901">video</a>)</li> </ul> <h3 id="group_actions_on_spheres">Group actions on spheres</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/free+actions">free</a> <a class="existingWikiWord" href="/nlab/show/group+actions+on+spheres">group actions on spheres</a> by <a class="existingWikiWord" href="/nlab/show/finite+groups">finite groups</a> includes</p> <ul> <li id="Wall78"> <p><a class="existingWikiWord" href="/nlab/show/C.+T.+C.+Wall">C. T. C. Wall</a>, <em>Free actions of finite groups on spheres</em>, Proceedings of Symposia in Pure Mathematics, Volume 32, 1978 (<a href="http://www.maths.ed.ac.uk/~aar/papers/wall7.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alejandro+Adem">Alejandro Adem</a>, <em>Constructing and deconstructing group actions</em> (<a href="http://arxiv.org/abs/math/0212280">arXiv:0212280</a>)</p> </li> </ul> <p>The subgroups of <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">SO(8)</a> which act freely on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">S^7</annotation></semantics></math> have been classified in</p> <ul> <li>J. A. Wolf, <em>Spaces of constant curvature</em>, Publish or Perish, Boston, Third ed., 1974</li> </ul> <p>and lifted to actions of <a class="existingWikiWord" href="/nlab/show/Spin+group">Spin(8)</a> in</p> <ul> <li id="Gadhia07"><a class="existingWikiWord" href="/nlab/show/Sunil+Gadhia">Sunil Gadhia</a>, <em>Supersymmetric quotients of M-theory and supergravity backgrounds</em>, PhD thesis, School of Mathematics, University of Edinburgh, 2007 (<a href="http://inspirehep.net/record/1393845/">spire:1393845</a>)</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/transitive+actions">transitive actions</a> on <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 by <a class="existingWikiWord" href="/nlab/show/compact+Lie+groups">compact Lie groups</a>:</p> <ul> <li id="MontgomerySamelson43"> <p><a class="existingWikiWord" href="/nlab/show/Deane+Montgomery">Deane Montgomery</a>, <a class="existingWikiWord" href="/nlab/show/Hans+Samelson">Hans Samelson</a>, <em>Transformation Groups of Spheres</em>, Annals of Mathematics Second Series <strong>44</strong> 3 (1943) 454-470 [<a href="https://www.jstor.org/stable/1968975">jstor:1968975</a>]</p> </li> <li id="GrayGreen70"> <p><a class="existingWikiWord" href="/nlab/show/Alfred+Gray">Alfred Gray</a>, Paul S. Green, <em>Sphere transitive structures and the triality automorphism</em>, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 (<a href="https://projecteuclid.org/euclid.pjm/1102976640">euclid:1102976640</a>)</p> </li> </ul> <p>Further discussion of these actions is in</p> <ul> <li id="MFFGME09"> <p><a class="existingWikiWord" href="/nlab/show/Paul+de+Medeiros">Paul de Medeiros</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Figueroa-O%27Farrill">José Figueroa-O'Farrill</a>, <a class="existingWikiWord" href="/nlab/show/Sunil+Gadhia">Sunil Gadhia</a>, <a class="existingWikiWord" href="/nlab/show/Elena+M%C3%A9ndez-Escobar">Elena Méndez-Escobar</a>, <em>Half-BPS quotients in M-theory: ADE with a twist</em>, JHEP 0910:038,2009 (<a href="http://arxiv.org/abs/0909.0163">arXiv:0909.0163</a>, <a href="http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf">pdf slides</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+de+Medeiros">Paul de Medeiros</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Figueroa-O%27Farrill">José Figueroa-O'Farrill</a>, <em>Half-BPS M2-brane orbifolds</em>, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (<a href="http://arxiv.org/abs/1007.4761">arXiv:1007.4761</a>, <a href="https://projecteuclid.org/euclid.atmp/1408561553">Euclid</a>)</p> </li> </ul> <p>where they are related to the <a class="existingWikiWord" href="/nlab/show/black+brane">black</a> <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <a class="existingWikiWord" href="/nlab/show/BPS+state">BPS</a>-solutions of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> at <a class="existingWikiWord" href="/nlab/show/ADE-singularities">ADE-singularities</a>.</p> <p>See also the <a class="existingWikiWord" href="/nlab/show/ADE+classification">ADE classification</a> of such actions on the <a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a> (as discussed there)</p> <p>Discussion of actions of <a class="existingWikiWord" href="/nlab/show/Lorentz+groups">Lorentz groups</a> on <a class="existingWikiWord" href="/nlab/show/celestial+spheres">celestial spheres</a>:</p> <ul> <li id="Toller03"> <p><a class="existingWikiWord" href="/nlab/show/Marco+Toller">Marco Toller</a>, <em>Homogeneous Spaces of the Lorentz Group</em> [<a href="https://arxiv.org/abs/math-ph/0301014">arXiv:math-ph/0301014</a>]</p> </li> <li id="Varlamov06"> <p>V. V. Varlamov, <em>Relativistic Spherical Functions on the Lorentz Group</em>, J. Phys. A: Math. Gen. <strong>39</strong> (2006) 805-822 [<a href="https://doi.org/10.1088/0305-4470/39/4/006">doi:10.1088/0305-4470/39/4/006</a>]</p> </li> </ul> <h3 id="geometric_structures_on_spheres">Geometric structures on spheres</h3> <p>Coset space structures on spheres:</p> <ul> <li id="BorelSerre53"><a class="existingWikiWord" href="/nlab/show/Armand+Borel">Armand Borel</a>, <a class="existingWikiWord" href="/nlab/show/Jean-Pierre+Serre">Jean-Pierre Serre</a>, <em>Groupes de Lie et Puissances Reduites de Steenrod</em>, American Journal of Mathematics, Vol. 75, No. 3 (Jul., 1953), pp. 409-448 (<a href="https://www.jstor.org/stable/2372495">jstor:2372495</a>)</li> </ul> <p>The following to be handled with care:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <em>The non-existent complex 6-sphere</em>, <a href="https://arxiv.org/abs/1610.09366">arxiv/1610.09366</a></li> </ul> <h3 id="embeddings_of_spheres">Embeddings of spheres</h3> <p>The (<a class="existingWikiWord" href="/nlab/show/isotopy">isotopy</a> <a class="existingWikiWord" href="/nlab/show/equivalence+class">class</a> of an) <a class="existingWikiWord" href="/nlab/show/embedding+of+differentiable+manifolds">embedding</a> of a <a class="existingWikiWord" href="/nlab/show/circle">circle</a> (1-sphere) into the <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a> is a <em><a class="existingWikiWord" href="/nlab/show/knot">knot</a></em>. Discussion of embeddings of spheres of more general dimensions into each other:</p> <ul> <li id="Haefliger66"><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Haefliger">André Haefliger</a>, <em>Differentiable Embeddings of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math> in <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><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">S^{n+q}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>></mo><mn>2</mn></mrow><annotation encoding="application/x-tex">q \gt 2</annotation></semantics></math></em>, Annals of Mathematics Second Series, Vol. 83, No. 3 (May, 1966), pp. 402-436 (<a href="https://www.jstor.org/stable/1970475">jstor:1970475</a>)</li> </ul> <h3 id="iterated_loop_spaces_2">Iterated loop spaces</h3> <ul> <li id="KallelSjerve99"><a class="existingWikiWord" href="/nlab/show/Sadok+Kallel">Sadok Kallel</a>, <a class="existingWikiWord" href="/nlab/show/Denis+Sjerve">Denis Sjerve</a>, <em>On Brace Products and the Structure of Fibrations with Section</em>, 1999 (<a href="https://www.math.ubc.ca/~sjer/brace.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/KallelSjerv99.pdf" title="pdf">pdf</a>)</li> </ul> <h3 id="topological_complexity">Topological complexity</h3> <p>On <a class="existingWikiWord" href="/nlab/show/topological+complexity">topological complexity</a> of spheres and <a class="existingWikiWord" href="/nlab/show/product+topological+space">products</a> of spheres (including <a class="existingWikiWord" href="/nlab/show/tori">tori</a> as special case):</p> <ul> <li id="Farber01"><a class="existingWikiWord" href="/nlab/show/Michael+Farber">Michael Farber</a>, <em>Topological complexity of motion planning</em> (2001), <a href="https://arxiv.org/abs/math/0111197">arXiv:math/0111197</a>;</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 31, 2025 at 08:55:15. See the <a href="/nlab/history/sphere" 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/sphere" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/9433/#Item_14">Discuss</a><span class="backintime"><a href="/nlab/revision/sphere/73" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/sphere" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/sphere" accesskey="S" class="navlink" id="history" rel="nofollow">History (73 revisions)</a> <a href="/nlab/show/sphere/cite" style="color: black">Cite</a> <a href="/nlab/print/sphere" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/sphere" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>