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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='#GeometricModel'>Geometric model via cobordism classes</a></li> <li><a href='#Examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesPontrjaginThomConstruction'>Pontrjagin-Thom construction</a></li> <ul> <li><a href='#ReferencesPontrjaginConstruction'>Pontrjagin’s construction</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#TwistedEquivariantPontrjaginConstruction'>Twisted/equivariant generalizations</a></li> <li><a href='#InNegativeCodimension'>In negative codimension</a></li> </ul> <li><a href='#thoms_construction'>Thom’s construction</a></li> <li><a href='#lashofs_construction'>Lashof’s construction</a></li> </ul> <li><a href='#relation_to_divisors'>Relation to divisors</a></li> </ul> </ul> </div> <h2 id="Idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>, a <a class="existingWikiWord" href="/nlab/show/Whitehead-generalized+cohomology+theory">Whitehead-generalized cohomology theory</a> <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">represented</a> by a <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> is called a <em>cobordism cohomology theory</em> (<a href="#Atiyah61">Atiyah 61</a>), in <a class="existingWikiWord" href="/nlab/show/duality">duality</a> with the corresponding <a class="existingWikiWord" href="/nlab/show/generalized+homology+theory">generalized homology theory</a> called <em><a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism homology theory</a></em>.</p> <p>In both cases, a version of the <a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+construction">Pontryagin-Thom construction</a> identifies the (co)homology classes of these (co)homology theories with <a class="existingWikiWord" href="/nlab/show/bordism">bordism</a>-<a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> (carrying some given <a class="existingWikiWord" href="/nlab/show/extra+structure">extra structure</a>), whence the name. For <a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism homology theory</a> this was understood since the very inception of the subject (<a href="#Thom54">Thom 54</a>), while for cobordism cohomology theory this identification is made explicit in <a href="#Atiyah61">Atiyah 61, Sec. 3</a>, <a href="#Quillen71">Quillen 71</a> (relying on results from <a href="#Thom54">Thom 54</a> nonetheless), see below at <em><a href="#GeometricModel">Geometric model via cobordism classes</a></em>.</p> <p>Accordingly, cobordism cohomology theories are fundamental concepts of <a class="existingWikiWord" href="/nlab/show/bordism+theory">bordism theory</a> in <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a>. But in addition they turn out to play a special role in the more abstract <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theories">complex oriented cohomology theories</a> (with its variants such as <a class="existingWikiWord" href="/nlab/show/quaternionic+oriented+cohomology">quaternionic-oriented theories</a>) and in the resulting <a class="existingWikiWord" href="/nlab/show/chromatic+homotopy+theory">chromatic homotopy theory</a>, see for instance the <a class="existingWikiWord" href="/nlab/show/universal+complex+orientation+on+MU">universal complex orientation on MU</a>. This way, cobordism cohomology embodies a remarkable confluence of the <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> with deep issues in abstract <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <p>There are many different flavours of cobordism cohomology theories (see the list of <a href="#Examples">Examples</a> below), depending on the <a class="existingWikiWord" href="/nlab/show/tangential+structure">tangential structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> encoded in the representing <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">M f</annotation></semantics></math>. Among the most commonly considered versions are these:</p> <ul> <li> <p>The <a class="existingWikiWord" href="/nlab/show/cohomology+theory">cohomology theory</a> <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">represented</a> by <a class="existingWikiWord" href="/nlab/show/MO">MO</a> is the base case, in the sense that the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalent</a> to the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mi>ℝ</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_{\mathbb{R}}(n)</annotation></semantics></math> (namely the <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a>), so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/tangential+structure">tangential structure</a> corresponds to no extra <a class="existingWikiWord" href="/nlab/show/tangential+structure">tangential structure</a>.</p> </li> <li> <p>The cohomology theory represented by <a class="existingWikiWord" href="/nlab/show/MU">MU</a> is <em><a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology">complex cobordism cohomology</a></em>. Its <a class="existingWikiWord" href="/nlab/show/periodic+cohomology+theory">periodic cohomology theory</a> version is sometimes denoted <a class="existingWikiWord" href="/nlab/show/MP">MP</a>. The cohomology theories <a class="existingWikiWord" href="/nlab/show/MO">MO</a> and <a class="existingWikiWord" href="/nlab/show/MU">MU</a> are unified by the <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology+theory">equivariant cohomology theory</a> called <a class="existingWikiWord" href="/nlab/show/MR-theory">MR-theory</a>.</p> </li> <li> <p>On the other hand, <a class="existingWikiWord" href="/nlab/show/framed+manifold">framed</a> cobordism cohomology theory <a class="existingWikiWord" href="/nlab/show/MFr">MFr</a> is equivalently <em><a class="existingWikiWord" href="/nlab/show/stable+Cohomotopy+theory">stable Cohomotopy theory</a></em> (by the <a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+theorem">Pontryagin-Thom theorem</a>). In its unstable version as plain <a class="existingWikiWord" href="/nlab/show/Cohomotopy+theory">Cohomotopy theory</a> this – and its relation to cobordism classes of <a class="existingWikiWord" href="/nlab/show/normally+framed+submanifolds">normally framed submanifolds</a> (the <a href="cohomotopy#RelationToCobordismGroup">Pontryagin construction</a>) – this was the historical origin of <a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a> (in <a href="#Pontrjagin38">Pontrjagin 38</a>, <a href="#Pontrjagin55">Pontrjagin 55</a>).</p> </li> </ul> <h2 id="GeometricModel">Geometric model via cobordism classes</h2> <p>We discuss a geometric model for the cobordism cohomology theory, due to <a href="#Quillen71">Quillen 71, Section 1</a>. We concentrate on the complex case, corresponding to the <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> <a class="existingWikiWord" href="/nlab/show/MU">MU</a>:</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For a <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><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, the cobordism <a class="existingWikiWord" href="/nlab/show/cohomology+group">cohomology group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">M</mi><msup><mi mathvariant="normal">U</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><msub><mi>X</mi> <mo>+</mo></msub><mo>,</mo><msup><mi>Σ</mi> <mi>q</mi></msup><mi>MU</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathrm{M} \mathrm{U}^q(X) \;\coloneqq\; [\Sigma^\infty X_+, \Sigma^q MU]</annotation></semantics></math> is equivalently the set of <a href="#CobordismClassesOfMaps">cobordism classes</a> of <a class="existingWikiWord" href="/nlab/show/proper+maps">proper</a> <a href="#ComplexOrientedMaps">complex-oriented</a> maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \colon Z \to X</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>.</p> </div> <p>(<a href="#Quillen71">Quillen 71, Prop. 1.2</a>)</p> <p>This uses the following definitions:</p> <div class="num_defn" id="ComplexOrientedMaps"> <h6 id="definition">Definition</h6> <p><strong>(complex-oriented maps)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \colon Z \to X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+map">smooth map</a>.</p> <p>If the relative <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is even at all points of <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>, then a <em>complex orientation</em> is an <a class="existingWikiWord" href="/nlab/show/equivalence+class">equivalence class</a> of factorizations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> in the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>i</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Z</mi><mo>⟶</mo><mi>E</mi><mo>⟶</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> p \circ i \;\colon\; Z \longrightarrow E \longrightarrow X \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p\colon E\to X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/complex+vector+bundle">complex vector bundle</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">i \colon Z\to E</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/embedding+of+topological+spaces">embedding</a> equipped with a <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a> on its <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a>.</p> <p>Two such factorizations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,p)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>′</mo><mo>,</mo><mi>p</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i',p')</annotation></semantics></math> are regarded as equivalent if there is another factorization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>″</mo><mo>,</mo><mi>p</mi><mo>″</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i'',p'')</annotation></semantics></math> together with embeddings of <a class="existingWikiWord" href="/nlab/show/complex+vector+bundles">complex vector bundles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>E</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">E\to E'</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>E</mi><mo>″</mo></mrow><annotation encoding="application/x-tex">E\to E''</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>″</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>E</mi><mo>″</mo><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i''\colon X\times[0,1]\to E''\times [0,1]</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a> on its <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a> that restricts to the corresponding complex structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">X \times \{0\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">X \times \{1\}</annotation></semantics></math>.</p> </div> <div class="num_defn" id="CobordismClassesOfMaps"> <h6 id="definition_2">Definition</h6> <p><strong>(cobordism classes of maps)</strong></p> <p>Here two <a class="existingWikiWord" href="/nlab/show/proper+map">proper</a> <a href="#ComplexOrientedMaps">complex-oriented</a> maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>Z</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f_i \colon Z_i \to X</annotation></semantics></math> are called <em>cobordant</em> if there is a proper complex-oriented map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo lspace="verythinmathspace">:</mo><mi>W</mi><mo>→</mo><mi>X</mi><mo>×</mo><mstyle mathvariant="bold"><mi>R</mi></mstyle></mrow><annotation encoding="application/x-tex">b\colon W\to X\times\mathbf{R}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">X\times\{0\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">X\times\{1\}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/transversal+map">transversal</a> to <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">b</annotation></semantics></math> and pulling back <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">b</annotation></semantics></math> to these submanifolds yields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">f_0</annotation></semantics></math> and <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>.</p> </div> <p>(<a href="#Quillen71">Quillen 71, p. 31</a>)</p> <h2 id="Examples">Examples</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="related_concepts">Related concepts</h2> <ul> <li> <p>the refinement of cobordism cohomology theory to <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> is <a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+theory+determining+homology+theory">cobordism theory determining homology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+complex+cobordism+cohomology+theory">equivariant complex cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Adams-Novikov+spectral+sequence">Adams-Novikov spectral sequence</a></p> </li> </ul> <p><br /></p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/chromatic+homotopy+theory">chromatic homotopy theory</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/chromatic+level">chromatic level</a></th><th><a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>/<a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></th><th><a class="existingWikiWord" href="/nlab/show/real+oriented+cohomology+theory">real oriented cohomology theory</a></th></tr></thead><tbody><tr><td style="text-align: left;">0</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">H \mathbb{Z}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/HZR-theory">HZR-theory</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">0th <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">Morava K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(0)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory+spectrum">complex K-theory spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KU</mi></mrow><annotation encoding="application/x-tex">KU</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">first <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">Morava K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(1)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">first <a class="existingWikiWord" href="/nlab/show/Morava+E-theory">Morava E-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(1)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+spectrum">elliptic spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ell</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">Ell_E</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">second <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">Morava K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(2)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">second <a class="existingWikiWord" href="/nlab/show/Morava+E-theory">Morava E-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(2)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a> of <a class="existingWikiWord" href="/nlab/show/KU">KU</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(KU)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">3 …10</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+cohomology">K3 cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+spectrum">K3 spectrum</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><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></td><td style="text-align: left;"><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>th <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">Morava K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(n)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><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>th <a class="existingWikiWord" href="/nlab/show/Morava+E-theory">Morava E-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(n)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/BPR-theory">BPR-theory</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a> applied to chrom. level <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></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(E_n)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/red-shift+conjecture">red-shift conjecture</a>)</td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology">complex cobordism cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/MU">MU</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/MR-theory">MR-theory</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Original articles introducing cobordism as a <a class="existingWikiWord" href="/nlab/show/Whitehead-generalized+cohomology+theory">Whitehead-generalized cohomology theory</a>:</p> <ul> <li id="Atiyah61"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <em>Bordism and Cobordism</em>, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 57, Issue 2, April 1961, pp. 200 - 208 (<a href="https://doi.org/10.1017/S0305004100035064">doi:10.1017/S0305004100035064</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahb.pdf">pdf</a>)</p> <blockquote> <p>(introducing the concept, with focus on<a class="existingWikiWord" href="/nlab/show/MSO">MSO</a>)</p> </blockquote> </li> <li id="ConnerFloyd66"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Conner">Pierre Conner</a>, <a class="existingWikiWord" href="/nlab/show/Edwin+Floyd">Edwin Floyd</a>, <em><a class="existingWikiWord" href="/nlab/show/The+Relation+of+Cobordism+to+K-Theories">The Relation of Cobordism to K-Theories</a></em>, Lecture Notes in Mathematics <strong>28</strong> Springer 1966 (<a href="https://link.springer.com/book/10.1007/BFb0071091">doi:10.1007/BFb0071091</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=216511">MR216511</a>)</p> <blockquote> <p>(relating <a class="existingWikiWord" href="/nlab/show/complex+cobordism+theory">complex cobordism theory</a> to <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> and the <a class="existingWikiWord" href="/nlab/show/e-invariant">e-invariant</a>)</p> </blockquote> </li> <li id="Quillen71"> <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 7 (1971), 29–56 (<a href="http://dx.doi.org/10.1016/0001-8708(71)90041-7">doi:10.1016/0001-8708(71)90041-7</a>)</p> </li> </ul> <p>Early survey:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Victor+Buchstaber">Victor Buchstaber</a>, <em>Cobordisms in problems of algebraic topology</em>, J Math Sci 7, 629–653 (1977) (<a href="https://doi.org/10.1007/BF01084983">doi:10.1007/BF01084983</a>)</p> </li> <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, Volume 100, Issue 2 September 1986, pp. 207-223 (<a href="https://doi.org/10.1017/S0305004100066032">doi:10.1017/S0305004100066032</a>)</p> </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="Rudyak98"> <p><a class="existingWikiWord" href="/nlab/show/Yuli+Rudyak">Yuli Rudyak</a>, <em>On Thom spectra, orientability and cobordism</em>, Springer Monographs in Mathematics (1998) [<a href="https://doi.org/10.1007/978-3-540-77751-9">doi:10.1007/978-3-540-77751-9</a>, <a href="http://www.maths.ed.ac.uk/~aar/papers/rudyakthom.pdf">pdf</a>]</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> and <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant</a> versions:</p> <ul> <li id="Cruickshank99"> <p><a class="existingWikiWord" href="/nlab/show/James+Cruickshank">James Cruickshank</a>, <em>Twisted Cobordism and its Relationship to Equivariant Homotopy Theory</em>, 1999 (<a href="http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape9/PQDD_0030/NQ46823.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Cruickshank99.pdf" title="pdf">pdf</a>)</p> <blockquote> <p>(<a class="existingWikiWord" href="/nlab/show/equivariant+Pontrjagin+theorem">equivariant Pontrjagin theorem</a> for <a class="existingWikiWord" href="/nlab/show/free+actions">free</a> <a class="existingWikiWord" href="/nlab/show/G-manifolds">G-manifolds</a>)</p> </blockquote> </li> <li id="Cruickshank03"> <p><a class="existingWikiWord" href="/nlab/show/James+Cruickshank">James Cruickshank</a>, <em>Twisted homotopy theory and the geometric equivariant 1-stem</em>, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (<a href="https://doi.org/10.1016/S0166-8641(02)00183-9">arXiv:10.1016/S0166-8641(02)00183-9</a>)</p> </li> </ul> <div> <h3 id="ReferencesPontrjaginThomConstruction">Pontrjagin-Thom construction</h3> <h4 id="ReferencesPontrjaginConstruction">Pontrjagin’s construction</h4> <h5 id="general">General</h5> <p>The <em><a class="existingWikiWord" href="/nlab/show/Pontryagin+theorem">Pontryagin theorem</a></em>, i.e. the unstable and <a class="existingWikiWord" href="/nlab/show/normally+framed+submanifold">framed</a> version of the <a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a>, identifying <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> with their <a class="existingWikiWord" href="/nlab/show/Cohomotopy+charge">Cohomotopy charge</a> in unstable <a class="existingWikiWord" href="/nlab/show/Karol+Borsuk">Borsuk</a>-<a class="existingWikiWord" href="/nlab/show/Edwin+Spanier">Spanier</a> <a class="existingWikiWord" href="/nlab/show/Cohomotopy+sets">Cohomotopy sets</a>, is due to:</p> <ul> <li id="Pontryagin38a"> <p><a class="existingWikiWord" href="/nlab/show/Lev+Pontrjagin">Lev Pontrjagin</a>, <em><a class="existingWikiWord" href="/nlab/show/Classification+of+continuous+maps+of+a+complex+into+a+sphere">Classification of continuous maps of a complex into a sphere</a></em>, <em>Communication I</em>, Doklady Akademii Nauk SSSR <strong>19</strong> 3 (1938) 147-149</p> </li> <li id="Pontryagin50"> <p><a class="existingWikiWord" href="/nlab/show/Lev+Pontryagin">Lev Pontryagin</a>, <em><a class="existingWikiWord" href="/nlab/show/Homotopy+classification+of+mappings+of+an+%28n%2B2%29-dimensional+sphere+on+an+n-dimensional+one">Homotopy classification of mappings of an (n+2)-dimensional sphere on an n-dimensional one</a></em>, Doklady Akad. Nauk SSSR (N.S.) 19 (1950), 957–959 (<a href="https://www.maths.ed.ac.uk/~v1ranick/papers/pont3.pdf">pdf</a>)</p> </li> </ul> <p>(both available in English translation in <a href="Revaz+Gamkrelidze#Gamkrelidze86">Gamkrelidze 86</a>),</p> <p>as presented more comprehensively in:</p> <ul> <li id="Pontryagin55"><a class="existingWikiWord" href="/nlab/show/Lev+Pontrjagin">Lev Pontrjagin</a>, <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>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/pont001.pdf">pdf</a>)</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/Pontrjagin+theorem">Pontrjagin theorem</a> must have been known to Pontrjagin at least by 1936, when he announced the computation of the <a class="existingWikiWord" href="/nlab/show/second+stable+homotopy+group+of+spheres">second stem of homotopy groups of spheres</a>:</p> <ul> <li id="Pontrjagin36"><a class="existingWikiWord" href="/nlab/show/Lev+Pontrjagin">Lev Pontrjagin</a>, <em>Sur les transformations des sphères en sphères</em> (<a class="existingWikiWord" href="/nlab/files/PontrjaginSurLesTransformationDesSpheres.pdf" title="pdf">pdf</a>) in: <em>Comptes Rendus du Congrès International des Mathématiques – Oslo 1936</em> (<a href="https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1936.2/ICM1936.2.ocr.pdf">pdf</a>)</li> </ul> <p>Review:</p> <ul> <li id="FreedUhlenbeck91"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Karen+Uhlenbeck">Karen Uhlenbeck</a>, Appendix B of: <em>Instantons and Four-Manifolds</em>, Mathematical Sciences Research Institute Publications, Springer 1991 (<a href="https://link.springer.com/book/10.1007/978-1-4613-9703-8">doi:10.1007/978-1-4613-9703-8</a>)</p> </li> <li id="Bredon93"> <p><a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a>, chapter II.16 of: <em>Topology and Geometry</em>, Graduate Texts in Mathematics <strong>139</strong>, Springer (1993) [<a href="https://link.springer.com/book/10.1007/978-1-4757-6848-0">doi:10.1007/978-1-4757-6848-0</a>, <a href="http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf">pdf</a>]</p> </li> </ul> <div style="margin: -30px 0px 20px 10px"> <img src="/nlab/files/Bredon-FigII13-PontryaginThom.jpg" width="450px" /> </div> <ul> <li id="Kosinski93"> <p><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>, <a href="https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/138/suppl/C">ISBN:978-0-12-421850-5</a>]</p> </li> <li id="Milnor97"> <p><a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, Chapter 7 of: <em>Topology from the differentiable viewpoint</em>, Princeton University Press, 1997. (<a href="https://press.princeton.edu/books/paperback/9780691048338/topology-from-the-differentiable-viewpoint">ISBN:9780691048338</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/milnortop.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mladen+Bestvina">Mladen Bestvina</a> (notes by <a class="existingWikiWord" href="/nlab/show/Adam+Keenan">Adam Keenan</a>), Chapter 16 in: <em>Differentiable Topology and Geometry</em>, 2002 (<a class="existingWikiWord" href="/nlab/files/BestvinaKeenanDifferentialTopology.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michel+Kervaire">Michel Kervaire</a>, <em>La méthode de Pontryagin pour la classification des applications sur une sphère</em>, in: E. Vesentini (ed.), <em>Topologia Differenziale</em>, CIME Summer Schools, vol. 26, Springer 2011 (<a href="https://doi.org/10.1007/978-3-642-10988-1_3">doi:10.1007/978-3-642-10988-1_3</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Rustam+Sadykov">Rustam Sadykov</a>, Section 1 of: <em>Elements of Surgery Theory</em>, 2013 (<a href="https://www.math.ksu.edu/~sadykov/Lecture%20Notes/Surgery%20Theory.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/SadykovSurgeryTheory.pdf" title="pdf">pdf</a>)</p> </li> <li id="Csepai20"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A1s+Cs%C3%A9pai">András Csépai</a>, <em>Stable Pontryagin-Thom construction for proper maps</em>, Period Math Hung 80, 259–268 (2020) (<a href="https://arxiv.org/abs/1905.07734">arXiv:1905.07734</a>, <a href="https://doi.org/10.1007/s10998-020-00327-0">doi:10.1007/s10998-020-00327-0</a>)</p> </li> </ul> <p>Discussion of the early history:</p> <ul> <li><a href="#Kosinski93">Kosinski 93, Section IX.9</a></li> </ul> <h5 id="TwistedEquivariantPontrjaginConstruction">Twisted/equivariant generalizations</h5> <p>The (fairly straightforward) generalization of the <a class="existingWikiWord" href="/nlab/show/Pontrjagin+theorem">Pontrjagin theorem</a> to the <a class="existingWikiWord" href="/nlab/show/twisted+Pontrjagin+theorem">twisted Pontrjagin theorem</a>, identifying <a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a> with <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of <a class="existingWikiWord" href="/nlab/show/normally+twisted-framed+submanifolds">normally twisted-framed submanifolds</a>, is made explicit in:</p> <ul> <li id="Cruickshank03"><a class="existingWikiWord" href="/nlab/show/James+Cruickshank">James Cruickshank</a>, Lemma 5.2 using Sec. 5.1 in: <em>Twisted homotopy theory and the geometric equivariant 1-stem</em>, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (<a href="https://doi.org/10.1016/S0166-8641(02)00183-9">doi:10.1016/S0166-8641(02)00183-9</a>)</li> </ul> <p>A general <a class="existingWikiWord" href="/nlab/show/equivariant+Pontrjagin+theorem">equivariant Pontrjagin theorem</a> – relating <a class="existingWikiWord" href="/nlab/show/equivariant+Cohomotopy">equivariant Cohomotopy</a> to normal equivariant framed submanifolds – remains elusive, but on <a class="existingWikiWord" href="/nlab/show/free+action">free</a> <a class="existingWikiWord" href="/nlab/show/G-manifolds">G-manifolds</a> it is again straightforward (and reduces to the <a class="existingWikiWord" href="/nlab/show/twisted+Pontrjagin+theorem">twisted Pontrjagin theorem</a> on the <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>), made explicit in:</p> <ul> <li id="Cruickshank99"><a class="existingWikiWord" href="/nlab/show/James+Cruickshank">James Cruickshank</a>, Thm. 5.0.6, Cor. 6.0.13 in: <em>Twisted Cobordism and its Relationship to Equivariant Homotopy Theory</em>, 1999 (<a href="http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape9/PQDD_0030/NQ46823.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Cruickshank99.pdf" title="pdf">pdf</a>)</li> </ul> <h5 id="InNegativeCodimension">In negative codimension</h5> <p>In <a class="existingWikiWord" href="/nlab/show/negative+number">negative</a> <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a>, the <a class="existingWikiWord" href="/nlab/show/Cohomotopy+charge+map">Cohomotopy charge map</a> from the <a class="existingWikiWord" href="/nlab/show/Pontrjagin+theorem">Pontrjagin theorem</a> gives the <a href="configuration+space+of+points#LoopSpacesOfSuspensions">May-Segal theorem</a>, now identifying <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a> <em><a class="existingWikiWord" href="/nlab/show/cocycle+spaces">cocycle spaces</a></em> with <a class="existingWikiWord" href="/nlab/show/configuration+spaces+of+points">configuration spaces of points</a>:</p> <ul> <li id="May72"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em>The geometry of iterated loop spaces</em>, Springer 1972 (<a href="https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf">pdf</a>)</p> </li> <li id="Segal73"> <p><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>Configuration-spaces and iterated loop-spaces</em>, Invent. Math. <strong>21</strong> (1973), 213–221. MR 0331377 (<a href="http://dodo.pdmi.ras.ru/~topology/books/segal.pdf">pdf</a>)</p> <p>c Generalization of these constructions and results is due to</p> </li> <li id="McDuff75"> <p><a class="existingWikiWord" href="/nlab/show/Dusa+McDuff">Dusa McDuff</a>, <em>Configuration spaces of positive and negative particles</em>, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (<a href="https://doi.org/10.1016/0040-9383(75)90038-5">doi:10.1016/0040-9383(75)90038-5</a>)</p> </li> <li id="Boedigheimer87"> <p><a class="existingWikiWord" href="/nlab/show/Carl-Friedrich+B%C3%B6digheimer">Carl-Friedrich Bödigheimer</a>, <em>Stable splittings of mapping spaces</em>, Algebraic topology. Springer 1987. 174-187 (<a href="http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BoedigheimerStableSplittings87.pdf" title="pdf">pdf</a>)</p> </li> </ul> <h4 id="thoms_construction">Thom’s construction</h4> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+theorem">Thom's theorem</a> i.e. the unstable and <em>oriented</em> version of the <a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a>, identifying <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> with <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of <a class="existingWikiWord" href="/nlab/show/maps">maps</a> to the <a class="existingWikiWord" href="/nlab/show/universal+vector+bundle">universal</a> <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal</a> <a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M SO(n)</annotation></semantics></math>, is due to:</p> <ul> <li id="Thom54"><a class="existingWikiWord" href="/nlab/show/Ren%C3%A9+Thom">René Thom</a>, <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>)</li> </ul> <p>Textbook accounts:</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> <h4 id="lashofs_construction">Lashof’s construction</h4> <p>The joint generalization of <a href="#Pontryagin38a">Pontryagin 38a</a>, <a href="#Pontryagin55">55</a> (framing structure) and <a href="#Thom54">Thom 54</a> (orientation structure) to any family of <a class="existingWikiWord" href="/nlab/show/tangential+structures">tangential structures</a> (“<a class="existingWikiWord" href="/nlab/show/%28B%2Cf%29-structure">(B,f)-structure</a>”) is first made explicit in</p> <ul> <li id="Lashof63"><a class="existingWikiWord" href="/nlab/show/Richard+Lashof">Richard Lashof</a>, <em>Poincaré duality and cobordism</em>, Trans. AMS 109 (1963), 257-277 (<a href="https://doi.org/10.1090/S0002-9947-1963-0156357-4">doi:10.1090/S0002-9947-1963-0156357-4</a>)</li> </ul> <p>and the general statement that has come to be known as the <em><a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+isomorphism">Pontryagin-Thom isomorphism</a></em> (identifying the stable <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of normally <a class="existingWikiWord" href="/nlab/show/tangential+structure">(B,f)-structured</a> <a class="existingWikiWord" href="/nlab/show/submanifolds">submanifolds</a> with <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of <a class="existingWikiWord" href="/nlab/show/maps">maps</a> to the <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> <a class="existingWikiWord" href="/nlab/show/Mf">Mf</a>) is really due to <a href="#Lashof63">Lashof 63, Theorem C</a>.</p> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Theodor+Br%C3%B6cker">Theodor Bröcker</a>, <a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, Satz 3.1 & 4.9 in: <em>Kobordismentheorie</em>, Lecture Notes in Mathematics <strong>178</strong>, Springer (1970) &lbrack;<a href="https://link.springer.com/book/9783540053415">ISBN:9783540053415</a>&rbrack;</p> </li> <li id="Kochman96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochman">Stanley Kochman</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 id="Rudyak98"> <p><a class="existingWikiWord" href="/nlab/show/Yuli+Rudyak">Yuli Rudyak</a>, <em>On Thom spectra, orientability and cobordism</em>, Springer Monographs in Mathematics (1998) [<a href="https://doi.org/10.1007/978-3-540-77751-9">doi:10.1007/978-3-540-77751-9</a>, <a href="http://www.maths.ed.ac.uk/~aar/papers/rudyakthom.pdf">pdf</a>]</p> </li> </ul> <p>Lecture notes:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Francis">John Francis</a>, <em>Topology of manifolds</em> course notes (2010) (<a href="http://math.northwestern.edu/~jnkf/classes/mflds/">web</a>), Lecture 3: <em>Thom’s theorem</em> (<a href="http://math.northwestern.edu/~jnkf/classes/mflds/3thom.pdf">pdf</a>), Lecture 4 <em>Transversality</em> (notes by I. Bobkova) (<a href="http://math.northwestern.edu/~jnkf/classes/mflds/4transversality.pdf">pdf</a>)</p> </li> <li id="Malkiewich11"> <p><a class="existingWikiWord" href="/nlab/show/Cary+Malkiewich">Cary Malkiewich</a>, Section 3 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>Tom Weston, Part I of <em>An introduction to cobordism theory</em> (<a href="http://people.math.umass.edu/~weston/oldpapers/cobord.pdf">pdf</a>)</p> </li> </ul> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Manifold+Atlas">Manifold Atlas</a>, <em><a href="http://www.map.mpim-bonn.mpg.de/B-Bordism#The_Pontrjagin-Thom_isomorphism">The Pontrjagin-Thom isomorphism</a></em></li> </ul> </div> <h3 id="relation_to_divisors">Relation to divisors</h3> <p>Relation of <a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology">complex cobordism cohomology</a> with <a class="existingWikiWord" href="/nlab/show/divisors">divisors</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+cycles">algebraic cycles</a> and <a class="existingWikiWord" href="/nlab/show/Chow+groups">Chow groups</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Burt+Totaro">Burt Totaro</a>, <em>Torsion algebraic cycles and complex cobordism</em>, J. Amer. Math. Soc. 10 (1997), 467-493 (<a href="https://doi.org/10.1090/S0894-0347-97-00232-4">doi:10.1090/S0894-0347-97-00232-4</a>)</p> </li> <li> <p><a href="https://mathoverflow.net/a/272131/381">MO discussion</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 1, 2024 at 13:27:35. 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