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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="cobordism_theory">Cobordism theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a></strong> = <a class="existingWikiWord" href="/nlab/show/manifolds+and+cobordisms+-+contents">manifolds and cobordisms</a> + <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>/<a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/equivariant+cobordism+theory">equivariant cobordism theory</a></li> </ul> <p><strong>Concepts of cobordism theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangential+structure">tangential structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>, <a class="existingWikiWord" href="/nlab/show/cobordism+class">cobordism class</a></p> <p><a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submanifold">submanifold</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin%27s+theorem">Pontrjagin's theorem</a> (<a class="existingWikiWord" href="/nlab/show/equivariant+Pontrjagin+theorem">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/twisted+Pontrjagin+theorem">twisted</a>):</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo>↔</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{\leftrightarrow}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\leftrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of <a class="existingWikiWord" href="/nlab/show/normally+framed+submanifolds">normally framed submanifolds</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+theorem">Thom's theorem</a>:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo>↔</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{\leftrightarrow}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of maps to <a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a> <a class="existingWikiWord" href="/nlab/show/MO">MO</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\leftrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of <a class="existingWikiWord" href="/nlab/show/normally+oriented+submanifolds">normally oriented submanifolds</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a></p> <p><a class="existingWikiWord" href="/nlab/show/Thom+isomorphism">Thom isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></p> <p><a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+collapse+construction">Pontryagin-Thom collapse construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology+theory">complex cobordism cohomology theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></p> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</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> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></p> </li> </ul> <div> <p><strong>flavors of <a class="existingWikiWord" href="/nlab/show/bordism+homology+theories">bordism homology theories</a>/<a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theories">cobordism cohomology theories</a>, their <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">representing</a> <a class="existingWikiWord" href="/nlab/show/Thom+spectra">Thom spectra</a> and <a class="existingWikiWord" href="/nlab/show/cobordism+rings">cobordism rings</a></strong>:</p> <p><a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/M%28B%2Cf%29">M(B,f)</a> (<a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/MFr">MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MO">MO</a>, <a class="existingWikiWord" href="/nlab/show/MSO">MSO</a>, <a class="existingWikiWord" href="/nlab/show/MSpin">MSpin</a>, <a class="existingWikiWord" href="/nlab/show/MString">MString</a>, …</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MU">MU</a>, <a class="existingWikiWord" href="/nlab/show/MSU">MSU</a>, …</p> <p><a class="existingWikiWord" href="/nlab/show/Ravenel%27s+spectrum">MΩΩSU(n)</a></p> <p><a class="existingWikiWord" href="/nlab/show/MP-theory">MP</a>, <a class="existingWikiWord" href="/nlab/show/MR-theory">MR</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSpin%5Ec">MSpin<sup><i>c</i></sup></a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSp">MSp</a></p> </li> </ul> <p>relative bordism theories:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/MOFr">MOFr</a>, <a class="existingWikiWord" href="/nlab/show/MUFr">MUFr</a>, <a class="existingWikiWord" href="/nlab/show/MSUFr">MSUFr</a></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bordism+homology+theory">equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MFr">equivariant MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MO">equivariant MO</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MU">equivariant MU</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+bordism+homology+theory">global equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mO">global equivariant mO</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mU">global equivariant mU</a></p> </li> </ul> <p>algebraic:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+cobordism">algebraic cobordism</a></li> </ul> </div></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='#examples'>Examples</a></li> <ul> <li><a href='#FramedCobordism'>Framed cobordism</a></li> <li><a href='#OrientedCobordismRing'>Oriented cobordism</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 <strong>(co)bordism ring</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mo>*</mo></msub><mo>=</mo><msub><mo>⊕</mo> <mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub><msub><mi>Ω</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\Omega_*=\oplus_{n\geq 0}\Omega_n</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/graded+ring">graded ring</a> whose</p> <ul> <li> <p>degree <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> elements are 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>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> modulo <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>;</p> </li> <li> <p>product operation is given by the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of manifolds;</p> </li> <li> <p>addition operation is given by the <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> of manifolds.</p> </li> </ul> <p>Instead of bare manifolds one may consider manifolds with extra structure, such as <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>, etc. and accordingly there is</p> <ul> <li> <p>the <em>oriented cobordism ring</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>*</mo> <mi>SO</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{SO}_*</annotation></semantics></math>,</p> </li> <li> <p>the <em>spin cobordism ring</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>*</mo> <mi>Spin</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{Spin}_*</annotation></semantics></math>,</p> </li> </ul> <p>etc.</p> <p>In this general context the bare cobordism ring is also denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>*</mo> <mi>O</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^O_*</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>*</mo> <mi>un</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{un}_*</annotation></semantics></math>, for emphasis.</p> <p>A ring <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> out of the cobordism ring is a (multiplicative) <em><a class="existingWikiWord" href="/nlab/show/genus">genus</a></em>.</p> <p>More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a fixed <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> there is a relative cobordism ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega_\bullet(X)</annotation></semantics></math> whose</p> <ul> <li> <p>elements are classes modulo cobordism 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> of manifolds equipped with <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> to <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> (“singular manifolds”);</p> </li> <li> <p>multiplication of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><msub><mi>Σ</mi> <mn>1</mn></msub><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[f_1 \colon \Sigma_1 \to X]</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><msub><mi>Σ</mi> <mn>2</mn></msub><mo>→</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[f_2 \colon \Sigma_2 \to X]</annotation></semantics></math> is given by <em>transversal intersection</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>1</mn></msub><msub><mo>∩</mo> <mi>X</mi></msub><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_1 \cap_X \Sigma_2</annotation></semantics></math> 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>: perturb <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">f_1</annotation></semantics></math> such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>′</mo><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f_1',f_2)</annotation></semantics></math> becomes a <a class="existingWikiWord" href="/nlab/show/transversal+maps">transversal maps</a> and then form the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>′</mo><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msub><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_1 \times_{(f_1',f_2)} \Sigma_2</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a>.</p> </li> </ul> <p>This product is graded in that it satisfies the <strong>dimension formula</strong></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><mi>dim</mi><mi>X</mi><mo>−</mo><mi>dim</mi><msub><mi>Σ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>dim</mi><mi>X</mi><mo>−</mo><mi>dim</mi><msub><mi>Σ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>dim</mi><mi>X</mi><mo>−</mo><mi>dim</mi><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mn>1</mn></msub><msub><mo>∩</mo> <mi>X</mi></msub><msub><mi>Σ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (dim X - dim \Sigma_1) + (dim X - dim \Sigma_2) = dim X - dim (\Sigma_1 \cap_X \Sigma_2) </annotation></semantics></math></div> <p>hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mn>1</mn></msub><msub><mo>∩</mo> <mi>X</mi></msub><msub><mi>Σ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>dim</mi><msub><mi>Σ</mi> <mn>1</mn></msub><mo>+</mo><mi>dim</mi><msub><mi>Σ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>dim</mi><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> dim (\Sigma_1 \cap_X \Sigma_2 ) = (dim \Sigma_1 + dim \Sigma_2) - dim X \,. </annotation></semantics></math></div> <p>Still more generally, this may be considered for <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> being <a class="existingWikiWord" href="/nlab/show/manifolds+with+boundary">manifolds with boundary</a>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega(X,A)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,A)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/CW+pair">CW pair</a> is the ring of cobordism classes, relative boundary, of singular manifolds <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math> such that 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> lands in in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>The resulting <a class="existingWikiWord" href="/nlab/show/functor">functor</a></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><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>↦</mo><msubsup><mi>Ω</mi> <mo>•</mo> <mi>G</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (X,A) \mapsto \Omega^G_\bullet(X,A) </annotation></semantics></math></div> <p>constitutes a <a class="existingWikiWord" href="/nlab/show/generalized+homology+theory">generalized homology theory</a> (see e.g. <a href="#Buchstaber">Buchstaber, II.8</a>). Accordingly this is called <em><a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism homology theory</a></em>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> that represents this under the <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a> is the universal <a class="existingWikiWord" href="/nlab/show/Thom+spectra">Thom spectra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">M G</annotation></semantics></math> (e.g. <a class="existingWikiWord" href="/nlab/show/MO">MO</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>O</mi></mrow><annotation encoding="application/x-tex">G=O</annotation></semantics></math> or <a class="existingWikiWord" href="/nlab/show/MU">MU</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">G = U</annotation></semantics></math>), which canonically is a <a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a> under <a class="existingWikiWord" href="/nlab/show/Whitney+sum">Whitney sum</a> of <a class="existingWikiWord" href="/nlab/show/universal+vector+bundles">universal vector bundles</a>. Accordingly the (co-bordism ring) itself is equivalently the bordism homology groups of the point, hence the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups">stable homotopy groups</a> of the <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> (this is <em><a class="existingWikiWord" href="/nlab/show/Thom%27s+theorem">Thom's theorem</a></em>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>•</mo> <mi>G</mi></msubsup><mo>≃</mo><mo lspace="0em" rspace="thinmathspace">M</mo><msub><mi>G</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>M</mi><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega_\bullet^G \simeq \M G_\bullet(\ast) \simeq \pi_\bullet(M G) \,. </annotation></semantics></math></div> <p>This remarkable relation between <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> and <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> is known as <em><a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a></em> (or “Thom theory”).</p> <p>On general grounds this is equivalently the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">M G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/generalized+cohomology">generalized cohomology</a> of the point (<a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>•</mo> <mi>G</mi></msubsup><mo>≃</mo><mi>M</mi><msup><mi>G</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega_\bullet^G \simeq M G^\bullet(\ast) </annotation></semantics></math></div> <p>which justifies calling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>•</mo> <mi>G</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega_\bullet^G</annotation></semantics></math> both the “bordism ring” as well as the “cobordism ring”.</p> <h2 id="properties">Properties</h2> <h3 id="RelationToCohomotopy">Relation to cohomotopy</h3> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \leq n</annotation></semantics></math>. Then the <a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+construction">Pontryagin-Thom construction</a> induces a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a></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><mi>X</mi><mo>,</mo><msup><mi>S</mi> <mi>k</mi></msup><mo stretchy="false">]</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [X, S^k] \overset{\simeq}{\longrightarrow} \Omega^{n-k}(X) </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/cohomotopy">cohomotopy</a> sets 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> to the <a class="existingWikiWord" href="/nlab/show/cobordism+group">cobordism group</a> of <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><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-k)</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/submanifolds">submanifolds</a> with <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal</a> <a class="existingWikiWord" href="/nlab/show/framed+manifold">framing</a> up to normally framed <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>.</p> <p>In particular, the natural group structure on <a class="existingWikiWord" href="/nlab/show/cobordism+group">cobordism group</a> (essentially given by <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> of submanifolds) this way induces a group structure on the cohomotopy sets.</p> <p>This is made explicit for instance in <a href="#Kosinski93">Kosinski 93, chapter IX</a>.</p> <h2 id="examples">Examples</h2> <h3 id="FramedCobordism">Framed cobordism</h3> <p>By <a class="existingWikiWord" href="/nlab/show/Thom%27s+theorem">Thom's theorem</a>, for any <a class="existingWikiWord" href="/nlab/show/%28B%2Cf%29-structure">(B,f)-structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}</annotation></semantics></math>, there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> (of <a class="existingWikiWord" href="/nlab/show/commutative+rings">commutative rings</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>•</mo> <mi>ℬ</mi></msubsup><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>M</mi><mi>ℬ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^{\mathcal{B}}_\bullet \overset{\simeq}{\longrightarrow} \pi_\bullet(M\mathcal{B}) </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a> of manifolds with stable normal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}</annotation></semantics></math>-structure to the <a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+spectrum">homotopy groups</a> of the universal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\mathcal{B}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a>.</p> <p>Now for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mo>=</mo><mi>Fr</mi></mrow><annotation encoding="application/x-tex">\mathcal{B} = Fr</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/framing">framing</a> structure, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>Fr</mi><mo>≃</mo><mi>𝕊</mi></mrow><annotation encoding="application/x-tex"> M Fr \simeq \mathbb{S} </annotation></semantics></math></div> <p>is equivalently the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a>. Hence in this case <a class="existingWikiWord" href="/nlab/show/Thom%27s+theorem">Thom's theorem</a> states that there is an isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>•</mo> <mi>fr</mi></msubsup><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^{fr}_\bullet \overset{\simeq}{\longrightarrow} \pi_\bullet(\mathbb{S}) </annotation></semantics></math></div> <p>between the framed cobordism ring and the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a>.</p> <p>For discussion of computation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(\mathbb{S})</annotation></semantics></math> this way, see for instance (<a href="#WangXu10">Wang-Xu 10, section 2</a>).</p> <p>For instance</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mn>0</mn> <mi>fr</mi></msubsup><mo>=</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\Omega^{fr}_0 = \mathbb{Z}</annotation></semantics></math> because there are two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-framings on a single point, corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0(O(k)) \simeq \mathbb{Z}_2</annotation></semantics></math>, the negative of a point with one framing is the point with the other framing, and so under disjoint union, the framed points form the group of integers;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mn>1</mn> <mi>fr</mi></msubsup><mo>=</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Omega^{fr}_1 = \mathbb{Z}_2</annotation></semantics></math> because the only compact connected 1-manifold is the circle, there are two framings on the circle, corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\pi_1(O(k)) \simeq \mathbb{Z}_2</annotation></semantics></math> and they are their own negatives.</p> </li> </ul> <h3 id="OrientedCobordismRing">Oriented cobordism</h3> <div class="num_prop" id="OrientedCobordismOverPoint"> <h6 id="proposition">Proposition</h6> <p>The cobordism ring over the point for <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> manifolds starts out as</p> <table><thead><tr><th><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></th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>9</mn></mrow><annotation encoding="application/x-tex">\geq 9</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>k</mi> <mi>SO</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{SO}_k</annotation></semantics></math></td><td style="text-align: left;"><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">\mathbb{Z}</annotation></semantics></math></td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;"><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">\mathbb{Z}</annotation></semantics></math></td><td style="text-align: left;"><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></td><td style="text-align: left;">0</td><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>⊕</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}\oplus \mathbb{Z}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\neq 0</annotation></semantics></math></td></tr> </tbody></table></div> <p>see e.g. (<a href="#ManifoldAtlas">ManifoldAtlas</a>)</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> (for instance a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>), then the oriented cobordism ring is expressed in terms of the <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msubsup><mi>Ω</mi> <mrow><mi>p</mi><mo>−</mo><mi>q</mi></mrow> <mi>SO</mi></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_q(X,\Omega^{SO}_{p-q})</annotation></semantics></math> 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> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the cobordism ring over the point, prop. <a class="maruku-ref" href="#OrientedCobordismOverPoint"></a>, as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>p</mi> <mi>SO</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>⊕</mo> <mrow><mi>q</mi><mo>=</mo><mn>0</mn></mrow> <mi>p</mi></msubsup><msub><mi>H</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msubsup><mi>Ω</mi> <mrow><mi>p</mi><mo>−</mo><mi>q</mi></mrow> <mi>SO</mi></msubsup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi>mod</mi><mspace width="thickmathspace"></mspace><mi>odd</mi><mspace width="thickmathspace"></mspace><mi>torsion</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega_p^{SO}(X) = \oplus_{q = 0}^p H_q(X,\Omega_{p-q}^{SO}) \; mod\; odd \; torsion \,. </annotation></semantics></math></div></div> <p>e.g. <a href="#ConnorFloyd62">Connor-Floyd 62, theorem 14.2</a></p> <h2 id="related_concepts">Related concepts</h2> <div> <p><strong>flavors of <a class="existingWikiWord" href="/nlab/show/bordism+homology+theories">bordism homology theories</a>/<a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theories">cobordism cohomology theories</a>, their <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">representing</a> <a class="existingWikiWord" href="/nlab/show/Thom+spectra">Thom spectra</a> and <a class="existingWikiWord" href="/nlab/show/cobordism+rings">cobordism rings</a></strong>:</p> <p><a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/M%28B%2Cf%29">M(B,f)</a> (<a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/MFr">MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MO">MO</a>, <a class="existingWikiWord" href="/nlab/show/MSO">MSO</a>, <a class="existingWikiWord" href="/nlab/show/MSpin">MSpin</a>, <a class="existingWikiWord" href="/nlab/show/MString">MString</a>, …</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MU">MU</a>, <a class="existingWikiWord" href="/nlab/show/MSU">MSU</a>, …</p> <p><a class="existingWikiWord" href="/nlab/show/Ravenel%27s+spectrum">MΩΩSU(n)</a></p> <p><a class="existingWikiWord" href="/nlab/show/MP-theory">MP</a>, <a class="existingWikiWord" href="/nlab/show/MR-theory">MR</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSpin%5Ec">MSpin<sup><i>c</i></sup></a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSp">MSp</a></p> </li> </ul> <p>relative bordism theories:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/MOFr">MOFr</a>, <a class="existingWikiWord" href="/nlab/show/MUFr">MUFr</a>, <a class="existingWikiWord" href="/nlab/show/MSUFr">MSUFr</a></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bordism+homology+theory">equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MFr">equivariant MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MO">equivariant MO</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MU">equivariant MU</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+bordism+homology+theory">global equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mO">global equivariant mO</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mU">global equivariant mU</a></p> </li> </ul> <p>algebraic:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+cobordism">algebraic cobordism</a></li> </ul> </div> <h2 id="references">References</h2> <p>Original articles:</p> <ul> <li id="Thom54"> <p><a class="existingWikiWord" href="/nlab/show/Ren%C3%A9+Thom">René Thom</a>, <em>Quelques propriétés globales des variétés différentiables</em> Comment. Math. Helv. 28, (1954). 17-86 (<a href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259">digiz:GDZPPN002056259</a>)</p> </li> <li id="Milnor60"> <p><a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, <em>On the cobordism ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">\Omega^\bullet</annotation></semantics></math> and a complex analogue</em>, Amer. J. Math. 82 (1960), 505–521 (<a href="https://www.jstor.org/stable/2372970">jstor:2372970</a>)</p> </li> <li id="Novikov60"> <p><a class="existingWikiWord" href="/nlab/show/Sergei+Novikov">Sergei Novikov</a>, <em>Some problems in the topology of manifolds connected with the theory of Thom spaces</em>, Dokl. Akad. Nauk. SSSR. 132 (1960), 1031–1034 (Russian).</p> </li> <li id="Novikov62"> <p><a class="existingWikiWord" href="/nlab/show/Sergei+Novikov">Sergei Novikov</a>, <em>Homotopy properties of Thom complexes</em>, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 (<a href="http://www.mi-ras.ru/~snovikov/6.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/NovikovThomComplexes.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>, <em>Elementary proofs of some results of cobordism theory using Steenrod operations</em>, Advances in Mathematics <strong>7</strong> 1 (1971) 29-56 [<a href="https://doi.org/10.1016/0001-8708(71)90041-7">doi:10.1016/0001-8708(71)90041-7</a>]</p> <blockquote> <p>(using <a class="existingWikiWord" href="/nlab/show/Steenrod+operations">Steenrod operations</a>)</p> </blockquote> </li> </ul> <p>Textbook accounts:</p> <ul> <li id="Stong68"> <p><a class="existingWikiWord" href="/nlab/show/Robert+Stong">Robert Stong</a>, <em>Notes on Cobordism theory</em>, Princeton University Press, 1968 (<a href="http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf">toc pdf</a>, <a href="http://press.princeton.edu/titles/6465.html">ISBN:9780691649016</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/stongcob.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> </ul> <p>Lecture notes:</p> <ul> <li id="Buchstaber"> <p><a class="existingWikiWord" href="/nlab/show/Victor+Buchstaber">Victor Buchstaber</a>, <em>Geometric cobordism theory</em> (<a href="http://www.maths.manchester.ac.uk/~peter/MATH41101notes07.pdf">pdf</a>)</p> </li> <li id="Francis"> <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/Gerald+H%C3%B6hn">Gerald Höhn</a>, <em>Komplexe elliptische Geschlechter und <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>-äquivariante Kobordismustheorie</em> (german) (<a href="http://arxiv.org/PS_cache/math/pdf/0405/0405232v1.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sander+Kupers">Sander Kupers</a>, <em>Oriented bordism: Calculation and application</em> (<a href="http://web.stanford.edu/~kupers/orientedbordism.pdf">pdf</a>)</p> </li> </ul> <p>Details for <a class="existingWikiWord" href="/nlab/show/framed+cobordism">framed cobordism</a>:</p> <ul> <li id="WangXu10"> <p><a class="existingWikiWord" href="/nlab/show/Guozhen+Wang">Guozhen Wang</a>, Zhouli Xu, section 2 of <em>A survey of computations of homotopy groups of Spheres and Cobordisms</em>, 2010 (<a href="http://math.mit.edu/~guozhen/homotopy%20groups.pdf">pdf</a>)</p> </li> <li id="Putnam"> <p><a class="existingWikiWord" href="/nlab/show/Andrew+Putman">Andrew Putman</a>, <em>Homotopy groups of spheres and low-dimensional topology</em> (<a href="http://www.math.rice.edu/~andyp/notes/HomotopySpheresLowDimTop.pdf">pdf</a>)</p> </li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/cohomotopy">cohomotopy</a> is made explicit in</p> <ul> <li id="Kosinski93"><a class="existingWikiWord" href="/nlab/show/Antoni+Kosinski">Antoni Kosinski</a>, chapter IX of <em>Differential manifolds</em>, Academic Press 1993 (<a href="http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf">pdf</a>)</li> </ul> <p>Further discussion of oriented cobordism includes</p> <ul> <li id="ManifoldAtlas"> <p>Manifold Atlas, <em><a href="http://www.map.mpim-bonn.mpg.de/Oriented_bordism">Oriented bordism</a></em></p> </li> <li id="ConnorFloyd62"> <p>P. E. Conner, E. E. Floyd, <em>Differentiable periodic maps</em>, Bull. Amer. Math. Soc. Volume 68, Number 2 (1962), 76-86. (<a href="http://projecteuclid.org/euclid.bams/1183524501">Euclid</a>, <a href="http://www.maths.ed.ac.uk/~aar/papers/cf1.pdf">pdf</a>)</p> </li> </ul> <p>A historical review in the context of <a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology+theory">complex cobordism cohomology theory</a>/<a class="existingWikiWord" href="/nlab/show/Brown-Peterson+theory">Brown-Peterson theory</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Doug+Ravenel">Doug Ravenel</a>, chapter 4, section 2 of <em><a class="existingWikiWord" href="/nlab/show/Complex+cobordism+and+stable+homotopy+groups+of+spheres">Complex cobordism and stable homotopy groups of spheres</a></em></li> </ul> <p>On fibered cobordism groups:</p> <ul> <li>Astey, Greenberg, Micha, Pastor, <em>Some fibered cobordisms groups are not finitely generated</em> (<a href="http://www.ams.org/journals/proc/1988-104-01/S0002-9939-1988-0958093-3/S0002-9939-1988-0958093-3.pdf">pdf</a>)</li> </ul> <p>Discussion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-equivariant complex coborism ring includes</p> <ul> <li> <p>G. Comezana and <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em>A completion theorem in complex cobordism</em>, in Equivariant Homotopy and Cohomology Theory, CBMS Regional conference series in Mathematics, American Mathematical Society Publications, Volume 91, Providence, 1996.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, <em>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/p</annotation></semantics></math>–equivariant complex cobordism ring</em>, from: “Homotopy invariant algebraic structures (Baltimore, MD, 1998)”, Amer. Math. Soc. Providence, RI (1999) 217–223</p> </li> <li id="Strickland01"> <p><a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, <em>Complex cobordism of involutions</em>, Geom. Topol. 5 (2001) 335-345 (<a href="http://arxiv.org/abs/math/0105020">arXiv:math/0105020</a>)</p> </li> <li> <p>William Abrams, <em>Equivariant complex cobordism</em>, 2013 (<a href="http://deepblue.lib.umich.edu/bitstream/handle/2027.42/99796/abramwc_1.pdf">pdf</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 8, 2023 at 15:07:22. 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