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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="linear_algebra">Linear algebra</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> </div> </div> <h1 id="vector_bundles">Vector bundles</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#standard'>Standard</a></li> <li><a href='#sheaftheoretic_version'>Sheaf-theoretic version</a></li> <li><a href='#virtual_vector_bundles'>Virtual vector bundles</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#Literature'>Literature</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Given some context of <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>, then a <em>vector bundle</em> is a collection of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> that varies in a geometric way over a given base <a class="existingWikiWord" href="/nlab/show/space">space</a> <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>: over each <a class="existingWikiWord" href="/nlab/show/generalized+element">element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">V_x</annotation></semantics></math>, called the <em><a class="existingWikiWord" href="/nlab/show/fiber">fiber</a></em> 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>, and as <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> varies in <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>, the fibers vary along in a geometric way. One also says that vector bundles are <em><a class="existingWikiWord" href="/nlab/show/fiber+bundles">fiber bundles</a></em> whose fiber carries <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>-structure. Hence the theory of vector bundles is <em>parameterized</em> <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a>. The vector bundles over a fixed base <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> form a category <a class="existingWikiWord" href="/nlab/show/Vect%28X%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>VectBund</mi> <mi>X</mi></msub> </mrow> <annotation encoding="application/x-tex">VectBund_X</annotation> </semantics> </math></a>, and as the base space is allowed to vary these fit into a global category <a class="existingWikiWord" href="/nlab/show/VectBund">VectBund</a>.</p> <p>For example</p> <ul> <li> <p>in <a class="existingWikiWord" href="/nlab/show/topology">topology</a> a <em><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a></em> is a collection of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> which “vary continuously” over a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <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> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> a <em><a class="existingWikiWord" href="/nlab/show/differentiable+vector+bundle">differentiable vector bundle</a></em> is a collection of vector space which “varies differentiably” over a <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>,</p> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a> an <em><a class="existingWikiWord" href="/nlab/show/algebraic+vector+bundle">algebraic vector bundle</a></em> is a collection of vector spaces which vary algebraically over a <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a>.</p> </li> </ul> <p>and so on.</p> <p>One requires that “locally”, on small enough patches of the base 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>, the variation of the fibers is constant up to isomorphism (one says the vector bundle is “locally trivial”), but the key point of vector bundles is that there may be non-trivial structure in how the collection of vector spaces “globally glues together”.</p> <p>For example if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">X = S^1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> regarded as a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> in the standard way, and if we consider <a class="existingWikiWord" href="/nlab/show/real+vector+spaces">real vector spaces</a>, then there are up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> two different <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>-vector bundles over <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> whose <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> look like the 1-dimensional real vector space <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> itself, namely</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a></p> <p><img src="https://ncatlab.org/nlab/files/cylinder.jpg" width="190" /></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/M%C3%B6bius+strip">Möbius strip</a>:</p> <p><img src="https://ncatlab.org/nlab/files/moebiusstrip.jpg" width="200" /></p> </li> </ol> <p>(In these pictures each vertical interval is to be thought of as a stand-in for a copy of the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> <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>.)</p> <p>Clearly for the cylinder nothing special happens to the fibers as one moves around the circle (one says this is a <em>trivial vector bundle</em>) while the Möbius strip is “locally trivial” but globally has a twist: as one moves once around the circle the original fiber comes back identified with its reflection at the origin.</p> <div style="float:right;margin:0 10px 10px 0;"> <img src="https://ncatlab.org/nlab/files/TangentSpaceToSphere.png" width="250" /> <blockquote> graphics grabbed from <a href="Hatcher">Hatcher</a> </blockquote> </div> <p>An important class of examples of vector bundles are <a class="existingWikiWord" href="/nlab/show/tangent+bundles">tangent bundles</a> of <a class="existingWikiWord" href="/nlab/show/differentiable+manifolds">differentiable manifolds</a> <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>. Here the <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> at each point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/tangent+space">tangent space</a> of that point, the space of all <a class="existingWikiWord" href="/nlab/show/tangent+vectors">tangent vectors</a> based at that point. The graphics on the right shows one of the tangent space of the <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a>.</p> <p>Dually, given an <a class="existingWikiWord" href="/nlab/show/embedding+of+differentiable+manifolds">embedding of differentiable manifolds</a> into a <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a>, then the <a class="existingWikiWord" href="/nlab/show/normal+vectors">normal vectors</a> to the tangent bundle span a vector bundle called the <em><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></em> of the embedding.</p> <p>All the usual operations on <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+spaces">finite dimensional vector spaces</a> in <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a> generalize to vector bundles by applying them <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a>-wise. For instance there is <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a> and the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product of vector bundles</a> over the same base space.</p> <p>To the extent that the base <a class="existingWikiWord" href="/nlab/show/space">space</a> <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 encoded in its <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of functions</a> (tautologically in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a> or via <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a> in <a class="existingWikiWord" href="/nlab/show/topology">topology</a>), the <a class="existingWikiWord" href="/nlab/show/Serre-Swan+theorem">Serre-Swan theorem</a> asserts that vector bundles 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> are equivalently encoded in the <a class="existingWikiWord" href="/nlab/show/projective+modules">projective modules</a> over these algebras constituted by their <a class="existingWikiWord" href="/nlab/show/sections">sections</a>.</p> <p>Vector bundles have various applications and uses:</p> <ol> <li> <p>their <a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a> under <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a> yields <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a>, an interesting <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology+theory">generalized (Eilenberg-Steenrod) cohomology theory</a>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/reduction+of+the+structure+group">reduction of the structure group</a> of vector bundles encodes actual <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> on the base space; when applied to <a class="existingWikiWord" href="/nlab/show/tangent+bundles">tangent bundles</a> such <em><a class="existingWikiWord" href="/nlab/show/G-structures">G-structures</a></em> on vector bundles encode for instance <a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a>, <a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a>, <a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a>, <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a>, <a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a> etc. (in general: <a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>); when applied to <a class="existingWikiWord" href="/nlab/show/normal+bundles">normal bundles</a> these <a class="existingWikiWord" href="/nlab/show/G-structures">G-structures</a> give rise, via <a class="existingWikiWord" href="/nlab/show/Thom%27s+theorem">Thom's theorem</a>, to <a class="existingWikiWord" href="/nlab/show/Thom+spectra">Thom spectra</a> and <a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a>;</p> </li> <li> <p>equipping differentiable vector bundles with <a class="existingWikiWord" href="/nlab/show/connection+on+a+vector+bundle">connection on a vector bundle</a> is the basis for <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> and for the application of vector bundles in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, where they model <a class="existingWikiWord" href="/nlab/show/gauge+fields">gauge fields</a> and <a class="existingWikiWord" href="/nlab/show/instanton+sectors">instanton sectors</a>; see also at <em><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></em>.</p> </li> </ol> <h2 id="definition">Definition</h2> <h3 id="standard">Standard</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a></em></p> <h3 id="sheaftheoretic_version">Sheaf-theoretic version</h3> <p>Vector bundles can also be defined via <a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf theory</a>, which permits easy transport to general <a class="existingWikiWord" href="/nlab/show/Grothendieck+toposes">Grothendieck toposes</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X)</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/category">category</a> of (<a class="existingWikiWord" href="/nlab/show/set">set</a>-valued) <a class="existingWikiWord" href="/nlab/show/sheaf">sheaves</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. The sheaf of continuous local sections of the product projection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>ℝ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times \mathbb{R} \to X</annotation></semantics></math></div> <p>forms a <a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a> object <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>; when interpreted in the <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(X)</annotation></semantics></math>, it is the Dedekind <a class="existingWikiWord" href="/nlab/show/real+numbers+object">real numbers object</a>. Then, according to a <a class="existingWikiWord" href="/nlab/show/Serre-Swan+theorem">theorem of Richard Swan</a>, in its sheaf-theoretic incarnation a vector bundle is the same thing as a <a class="existingWikiWord" href="/nlab/show/projective+module">projective R-module</a>.</p> <ul> <li>A theorem of Kaplansky states “every <a class="existingWikiWord" href="/nlab/show/projective+module">projective module</a> over a <a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a> is <a class="existingWikiWord" href="/nlab/show/free+module">free</a>”. When interpreted in <span class="newWikiWord">sheaf semantics<a href="/nlab/new/sheaf+semantics">?</a></span> (<a class="existingWikiWord" href="/nlab/show/Kripke-Joyal+semantics">Kripke-Joyal semantics</a>), the <a class="existingWikiWord" href="/nlab/show/existential+quantifier">existential quantifier</a> implicit in “free” is interpreted <em>locally</em>, so we can consider a vector bundle as a locally free module over the Dedekind reals.</li> </ul> <h3 id="virtual_vector_bundles">Virtual vector bundles</h3> <p>In one class of models for <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> – <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> theory – cocycles are represented by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-graded vector bundles (pairs of vector bundles, essentially) modulo a certain equivalence relation. In that context it is sometimes useful to consider a certain variant of infinite-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-graded vector bundles called <a class="existingWikiWord" href="/nlab/show/vectorial+bundle"> vectorial bundles</a>.</p> <p>Much else to be discussed…</p> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+bundle">canonical bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/valence+bundle">valence bundle</a></p> </li> <li> <p>…</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a></p> </li> <li> <p><strong>vector bundle</strong>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/VectBund%28X%29">VectBund(X)</a>, <a class="existingWikiWord" href="/nlab/show/VectBund">VectBund</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/real+vector+bundle">real vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+vector+bundle">complex vector bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/holomorphic+vector+bundle">holomorphic vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/pseudoholomorphic+vector+bundle">pseudoholomorphic vector bundle</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+vector+bundle">universal vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rank">rank</a> of a vector bundle</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/module+bundle">module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/short+exact+sequence+of+vector+bundles">short exact sequence of vector bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+vector+bundle">connection on a vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+vector+bundle">flat vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/real+vector+bundle">real vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/complex+vector+bundle">complex vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+bundle">super vector bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measurable+field+of+Hilbert+spaces">measurable field of Hilbert spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-vector+bundle">2-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> </ul> <h2 id="Literature">Literature</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Glenys+Luke">Glenys Luke</a>, <a class="existingWikiWord" href="/nlab/show/Alexandr+S.+Mishchenko">Alexandr S. Mishchenko</a>, <em>Vector bundles and their applications</em>, Math. and its Appl. <strong>447</strong> Kluwer (1998) [<a href="https://doi.org/10.1007/978-1-4757-6923-4">doi:10.1007/978-1-4757-6923-4</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=99m:55019">MR99m:55019</a>]</p> </li> <li> <p>А. С. Мищенко, <em>Векторные расслоения и их применения</em> (Russian; A. S. Mishchenko, Vector bundles and their applications) Nauka, Moscow, 1984. 208 pp.</p> </li> <li> <p>Howard Osborn, <em>Vector bundles. Vol. 1. Foundations and Stiefel-Whitney classes</em>, Pure and Appl. Math. <strong>101</strong>, Academic Press 1982. xii+371 pp. <a href="http://www.ams.org/mathscinet-getitem?mr=85e:55001">MR85e:55001</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dale+Husem%C3%B6ller">Dale Husemöller</a>, <em>Fibre bundles</em>, McGraw-Hill 1966 (300 p.); Springer GTM 1975 (327 p.), 1994 (353 p.).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dale+Husem%C3%B6ller">Dale Husemöller</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Joachim">Michael Joachim</a>, <a class="existingWikiWord" href="/nlab/show/Branislav+Jurco">Branislav Jurco</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Schottenloher">Martin Schottenloher</a>, <em><a class="existingWikiWord" href="/nlab/show/Basic+Bundle+Theory+and+K-Cohomology+Invariants">Basic Bundle Theory and K-Cohomology Invariants</a></em>,</p> <p>Lecture Notes in Physics, Springer 2008 (<a href="http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf">pdf</a>)</p> </li> </ul> <p>An exposition with an eye towards <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> is in section 16.1 of</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Theodore+Frankel">Theodore Frankel</a>, <em><a class="existingWikiWord" href="/nlab/show/The+Geometry+of+Physics+-+An+Introduction">The Geometry of Physics - An Introduction</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>, <a class="existingWikiWord" href="/nlab/show/Loring+Tu">Loring Tu</a>, <em>Differential forms in algebraic topology</em>, Graduate Texts in Mathematics <strong>82</strong>, Springer 1982. xiv+331 pp.</p> </li> </ul> <p>Discussion with an eye towards <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> is in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Max+Karoubi">Max Karoubi</a>, <em>K-theory. An introduction</em>, Grundlehren der Mathematischen Wissenschaften <strong>226</strong>, Springer 1978. xviii+308 pp.</p> </li> <li id="Hatcher"> <p><a class="existingWikiWord" href="/nlab/show/Allen+Hatcher">Allen Hatcher</a>, <em>Vector bundles and K-Theory</em>, (partly finished book) <a href="https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html">web</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 12, 2024 at 15:04:43. 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