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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> <h4 id="monoidal_categories">Monoidal categories</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+monoidal+category">enriched monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a></p> </li> </ul> <p><strong>With braiding</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/balanced+monoidal+category">balanced monoidal category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twist">twist</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></p> </li> </ul> <p><strong>With duals for objects</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+duals">category with duals</a> (list of them)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> (what they have)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/autonomous+category">autonomous category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pivotal+category">pivotal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+category">spherical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ribbon+category">ribbon category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/tortile+category">tortile category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+closed+category">compact closed category</a></p> </li> </ul> <p><strong>With duals for morphisms</strong></p> <ul> <li> <p><span class="newWikiWord">monoidal dagger-category<a href="/nlab/new/monoidal+dagger-category">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+dagger-category">symmetric monoidal dagger-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dagger+compact+category">dagger compact category</a></p> </li> </ul> <p><strong>With traces</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/traced+monoidal+category">traced monoidal category</a></p> </li> </ul> <p><strong>Closed structure</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/star-autonomous+category">star-autonomous category</a></p> </li> </ul> <p><strong>Special sorts of products</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semicartesian+monoidal+category">semicartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal category with diagonals</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a></p> </li> </ul> <p><strong>Semisimplicity</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/semisimple+category">semisimple category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax</a>, <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax</a>, <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong</a> <a class="existingWikiWord" href="/nlab/show/bilax+monoidal+functor">bilax</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+monoidal+functor">Frobenius</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a></p> </li> </ul> <p><strong>Internal monoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in a symmetric monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a></p> </li> </ul> <p><strong id="_examples">Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <p><strong>In higher category theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/braided+monoidal+2-category">braided monoidal 2-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little cubes operad</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+double+category">compact double category</a></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='#properties'>Properties</a></li> <ul> <li><a href='#RelationToCohomotopy'>Relation to Cohomotopy</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>An (oriented) <em>cobordism</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> from an (oriented) <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>in</mi></msub></mrow><annotation encoding="application/x-tex">X_{in}</annotation></semantics></math> to an (oriented) smooth manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>out</mi></msub></mrow><annotation encoding="application/x-tex">X_{out}</annotation></semantics></math> is a smooth <a class="existingWikiWord" href="/nlab/show/manifold+with+boundary">manifold with boundary</a> such that its <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> is the <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Σ</mi><mo>≃</mo><msub><mi>X</mi> <mi>in</mi></msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><msub><mi>X</mi> <mi>out</mi></msub></mrow><annotation encoding="application/x-tex"> \partial \Sigma \simeq X_{in} \coprod X_{out} </annotation></semantics></math></div> <p>with induced <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> agreeing with the given one on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>in</mi></msub></mrow><annotation encoding="application/x-tex">X_{in}</annotation></semantics></math> and being the opposite of that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>out</mi></msub></mrow><annotation encoding="application/x-tex">X_{out}</annotation></semantics></math> (<a href="#Thom54">Thom 54, Chapter IV, p. 64</a>). Hence by labelling disjoint components of the boundary of any <a class="existingWikiWord" href="/nlab/show/manifold+with+boundary">manifold with boundary</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> as either “incoming” or “outgoing”, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> becomes a cobordism from its incoming to its outgoing boundary components.</p> <p>(While <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>in</mi></msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><msub><mi>X</mi> <mi>out</mi></msub></mrow><annotation encoding="application/x-tex">X_{in} \coprod X_{out}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>, conversely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is the “co-boundary” of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>in</mi></msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><msub><mi>X</mi> <mi>out</mi></msub></mrow><annotation encoding="application/x-tex">X_{in}\coprod X_{out}</annotation></semantics></math>. This is part of the reason for the “co-” in “cobordism”, but sometimes one just says <em>bordism</em>. The difference is more pronounced when distinguishing between <a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism homology theory</a> and <a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a>.)</p> <p>With a few technical conditions on the boundary inclusions added, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-dimensional manifolds with <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> classes of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional cobordisms between them form a <a class="existingWikiWord" href="/nlab/show/category">category</a> (a <a class="existingWikiWord" href="/nlab/show/category+of+cobordisms">category of cobordisms</a>) whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are the manifolds, whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are the cobordisms, and whose <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operation is the operation of gluing two cobordisms along a common boundary component. Such a <a class="existingWikiWord" href="/nlab/show/category+of+cobordisms">category of cobordisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bord</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Bord_n</annotation></semantics></math> of some <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is naturally a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mi>n</mi> <mo>⊔</mo></msubsup></mrow><annotation encoding="application/x-tex">Bord_n^{\sqcup}</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> being the <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊔</mo></mrow><annotation encoding="application/x-tex">\sqcup</annotation></semantics></math> of manifolds.</p> <p>The <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a> in this category are called <em>cobordism classes</em> of manifolds. Under <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> and <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a>, these form what is called the <a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a> in the given dimension. The study of these classes, hence of manifolds “up to cobordisms”, is a central topic in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>.</p> <p>A central insight connecting <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> with <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a> is that a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a> on a <a class="existingWikiWord" href="/nlab/show/category+of+cobordisms">category of cobordisms</a> with values in something like the category <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> (with its standard <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a>) may be thought of as a formalized incarnation of what in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> is called a <a class="existingWikiWord" href="/nlab/show/topological+quantum+field+theory">topological quantum field theory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo lspace="verythinmathspace">:</mo><msubsup><mi>Bord</mi> <mi>n</mi> <mo>⊔</mo></msubsup><mo>⟶</mo><msup><mi>Vect</mi> <mo>⊗</mo></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Z \colon Bord_n^\sqcup \longrightarrow Vect^\otimes \,. </annotation></semantics></math></div> <p>Here one thinks of a cobordism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> as a piece of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> (or <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>) of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, and of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-dimensional manifolds that this goes between as a piece of <a class="existingWikiWord" href="/nlab/show/space">space</a> (or <a class="existingWikiWord" href="/nlab/show/brane">brane</a>). A functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> as above is then thought of as sending each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-dimensional 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> to its <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(X)</annotation></semantics></math> and each spacetime <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>in</mi></msub></mrow><annotation encoding="application/x-tex">X_{in}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>out</mi></msub></mrow><annotation encoding="application/x-tex">X_{out}</annotation></semantics></math> to a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>in</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>Z</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>out</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(\Sigma)\colon Z(X_{in}) \longrightarrow Z(X_{out})</annotation></semantics></math>.</p> <p>From the perspective of <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a>, such a functor is a way to break up <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> <a class="existingWikiWord" href="/nlab/show/invariants">invariants</a> of <a class="existingWikiWord" href="/nlab/show/closed+manifolds">closed manifolds</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> into pieces and being able to reconstruct them from gluing of data associated to manifolds with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-dimensional boundary.</p> <p>If one considers <a class="existingWikiWord" href="/nlab/show/manifolds+with+corners">manifolds with corners</a>, then there is a fairly evident extension of the concept of cobordism that allows refinement of this gluing process to <a class="existingWikiWord" href="/nlab/show/extended+cobordisms">extended cobordisms</a> of any dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> going between <a class="existingWikiWord" href="/nlab/show/extended+cobordisms">extended cobordisms</a> of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k-1</annotation></semantics></math>. Such extended cobordisms of maximal dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> form a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28infinity%2Cn%29-category">symmetric monoidal (infinity,n)-category</a> called, naturally, the <a class="existingWikiWord" href="/nlab/show/%28infinity%2Cn%29-category+of+cobordisms">(infinity,n)-category of cobordisms</a>. The <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a> asserts that this is a most fundamental object in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> and <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a>, namely that it is the <em><a class="existingWikiWord" href="/nlab/show/free+construction">free</a></em> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28infinity%2Cn%29-category">symmetric monoidal (infinity,n)-category</a> <a class="existingWikiWord" href="/nlab/show/%28infinity%2Cn%29-category+with+duals">with duals</a>. The corresponding extended concept of <a class="existingWikiWord" href="/nlab/show/topological+quantum+field+theory">topological quantum field theory</a> is accordingly called <a class="existingWikiWord" href="/nlab/show/extended+TQFT">extended TQFT</a> or similar.</p> <h2 id="properties">Properties</h2> <h3 id="RelationToCohomotopy">Relation to Cohomotopy</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+isomorphism">Pontrjagin-Thom isomorphism</a> says that assigning <a class="existingWikiWord" href="/nlab/show/Cohomotopy+charge">Cohomotopy charge</a> identifies suitable <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of <a class="existingWikiWord" href="/nlab/show/submanifolds">submanifolds</a> or abstract <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> with <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> in <a class="existingWikiWord" href="/nlab/show/Cohomotopy">unstable Cohomotopy</a> (see <a href="cohomotopy#RelationToCobordismGroup">here</a>) or <a class="existingWikiWord" href="/nlab/show/stable+Cohomotopy">stable Cohomotopy</a>.</p> <p>The following graphics illustrates the <a class="existingWikiWord" href="/nlab/show/Cohomotopy+charge+map">Cohomotopy charge map</a> of the pair creation/annihilation cobordism for 0-dimensional <a class="existingWikiWord" href="/nlab/show/submanifolds">submanifolds</a>:</p> <center> <a href="https://arxiv.org/pdf/1909.12277.pdf#page=11"> <img src="https://ncatlab.org/schreiber/files/BraneAntibranePairCreationInCohomotopy.jpg" width="700" /> </a> </center> <blockquote> <p>graphics grabbed from <a href="cohomotopy+charge#SatiSchreiber19">SS 19</a></p> </blockquote> <p>(See also at <em><a class="existingWikiWord" href="/nlab/show/Cohomotopy+charge">Cohomotopy charge</a> – <a href="Cohomotopy+charge#ForChargedPoints">For charged points</a></em>.)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/handlebody">handlebody</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/h-cobordism">h-cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/movie">movie</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+group">cobordism group</a>, <a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a>, <a class="existingWikiWord" href="/nlab/show/Thom%27s+theorem">Thom's theorem</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/%28infinity%2Cn%29-category+of+cobordisms">(infinity,n)-category of cobordisms</a>, <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</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/B-bordism">B-bordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold+cobordism">orbifold cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+cobordism">Lagrangian cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+cobordism">algebraic cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+chord+diagram">Sullivan chord diagram</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/link+cobordism">link cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braid+cobordism">braid cobordism</a></p> </li> </ul> <h2 id="references">References</h2> <p>The notion originates with</p> <ul> <li id="Thom54"> <p><a class="existingWikiWord" href="/nlab/show/Ren%C3%A9+Thom">René Thom</a>, p. 64 of: <em><a class="existingWikiWord" href="/nlab/show/Quelques+propri%C3%A9t%C3%A9s+globales+des+vari%C3%A9t%C3%A9s+diff%C3%A9rentiables">Quelques propriétés globales des variétés différentiables</a></em>, Comment. Math. Helv. 28, (1954). 17-86 (<a href="https://doi.org/10.1007/BF02566923">doi:10.1007/BF02566923</a>, <a href="https://eudml.org/doc/139072">dml:139072</a>, <a href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259">digiz:GDZPPN002056259</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf">pdf</a>)</p> </li> <li id="Pontrjagin55"> <p><a class="existingWikiWord" href="/nlab/show/Lev+Pontrjagin">Lev Pontrjagin</a>, Section III.6, p. 41 of: <em><a class="existingWikiWord" href="/nlab/show/Smooth+manifolds+and+their+applications+in+homotopy+theory">Smooth manifolds and their applications in homotopy theory</a></em>, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (<a href="https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001">doi:10.1142/9789812772107_0001</a>)</p> </li> </ul> <p>(Beware that Pontrjagin still wrote “homology” for “cobordism”, following an earlier tradition that got abandoned.)</p> <p>An early textbook account:</p> <ul> <li id="Stong68"><a class="existingWikiWord" href="/nlab/show/Robert+Stong">Robert Stong</a>, <em>Notes on Cobordism theory</em>, 1968 (<a href="http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf">toc pdf</a>, <a href="http://press.princeton.edu/titles/6465.html">publisher page</a>)</li> </ul> <p>Further review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Landweber">Peter Landweber</a>, <em>A survey of bordism and cobordism</em>, Mathematical Proceedings of the Cambridge Philosophical Society <strong>100</strong> 2 (1986) 207-223 [<a href="https://doi.org/10.1017/S0305004100066032">doi:10.1017/S0305004100066032</a>]</p> </li> <li id="HirzebruchGergerJung92"> <p><a class="existingWikiWord" href="/nlab/show/Friedrich+Hirzebruch">Friedrich Hirzebruch</a>, Thomas Berger, Rainer Jung, Section 1.1 of: <em>Manifolds and Modular Forms</em>, Aspects of Mathematics <strong>20</strong>, Viehweg (1992), Springer (1994) [<a href="https://doi.org/10.1007/978-3-663-10726-2">doi:10.1007/978-3-663-10726-2</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/hirzjung.pdf">pdf</a>]</p> </li> <li id="Kochmann96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochmann">Stanley Kochmann</a>, section 1.5 of <em><a class="existingWikiWord" href="/nlab/show/Bordism%2C+Stable+Homotopy+and+Adams+Spectral+Sequences">Bordism, Stable Homotopy and Adams Spectral Sequences</a></em>, AMS 1996</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Francis">John Francis</a> (notes by <a class="existingWikiWord" href="/nlab/show/Owen+Gwilliam">Owen Gwilliam</a>), <em><a href="http://math.northwestern.edu/~jnkf/classes/mflds/">Topology of manifolds</a></em>, <em>Lecture 2: Cobordism</em> (<a href="http://math.northwestern.edu/~jnkf/classes/mflds/2cobordism.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Manifold+Atlas">Manifold Atlas</a>, <em><a href="http://www.map.mpim-bonn.mpg.de/Bordism">Bordism</a></em></p> </li> <li> <p>Wikipedia, <em><a href="http://en.wikipedia.org/wiki/Cobordism">Cobordism</a></em></p> </li> </ul> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></em>.</p> <p>A relation to <a class="existingWikiWord" href="/nlab/show/fixed+point+spaces">fixed point spaces</a>:</p> <ul> <li id="Prieto03">Carlos Prieto, <em>Fixed point theory and framed cobordism</em>, Topol. Methods Nonlinear Anal. Volume 21, Number 1 (2003), 155-169. (<a href="https://projecteuclid.org/euclid.tmna/1475266278">euclid:tmna/1475266278</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 24, 2024 at 12:45:28. 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