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Hopf fibration in nLab
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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 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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="bundles">Bundles</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/bundles">bundles</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/parameterized+stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> </ul> <h2 id="sidebar_context">Context</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/slice+topos">slice topos</a>, <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">linear</a>) <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a></p> </li> </ul> <h2 id="sidebar_classes_of_bundles">Classes of bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/retractive+space">retractive space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>, <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/numerable+bundle">numerable bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+bundle">sphere bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+bundle">projective bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+3-bundle">principal 3-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation+bundle">orientation bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a>, <a class="existingWikiWord" href="/nlab/show/stringor+bundle">stringor bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a>, <a class="existingWikiWord" href="/nlab/show/2-gerbe">2-gerbe</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-gerbe">∞-gerbe</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+coefficient+bundle">local coefficient bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/2-vector+bundle">2-vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/real+vector+bundle">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+vector+bundle">complex</a>/<a class="existingWikiWord" href="/nlab/show/holomorphic+vector+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/quaternionic+vector+bundle">quaternionic</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+vector+bundle">differentiable</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+vector+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+vector+bundle">with connection</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+line+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/cubical+structure+on+a+line+bundle">cubical structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a><a class="existingWikiWord" href="/nlab/show/Vect%28X%29">of vector bundles</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/VectBund">VectBund</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum</a>, <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product</a>, <a class="existingWikiWord" href="/nlab/show/external+tensor+product+of+vector+bundles">external tensor product</a>, <a class="existingWikiWord" href="/nlab/show/inner+product+of+vector+bundles">inner product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+vector+bundle">dual vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+vector+bundle">stable vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+of+spectra">bundle of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+bundle">natural bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bundle">equivariant bundle</a></p> </li> </ul> <h2 id="sidebar_universal_bundles">Universal bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+vector+bundle">universal vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+complex+line+bundle">universal complex line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a>, <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> </ul> <h2 id="sidebar_presentations">Presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microbundle">microbundle</a></p> </li> </ul> <h2 id="sidebar_examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+bundle">empty bundle</a>, <a class="existingWikiWord" href="/nlab/show/zero+bundle">zero bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tautological+line+bundle">tautological line bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/basic+line+bundle+on+the+2-sphere">basic line bundle on the 2-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+fibration">Hopf fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+line+bundle">canonical line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></p> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/clutching+construction">clutching construction</a></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="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <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/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen 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/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</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='#idea'>Idea</a></li> <li><a href='#DefinitionFor3Sphere'>On the 3-sphere</a></li> <ul> <li><a href='#HomotopyTheoreticCharacterization'>Homotopy-theoretic characterization</a></li> <li><a href='#realization_via_the_complex_numbers'>Realization via the complex numbers</a></li> <li><a href='#RealizationViaQuaternions'>Realization via quaternions</a></li> <li><a href='#realization_via_the_hopf_construction'>Realization via the Hopf construction</a></li> <li><a href='#'><a class="existingWikiWord" href="/nlab/show/Spin%283%29">Spin(3)</a>-<a class="existingWikiWord" href="/nlab/show/action">equivariance</a></a></li> </ul> <li><a href='#OnAllFourSpheres'>On the 1-sphere, 3-sphere, 7-sphere and 15-sphere</a></li> <ul> <li><a href='#via_norms_and_projections'>Via norms and projections</a></li> <li><a href='#via_the_hopf_construction'>Via the Hopf construction</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#RelationToStableHomotopyGroupsOfSpheres'>Relation to stable homotopy groups of spheres</a></li> </ul> <li><a href='#applications'>Applications</a></li> <ul> <li><a href='#magnetic_monopoles'>Magnetic monopoles</a></li> <li><a href='#ktheory'>K-theory</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>complex Hopf fibration</em> [<a href="#Hopf31">Hopf (1931)</a>] is a canonical nontrivial <a class="existingWikiWord" href="/nlab/show/circle+principal+bundle">circle principal bundle</a> over the <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a> whose total space is the <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>↪</mo><msup><mi>S</mi> <mn>3</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>2</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^1 \hookrightarrow S^3 \to S^2 \,. </annotation></semantics></math></div> <p>Its canonically <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> <a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex line bundle</a> is the <a class="existingWikiWord" href="/nlab/show/basic+line+bundle+on+the+2-sphere">basic line bundle on the 2-sphere</a>.</p> <p>This we discuss below in</p> <ul> <li><em><a href="#DefinitionFor3Sphere">On the 3-sphere</a></em></li> </ul> <p>More generally, there are four Hopf fibrations, on the 1-sphere, the 3-sphere, the 7-sphere and the 15-sphere, respectively. This we discuss in</p> <ul> <li><em><a href="#OnAllFourSpheres">On the 1-sphere, 3-sphere, 7-sphere and 15-sphere</a></em>.</li> </ul> <h2 id="DefinitionFor3Sphere">On the 3-sphere</h2> <h3 id="HomotopyTheoreticCharacterization">Homotopy-theoretic characterization</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+space">Eilenberg-MacLane space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>B</mi><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">K(\mathbb{Z},2) \simeq B S^1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> for <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> <a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a>. By its very nature, it has a single nontrivial <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>, the second, and this is isomorphic to the group of <a class="existingWikiWord" href="/nlab/show/integers">integers</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_2(K(\mathbb{Z},2)) \simeq \mathbb{Z} \,. </annotation></semantics></math></div> <p>This means that there is, up to <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, a canonical (up to sign), continuous map from the 2-sphere</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \phi : S^2 \to K(\mathbb{Z},2) \,, </annotation></semantics></math></div> <p>such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ϕ</mi><mo stretchy="false">]</mo><mo>∈</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">[\phi] \in \pi_2(K(\mathbb{Z},2)) = \pm 1 \in \mathbb{Z}</annotation></semantics></math>.</p> <p>As any map into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(\mathbb{Z},2)</annotation></semantics></math> this classifies a <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> <a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> over its <a class="existingWikiWord" href="/nlab/show/domain">domain</a>. This is the Hopf fibration, fitting into the long <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mn>1</mn></msup></mtd> <mtd><mo>↪</mo></mtd> <mtd><msup><mi>S</mi> <mn>3</mn></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>S</mi> <mn>2</mn></msup></mtd> <mtd><mover><mo>→</mo><mi>ϕ</mi></mover></mtd> <mtd><mi>B</mi><msup><mi>S</mi> <mn>1</mn></msup><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ S^1 &\hookrightarrow& S^3 \\ && \downarrow \\ && S^2 &\stackrel{\phi}{\to}& B S^1 \simeq K(\mathbb{Z},2) } \,. </annotation></semantics></math></div> <p>In other words, the Hopf fibration is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-bundle with unit first <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a> on <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>.</p> <h3 id="realization_via_the_complex_numbers">Realization via the complex numbers</h3> <p>An explicit <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> presenting the Hopf fibration may be obtained as follows. Identify</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">{</mo><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>z</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>ℂ</mi><mo>×</mo><mi>ℂ</mi><mspace width="thinmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">|</mo><mspace width="thinmathspace"></mspace><msup><mrow><mo stretchy="false">|</mo><msub><mi>z</mi> <mn>0</mn></msub><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">|</mo><msub><mi>z</mi> <mn>1</mn></msub><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>=</mo><mn>1</mn><mo maxsize="1.2em" minsize="1.2em">}</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>S</mi><mo stretchy="false">(</mo><msup><mi>ℂ</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S^3 \;\simeq\; \big\{ (z_0, z_1) \in \mathbb{C}\times \mathbb{C} \,\big\vert\, {|z_0|}^2 + {|z_1|}^2 = 1 \big\} \;=\; S(\mathbb{C}^2) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mi>ℂℙ</mi> <mn>1</mn></msup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℂ</mi><mo>⊔</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S^2 \;\simeq\; \mathbb{C P}^1 \;\simeq\; \mathbb{C} \sqcup \{\infty\} \,. </annotation></semantics></math></div> <p>Then the <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^3 \to S^2</annotation></semantics></math> defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>0</mn></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>z</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mfrac><mrow><msub><mi>z</mi> <mn>0</mn></msub></mrow><mrow><msub><mi>z</mi> <mn>1</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex"> (z_0, \,z_1) \;\mapsto\; \frac{z_0}{z_1} </annotation></semantics></math></div> <p>gives the Hopf fibration. (Thus, the Hopf fibration is the circle bundle naturally <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> with the <a class="existingWikiWord" href="/nlab/show/canonical+line+bundle">canonical line bundle</a>.)</p> <p>Alternatively, if we identify</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">{</mo><mo stretchy="false">(</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℂ</mi><mo>×</mo><mi>ℝ</mi><mspace width="thinmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">|</mo><mspace width="thinmathspace"></mspace><msup><mrow><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>+</mo><msup><mi>x</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn><mo maxsize="1.2em" minsize="1.2em">}</mo><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mi>S</mi><mo stretchy="false">(</mo><mi>ℂ</mi><mo>×</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S^2 \;\simeq\; \big\{ (z, x) \in \mathbb{C} \times \mathbb{R} \,\big\vert\, {|z|}^2 + x^2 = 1 \big\} \,=\, S(\mathbb{C} \times \mathbb{R}) </annotation></semantics></math></div> <p>and identify this presentation of the 2-sphere with the complex projective line via <a class="existingWikiWord" href="/nlab/show/stereographic+projection">stereographic projection</a>, then the Hopf fibration is identified with the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>z</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mn>2</mn><msub><mi>z</mi> <mn>0</mn></msub><mover><mrow><msub><mi>z</mi> <mn>1</mn></msub></mrow><mo>¯</mo></mover><mo>,</mo><mspace width="thinmathspace"></mspace><msup><mrow><mo stretchy="false">|</mo><msub><mi>z</mi> <mn>0</mn></msub><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">|</mo><msub><mi>z</mi> <mn>1</mn></msub><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (z_0, z_1) \;\mapsto\; \big( 2 z_0 \overline{z_1} ,\, {|z_0|}^2 - {|z_1|}^2 \big) \,. </annotation></semantics></math></div> <h3 id="RealizationViaQuaternions">Realization via quaternions</h3> <p>Alternatively, we may regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup><mo>≃</mo><mi>S</mi><mo stretchy="false">(</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^3 \simeq S(\mathbb{H})</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/unit+sphere">unit sphere</a> in the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo>≃</mo><mi>S</mi><mrow><mo>(</mo><msub><mi>ℍ</mi> <mi mathvariant="normal">im</mi></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">S^2 \simeq S\left( \mathbb{H}_{\mathrm{im}}\right)</annotation></semantics></math> as the unit sphere in the <a class="existingWikiWord" href="/nlab/show/imaginary+part">imaginary</a> <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a>. Under this identification, the complex Hopf fibration is equivalently represented by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>S</mi><mo stretchy="false">(</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>S</mi><mrow><mo>(</mo><msub><mi>ℍ</mi> <mi mathvariant="normal">im</mi></msub><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mi>q</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>q</mi><mo>⋅</mo><mstyle mathvariant="bold"><mi>i</mi></mstyle><mo>⋅</mo><mover><mi>q</mi><mo>¯</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S(\mathbb{H}) &\longrightarrow& S\left( \mathbb{H}_{\mathrm{im}}\right) \\ q &\mapsto& q \cdot \mathbf{i} \cdot \overline{q} } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>i</mi></mstyle><mo>∈</mo><mi>S</mi><mrow><mo>(</mo><msub><mi>ℍ</mi> <mi mathvariant="normal">im</mi></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathbf{i} \in S\left( \mathbb{H}_{\mathrm{im}}\right)</annotation></semantics></math> is any unit imaginary quaternion.</p> <h3 id="realization_via_the_hopf_construction">Realization via the Hopf construction</h3> <p>Regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^1 = U(1)</annotation></semantics></math> as equipped with its <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> structure. This makes <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> in particular an <a class="existingWikiWord" href="/nlab/show/H-space">H-space</a>. The Hopf fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>3</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^1 \to S^3 \to S^2</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Hopf+construction">Hopf construction</a> applied to this H-space.</p> <h3 id=""><a class="existingWikiWord" href="/nlab/show/Spin%283%29">Spin(3)</a>-<a class="existingWikiWord" href="/nlab/show/action">equivariance</a></h3> <p>Consider</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Spin%283%29">Spin(3)</a>-<a class="existingWikiWord" href="/nlab/show/action">action</a> on the <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a> <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> which is induced by the defining action on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^3</annotation></semantics></math> under the identification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo>≃</mo><mi>S</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^2 \simeq S(\mathbb{R}^3)</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Spin%283%29">Spin(3)</a>-action on the <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^3</annotation></semantics></math> which is induced under the exceptional <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(3) \simeq Sp(1) = U(1,\mathbb{H})</annotation></semantics></math> by the canonical left action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1,\mathbb{H})</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">\mathbb{H}</annotation></semantics></math> via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup><mo>≃</mo><mi>S</mi><mo stretchy="false">(</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^3 \simeq S(\mathbb{H})</annotation></semantics></math>.</p> </li> </ol> <p>Then the complex Hopf fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup><mover><mo>⟶</mo><mrow><msub><mi>h</mi> <mi>ℂ</mi></msub></mrow></mover><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^3 \overset{h_{\mathbb{C}}}{\longrightarrow} S^2</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/equivariant">equivariant</a> with respect to these <a class="existingWikiWord" href="/nlab/show/actions">actions</a>.</p> <p>A way to make the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(3)</annotation></semantics></math>-equivariance of the complex Hopf fibration fully explicit is to observe that it is equivalently the following map of <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><mtr><mtd><msup><mi>S</mi> <mn>1</mn></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>fib</mi><mo stretchy="false">(</mo><msub><mi>h</mi> <mi>ℂ</mi></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mi>S</mi> <mn>3</mn></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>h</mi> <mi>ℂ</mi></msub></mrow></mover></mtd> <mtd><msup><mi>S</mi> <mn>2</mn></msup></mtd></mtr> <mtr><mtd><mo>=</mo></mtd> <mtd></mtd> <mtd><mo>=</mo></mtd> <mtd></mtd> <mtd><mo>=</mo></mtd></mtr> <mtr><mtd><mfrac><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mfrac><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mfrac><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^1 &\overset{fib(h_{\mathbb{C}})}{\longrightarrow}& S^{3} &\overset{h_{\mathbb{C}}}{\longrightarrow}& S^2 \\ = && = && = \\ \frac{Spin(2)}{Spin(1)} &\longrightarrow& \frac{Spin(3)}{Spin(1)} &\longrightarrow& \frac{Spin(3)}{Spin(2)} } </annotation></semantics></math></div> <h2 id="OnAllFourSpheres">On the 1-sphere, 3-sphere, 7-sphere and 15-sphere</h2> <h3 id="via_norms_and_projections">Via norms and projections</h3> <p>For each of the <a class="existingWikiWord" href="/nlab/show/normed+division+algebra">normed division algebras</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>, <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>, <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a>, <a class="existingWikiWord" href="/nlab/show/octonions">octonions</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>ℝ</mi><mo>,</mo><mi>ℂ</mi><mo>,</mo><mi>ℍ</mi><mo>,</mo><mi>𝕆</mi><mo>,</mo></mrow><annotation encoding="application/x-tex">A = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O},</annotation></semantics></math></div> <p>there is a corresponding Hopf fibration of <a class="existingWikiWord" href="/nlab/show/Hopf+invariant+one">Hopf invariant one</a>.</p> <p>The total space of the fibration is the space of pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>A</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">(\alpha, \beta) \in A^2</annotation></semantics></math> of unit norm: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow><mo stretchy="false">|</mo><mi>α</mi><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">|</mo><mi>β</mi><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">{|\alpha|}^2 + {|\beta|}^2 = 1</annotation></semantics></math>. This gives <a class="existingWikiWord" href="/nlab/show/spheres">spheres</a> of dimension 1, <a class="existingWikiWord" href="/nlab/show/3-sphere">3</a>, <a class="existingWikiWord" href="/nlab/show/7-sphere">7</a>, and 15 respectively. The base space of the fibration is <a class="existingWikiWord" href="/nlab/show/projective+space">projective</a> 1-space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℙ</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{P}^1(A)</annotation></semantics></math>, giving spheres of dimension 1, 2, 4, and 8, respectively. In each case, the Hopf fibration is the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><msup><mn>2</mn> <mi>n</mi></msup><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>S</mi> <mrow><msup><mn>2</mn> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></msup></mrow><annotation encoding="application/x-tex">S^{2^n - 1} \to S^{2^{n-1}}</annotation></semantics></math></div> <p>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">n = 1, 2, 3, 4</annotation></semantics></math>) which sends the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\alpha, \beta)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo stretchy="false">/</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">\alpha/\beta</annotation></semantics></math>.</p> <h3 id="via_the_hopf_construction">Via the Hopf construction</h3> <p>When <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> is a <a class="existingWikiWord" href="/nlab/show/sphere">sphere</a> that is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>-space, namely, one of the <a class="existingWikiWord" href="/nlab/show/groups">groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">S^0 = 1</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/trivial+group">trivial group</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>=</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">S^1 = \mathbb{Z}/2</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/group+of+order+2">group of order 2</a>, the <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a> <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><msup><mi>S</mi> <mn>3</mn></msup><mo>=</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^3 = SU(2)</annotation></semantics></math>; or the <a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a> <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> with its <a class="existingWikiWord" href="/nlab/show/Moufang+loop">Moufang loop</a> structure, then the Hopf construction produces the above four Hopf fibrations:</p> <ol> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup><mo>↪</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^0 \hookrightarrow S^1 \to S^1 </annotation></semantics></math> – <a class="existingWikiWord" href="/nlab/show/real+Hopf+fibration">real Hopf fibration</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>↪</mo><msup><mi>S</mi> <mn>3</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex"> S^1 \hookrightarrow S^3 \to S^2 </annotation></semantics></math> – <a class="existingWikiWord" href="/nlab/show/complex+Hopf+fibration">complex Hopf fibration</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup><mo>↪</mo><msup><mi>S</mi> <mn>7</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex"> S^3 \hookrightarrow S^7 \to S^4 </annotation></semantics></math> – <a class="existingWikiWord" href="/nlab/show/quaternionic+Hopf+fibration">quaternionic Hopf fibration</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><mo>↪</mo><msup><mi>S</mi> <mn>15</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>8</mn></msup></mrow><annotation encoding="application/x-tex"> S^7 \hookrightarrow S^{15} \to S^8 </annotation></semantics></math> – <a class="existingWikiWord" href="/nlab/show/octonionic+Hopf+fibration">octonionic Hopf fibration</a></li> </ol> <h2 id="properties">Properties</h2> <h3 id="RelationToStableHomotopyGroupsOfSpheres">Relation to stable homotopy groups of spheres</h3> <p>Let</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>ℂ</mi></msub><mo>∈</mo><msub><mi>π</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_{\mathbb{C}} \in \pi_3(S^2)</annotation></semantics></math>,</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>ℍ</mi></msub><mo>∈</mo><msub><mi>π</mi> <mn>7</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_{\mathbb{H}} \in \pi_7(S^4)</annotation></semantics></math></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>𝕆</mi></msub><mo>∈</mo><msub><mi>π</mi> <mn>15</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>8</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_{\mathbb{O}} \in \pi_{15}(S^8)</annotation></semantics></math></p> <p>be the <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of the <a class="existingWikiWord" href="/nlab/show/complex+Hopf+fibration">complex Hopf fibration</a>, the <a class="existingWikiWord" href="/nlab/show/quaternionic+Hopf+fibration">quaternionic Hopf fibration</a> and the <a class="existingWikiWord" href="/nlab/show/octonionic+Hopf+fibration">octonionic Hopf fibration</a>, respectively. Then their <a class="existingWikiWord" href="/nlab/show/suspensions">suspensions</a> are the generators of the corresponding <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</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>Σ</mi><msub><mi>H</mi> <mi>ℂ</mi></msub></mtd> <mtd><mo>=</mo><mo>±</mo><mn>1</mn><mo>∈</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>≃</mo><msubsup><mi>π</mi> <mn>1</mn> <mi>st</mi></msubsup></mtd></mtr> <mtr><mtd><mi>Σ</mi><msub><mi>H</mi> <mi>ℍ</mi></msub></mtd> <mtd><mo>=</mo><mo>±</mo><mn>1</mn><mo>∈</mo><msub><mi>ℤ</mi> <mn>24</mn></msub><mo>≃</mo><msubsup><mi>π</mi> <mn>3</mn> <mi>st</mi></msubsup></mtd></mtr> <mtr><mtd><mi>Σ</mi><msub><mi>H</mi> <mi>𝕆</mi></msub></mtd> <mtd><mo>=</mo><mo>±</mo><mn>1</mn><mo>∈</mo><msub><mi>ℤ</mi> <mn>240</mn></msub><mo>≃</mo><msubsup><mi>π</mi> <mn>7</mn> <mi>st</mi></msubsup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Sigma H_{\mathbb{C}} & = \pm 1 \in \mathbb{Z}_2 \simeq \pi_1^{st} \\ \Sigma H_{\mathbb{H}} & = \pm 1 \in \mathbb{Z}_{24} \simeq \pi_3^{st} \\ \Sigma H_{\mathbb{O}} & = \pm 1 \in \mathbb{Z}_{240} \simeq \pi_7^{st} \end{aligned} </annotation></semantics></math></div> <p>see also at</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/first+stable+homotopy+group+of+spheres">first stable homotopy group of spheres</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/third+stable+homotopy+group+of+spheres">third stable homotopy group of spheres</a></p> </li> </ul> <p>and <a href="https://mathoverflow.net/a/224082/381">this MO comment</a></p> <h2 id="applications">Applications</h2> <h3 id="magnetic_monopoles">Magnetic monopoles</h3> <p>When <a class="existingWikiWord" href="/nlab/show/line+bundles">line bundles</a> are regarded as models for the topological structure underlying the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> the Hopf fibration is often called “the <a class="existingWikiWord" href="/nlab/show/magnetic+monopole">magnetic monopole</a>”. We may think of the <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> homotopically as being the 3-dimensional <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> with origin removed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>3</mn></msup><mo>−</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^3 - \{0\}</annotation></semantics></math> and think of this as being 3-dimensional physical space with a unit point <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> at the origin removed. The corresponding <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> away from the origin is given by a <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> on the corresponding Hopf fibration bundle.</p> <h3 id="ktheory">K-theory</h3> <p>In complex <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a>, the Hopf fibration represents a class <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> which generates the <a class="existingWikiWord" href="/nlab/show/cohomology+ring">cohomology ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>U</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K_U(S^2)</annotation></semantics></math>, and satisfying the relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>2</mn></msup><mo>=</mo><mn>2</mn><mo>⋅</mo><mi>H</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">H^2 = 2 \cdot H - 1</annotation></semantics></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>H</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(H-1)^2 = 0</annotation></semantics></math>. (So in particular <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> has an <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>2</mn><mo>−</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">H^{-1} = 2- H </annotation></semantics></math>, see at <a class="existingWikiWord" href="/nlab/show/Bott+generator">Bott generator</a>.)</p> <p>A succinct formulation of <a class="existingWikiWord" href="/nlab/show/Bott+periodicity">Bott periodicity</a> for <a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a> is that for a space <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 <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> is that of a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>×</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>K</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>⊗</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(S^2 \times X) \cong K(S^2) \otimes K(X)</annotation></semantics></math></div> <p>(It would be interesting to see whether this can be proved by internalizing the (classically easy) calculation for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(S^2)</annotation></semantics></math> to the topos of sheaves over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.)</p> <p>The Hopf fibrations over other <a class="existingWikiWord" href="/nlab/show/normed+division+algebras">normed division algebras</a> also figure in the more complicated case of <a class="existingWikiWord" href="/nlab/show/real+K-theory">real K-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>O</mi></msub></mrow><annotation encoding="application/x-tex">K_O</annotation></semantics></math>: they can be used to provide generators for the non-zero <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mi>O</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(B O)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of the <a class="existingWikiWord" href="/nlab/show/stable+orthogonal+group">stable orthogonal group</a>, which are periodic of period 8 (not coincidentally, 8 is the dimension of the largest normed division algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕆</mi></mrow><annotation encoding="application/x-tex">\mathbb{O}</annotation></semantics></math>). [To be followed up on.]</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+construction">Hopf construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+invariant+one">Hopf invariant one</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/real+Hopf+fibration">real Hopf fibration</a>, <a class="existingWikiWord" href="/nlab/show/quaternionic+Hopf+fibration">quaternionic Hopf fibration</a>, <a class="existingWikiWord" href="/nlab/show/octonionic+Hopf+fibration">octonionic Hopf fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex projective space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf+construction+in+homotopy+type+theory">Hopf construction in homotopy type theory</a></p> </li> </ul> <h2 id="References">References</h2> <p>Original articles:</p> <ul> <li id="Hopf31"> <p><a class="existingWikiWord" href="/nlab/show/Heinz+Hopf">Heinz Hopf</a>, <em>Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche</em>, Mathematische Annalen <strong>104</strong> (1931) 637–665 [<a href="https://doi.org/10.1007/BF01457962">doi:10.1007/BF01457962</a>]</p> </li> <li id="Hopf35"> <p><a class="existingWikiWord" href="/nlab/show/Heinz+Hopf">Heinz Hopf</a>, <em>Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension</em>, Fundamenta Mathematicae <strong>25</strong> 1 (1935) 427-440 [<a href="https://eudml.org/doc/212801">eudml:212801</a>]</p> </li> </ul> <p>Exposition:</p> <ul> <li>Saifuddin Syed, <em>Group structure on spheres and the Hopf fibration</em> [<a class="existingWikiWord" href="/nlab/files/SyedHopfFibration.pdf" title="pdf">pdf</a>]</li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Herman+Gluck">Herman Gluck</a>, <a class="existingWikiWord" href="/nlab/show/Frank+Warner">Frank Warner</a>, <a class="existingWikiWord" href="/nlab/show/Chung+Tao+Yang">Chung Tao Yang</a>, Section 6 of: <em>Division algebras, fibrations of spheres by great spheres and the topological determination of space by the gross behavior of its geodesics</em>, Duke Math. J. Volume 50, Number 4 (1983), 1041-1076 (<a href="https://projecteuclid.org/euclid.dmj/1077303489">euclid:dmj/1077303489</a>)</p> </li> <li id="GluckWarnerZiller86"> <p><a class="existingWikiWord" href="/nlab/show/Herman+Gluck">Herman Gluck</a>, <a class="existingWikiWord" href="/nlab/show/Frank+Warner">Frank Warner</a>, <a class="existingWikiWord" href="/nlab/show/Wolfgang+Ziller">Wolfgang Ziller</a>, <em>The geometry of the Hopf fibrations</em>, L’Enseignement Mathématique, <strong>32</strong> (1986), 173-198 [<a href="https://www.researchgate.net/publication/266548925_The_geometry_of_the_Hopf_fibrations">ResearchGate</a>, <a class="existingWikiWord" href="/nlab/files/GluckWarnerZiller-HopfFibrations.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><em><a href="https://chiasme.wordpress.com/2014/01/04/third-homotopy-group-of-the-sphere-and-hopf-fibration/">Third homotopy group of the sphere and Hopf fibration</a></em></p> </li> </ul> <p>and via the <a class="existingWikiWord" href="/nlab/show/Pontrjagin+theorem">Pontrjagin theorem</a>:</p> <ul> <li id="Kosinski93"><a class="existingWikiWord" href="/nlab/show/Antoni+Kosinski">Antoni Kosinski</a>, p. 185 of: <em>Differential manifolds</em>, Academic Press (1993) [<a href="http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf">pdf</a>, <a href="https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/138/suppl/C">ISBN:978-0-12-421850-5</a>]</li> </ul> <p>Formulation in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <ul> <li><a href="https://ncatlab.org/homotopytypetheory/show/HomePage">HoTT Wiki</a>, <em><a class="existingWikiWord" href="/homotopytypetheory/show/Hopf+fibration">Hopf fibration</a></em></li> </ul> <p>Relation to <a class="existingWikiWord" href="/nlab/show/skyrmions">skyrmions</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Sven+Bjarke+Gudnason">Sven Bjarke Gudnason</a>, <a class="existingWikiWord" href="/nlab/show/Muneto+Nitta">Muneto Nitta</a>, <em>Linking number of vortices as baryon number</em> (<a href="https://arxiv.org/abs/2002.01762">arXiv:2002.01762</a>)</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetric</a> Hopf fibrations:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A.+P.+Balachandran">A. P. Balachandran</a>, G. Marmo, B.-S. Skagerstam and A. Stern, section 9.3 of <em>Gauge Symmetries and Fibre Bundles</em>, Lect. Notes in Physics 188, Springer-Verlag, Berlin, 1983 (<a href="https://arxiv.org/abs/1702.08910">arXiv:1702.08910</a>)</p> </li> <li> <p>Simon Davis, section 3 of <em>Supersymmetry and the Hopf fibration</em> (<a href="https://doi.org/10.4995/agt.2012.1623">doi:10.4995/agt.2012.1623</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 17, 2024 at 18:51:05. 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