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<fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="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='#ThomTheoremOnMO'>Thom’s theorem on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>O</mi></mrow><annotation encoding="application/x-tex">M O</annotation></semantics></math></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 universal <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> (see there for more) of the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a>. (…) Abstractly, this is the <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a> of the <a class="existingWikiWord" href="/nlab/show/J-homomorphism">J-homomorphism</a> in <a class="existingWikiWord" href="/nlab/show/Spectra">Spectra</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>MO</mi><mo>=</mo><munder><mi>lim</mi><mo>→</mo></munder><mo stretchy="false">(</mo><mi>B</mi><mi>O</mi><mover><mo>→</mo><mi>J</mi></mover><mi>B</mi><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Spectra</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> MO = \underset{\rightarrow}{\lim}(B O \stackrel{J}{\to} B GL_1(\mathbb{S}) \to Spectra) </annotation></semantics></math></div> <h2 id="properties">Properties</h2> <h3 id="ThomTheoremOnMO">Thom’s theorem on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>O</mi></mrow><annotation encoding="application/x-tex">M O</annotation></semantics></math></h3> <p>By <a class="existingWikiWord" href="/nlab/show/Thom%27s+theorem">Thom's theorem</a> the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups">stable homotopy groups</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>O</mi></mrow><annotation encoding="application/x-tex">M O</annotation></semantics></math> form the <a class="existingWikiWord" href="/nlab/show/bordism+ring">bordism ring</a> of unoriented manifolds</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>M</mi><mi>O</mi><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>Ω</mi> <mo>•</mo> <mi>O</mi></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_\bullet(M O) \simeq \Omega^O_\bullet \,. </annotation></semantics></math></div> <p>Moreover, this is the <a class="existingWikiWord" href="/nlab/show/polynomial+algebra">polynomial algebra</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>M</mi><mi>O</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msub><mi>x</mi> <mi>n</mi></msub><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>n</mi><mo>≥</mo><mn>2</mn><mo>,</mo><mspace width="thinmathspace"></mspace><mi>n</mi><mo>≠</mo><msup><mn>2</mn> <mi>t</mi></msup><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_\bullet(M O) \simeq (\mathbb{Z}/2\mathbb{Z})[ x_n \;|\; n \in \mathbb{N}, \,n \geq 2, \, n \neq 2^t-1] \,. </annotation></semantics></math></div> <p>Due to (<a href="#Thom54">Thom 54</a>). See for instance (<a href="#Kochman96">Kochman 96, theorem 3.7.6</a>)</p> <p>The corresponding statement for <a class="existingWikiWord" href="/nlab/show/MU">MU</a> is considerably more subtle, see <em><a class="existingWikiWord" href="/nlab/show/Milnor-Quillen+theorem+on+MU">Milnor-Quillen theorem on MU</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homology+of+MO">homology of MO</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></p> </li> </ul> <p><br /></p> <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> <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</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> </ul> <p>Textbook accounts:</p> <ul> <li id="Stong68"> <p><a class="existingWikiWord" href="/nlab/show/Robert+Stong">Robert Stong</a>, Chapter VI of: <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="Kochman96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochman">Stanley Kochman</a>, section 1.5 and section 3.7 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>Review:</p> <ul> <li id="Malkiewich11"> <p><a class="existingWikiWord" href="/nlab/show/Cary+Malkiewich">Cary Malkiewich</a>, section 2 of: <em>Unoriented cobordism and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>O</mi></mrow><annotation encoding="application/x-tex">M O</annotation></semantics></math></em>, 2011 (<a href="http://math.uiuc.edu/~cmalkiew/cobordism.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Branko+Juran">Branko Juran</a>, <em>Thom spaces and the Oriented Cobordism Ring</em>, 2020 (<a href="http://www.math.uni-bonn.de/people/daniel/2020/CharClasses/Vortrag4.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/JuranMO.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/MO">MO</a>-bordism with <a class="existingWikiWord" href="/nlab/show/MSO">MSO</a>-<a class="existingWikiWord" href="/nlab/show/manifold+with+boundary">boundaries</a>:</p> <ul> <li>G. E. Mitchell, <em>Bordism of Manifolds with Oriented Boundaries</em>, Proceedings of the American Mathematical Society Vol. 47, No. 1 (Jan., 1975), pp. 208-214 (<a href="https://doi.org/10.2307/2040234">doi:10.2307/2040234</a>)</li> </ul> <p>In the incarnation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MO</mi></mrow><annotation encoding="application/x-tex">MO</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/symmetric+spectrum">symmetric spectrum</a>:</p> <ul> <li id="Schwede12"><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, Example I.2.8 in <em>Symmetric spectra</em>, 2012 (<a href="http://www.math.uni-bonn.de/~schwede/SymSpec.pdf">pdf</a>)</li> </ul> <p>In the incarnation as an <a class="existingWikiWord" href="/nlab/show/orthogonal+spectrum">orthogonal spectrum</a> (in fact as an <a class="existingWikiWord" href="/nlab/show/equivariant+spectrum">equivariant spectrum</a> in <a class="existingWikiWord" href="/nlab/show/global+equivariant+stable+homotopy+theory">global equivariant stable homotopy theory</a>):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, chapter V.4 of <em><a class="existingWikiWord" href="/nlab/show/Global+homotopy+theory">Global homotopy theory</a></em>, 2015</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 9, 2021 at 12:28:02. 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