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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='#relation_to_'>Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math></a></li> <li><a href='#RelationToCalabiYauManifolds'>Relation to Calabi-Yau manifolds</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 <a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a> for <a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>-<a class="existingWikiWord" href="/nlab/show/G-structure">structure</a>.</p> <h2 id="properties">Properties</h2> <p>We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>•</mo> <mi>SU</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{SU}_\bullet</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/bordism+ring">bordism ring</a> for <a class="existingWikiWord" href="/nlab/show/stable+tangent+bundle">stable</a> <a class="existingWikiWord" href="/nlab/show/special+unitary+group">SU</a>-<a class="existingWikiWord" href="/nlab/show/G-structure">structure</a>.</p> <h3 id="relation_to_">Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math></h3> <p>The canonical <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>-<a class="existingWikiWord" href="/nlab/show/subgroup">inclusions</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mspace width="thickmathspace"></mspace><mo>⊂</mo><mspace width="thickmathspace"></mspace><mi>Sp</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⊂</mo><mspace width="thickmathspace"></mspace><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⊂</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> 1 \;\subset\; Sp(k) \;\subset\; SU(2k) \;\subset\; U(2k) </annotation></semantics></math></div> <p>(<a class="existingWikiWord" href="/nlab/show/trivial+group">trivial group</a> into <a class="existingWikiWord" href="/nlab/show/quaternionic+unitary+group">quaternionic unitary group</a> into <a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a> into <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a>) induce <a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>-<a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/Thom+spectra">Thom spectra</a></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><mspace width="thickmathspace"></mspace><mo>⟶</mo><mspace width="thickmathspace"></mspace><mi>M</mi><mi>Sp</mi><mspace width="thickmathspace"></mspace><mo>⟶</mo><mspace width="thickmathspace"></mspace><mi>M</mi><mi>SU</mi><mspace width="thickmathspace"></mspace><mo>⟶</mo><mspace width="thickmathspace"></mspace><mi>M</mi><mi mathvariant="normal">U</mi></mrow><annotation encoding="application/x-tex"> M Fr \;\longrightarrow\; M Sp \;\longrightarrow\; M SU \;\longrightarrow\; M \mathrm{U} </annotation></semantics></math></div> <p>(from <a class="existingWikiWord" href="/nlab/show/MFr">MFr</a> to <a class="existingWikiWord" href="/nlab/show/MSp">MSp</a> to <a class="existingWikiWord" href="/nlab/show/MSU">MSU</a> to <a class="existingWikiWord" href="/nlab/show/MU">MU</a>)</p> <p>and hence corresponding <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a>-<a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theories">cobordism cohomology theories</a>, so in particular <a class="existingWikiWord" href="/nlab/show/ring+homomorphisms">ring homomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/bordism+rings">bordism rings</a></p> <div class="maruku-equation" id="eq:HomomorphismsOfCobordismRings"><span class="maruku-eq-number">(1)</span><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><mo>⟶</mo><msubsup><mi>Ω</mi> <mo>•</mo> <mi>Sp</mi></msubsup><mo>⟶</mo><msubsup><mi>Ω</mi> <mo>•</mo> <mi>SU</mi></msubsup><mo>⟶</mo><msubsup><mi>Ω</mi> <mo>•</mo> <mi>U</mi></msubsup></mrow><annotation encoding="application/x-tex"> \Omega^{fr}_{\bullet} \longrightarrow \Omega^{Sp}_{\bullet} \longrightarrow \Omega^{SU}_{\bullet} \longrightarrow \Omega^{U}_{\bullet} </annotation></semantics></math></div> <p>(e.g. <a href="#ConnerFloyd66">Conner-Floyd 66, p. 27 (34 of 120)</a>)</p> <div class="num_prop" id="KernelOfMapFromSUBordismToUBordismIsTorsion"> <h6 id="proposition">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful</a> morphism <a class="maruku-eqref" href="#eq:HomomorphismsOfCobordismRings">(1)</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>SU</mi></msubsup><mo>⟶</mo><msubsup><mi>Ω</mi> <mo>•</mo> <mi mathvariant="normal">U</mi></msubsup></mrow><annotation encoding="application/x-tex"> \Omega^{SU}_\bullet \longrightarrow \Omega^{\mathrm{U}}_\bullet </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> to the <a class="existingWikiWord" href="/nlab/show/complex+bordism+ring">complex bordism ring</a>, is pure <a class="existingWikiWord" href="/nlab/show/torsion+subgroup">torsion</a>.</p> </div> <p>(<a href="#CLP19">CLP 19, Thm. 5.8a</a>)</p> <div class="num_prop" id="TorsionInSUBordismConcentratedInDegrees1And2Mod8"> <h6 id="proposition_2">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/torsion+subgroup">torsion subgroup</a> of the <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> is concentrated in degrees <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>8</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">8k+1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>8</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">8k+2</annotation></semantics></math>, for <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 \in \mathbb{N}</annotation></semantics></math>.</p> </div> <p>(<a href="#CLP19">CLP 19, Thm. 5.11a</a>)</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/torsion+element">torsion element</a> in the <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>•</mo> <mi>SU</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{SU}_\bullet</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/order+of+an+element">order</a> 2.</p> </div> <p>(<a href="#CLP19">CLP 19, Thm. 5.8b</a>)</p> <div class="num_prop" id="SUBordismRingAwayFromTwo"> <h6 id="proposition_4">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">away from 2</a> is <a class="existingWikiWord" href="/nlab/show/polynomial+algebra">polynomial algebra</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> with <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">2 inverted</a> is the <a class="existingWikiWord" href="/nlab/show/polynomial+algebra">polynomial algebra</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}\big[\tfrac{1}{2}\big]</annotation></semantics></math> on one generator in every even degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">\geq 4</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>SU</mi></msup><mo maxsize="1.2em" minsize="1.2em">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">]</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><mo stretchy="false">{</mo><msub><mi>y</mi> <mrow><mn>2</mn><mi>i</mi><mo>+</mo><mn>4</mn></mrow></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^{SU}\big[\tfrac{1}{2}\big] \;\simeq\; \mathbb{Z} \big[ \tfrac{1}{2} \big] \big[ \{ y_{2i+4} \}_{i \in \mathbb{N}} \big] \,. </annotation></semantics></math></div></div> <p>(due to <a href="#Novikov62">Novikov 62</a>, review in <a href="#LLP17">LLP 17, Thm. 1.2</a>)</p> <h3 id="RelationToCalabiYauManifolds">Relation to Calabi-Yau manifolds</h3> <p>We discuss the classes of <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds">Calabi-Yau manifolds</a> in the <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a>. For more see at <em><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds+in+SU-bordism+theory">Calabi-Yau manifolds in SU-bordism theory</a></em>.</p> <div class="num_prop" id="K3SurfaceSpansSUBordismRingInDegree4"> <h6 id="proposition_5">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/K3-surface">K3-surface</a> spans <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> in degree 4)</strong></p> <p>The degree-4 generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>y</mi> <mn>4</mn></msub><mo>∈</mo><msubsup><mi>Ω</mi> <mn>4</mn> <mi>SU</mi></msubsup></mrow><annotation encoding="application/x-tex">y_4 \in \Omega^{SU}_4</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> (Prop. <a class="maruku-ref" href="#SUBordismRingAwayFromTwo"></a>) is represented by minus the class of any (non-<a class="existingWikiWord" href="/nlab/show/torus">torus</a>) <a class="existingWikiWord" href="/nlab/show/K3-surface">K3-surface</a>:</p> <div class="maruku-equation" id="eq:K3GeneratesNonTorsionSUBordimsRing"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mn>4</mn> <mi>SU</mi></msubsup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^{SU}_4 \;\simeq\; \mathbb{Z}\big\langle -[K3] \big\rangle \,. </annotation></semantics></math></div></div> <p>(<a href="#LLP17">LLP 17, Example 3.1</a>, <a href="#CLP19">CLP 19, Theorem 13.5a</a>)</p> <div class="num_cor" id="K3SurfaceInUBordismRing"> <h6 id="corollary">Corollary</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/K3-surface">K3-surface</a> represents non-trivial element in <a class="existingWikiWord" href="/nlab/show/U-bordism+ring">U-bordism ring</a>)</strong></p> <p>The image in the <a class="existingWikiWord" href="/nlab/show/MU">MU</a>-<a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a> of the class of the <a class="existingWikiWord" href="/nlab/show/K3-surface">K3-surface</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mo>∈</mo><msubsup><mi>Ω</mi> <mn>4</mn> <mi>SU</mi></msubsup></mrow><annotation encoding="application/x-tex">[K3] \in \Omega^{SU}_4</annotation></semantics></math> <a class="maruku-eqref" href="#eq:K3GeneratesNonTorsionSUBordimsRing">(2)</a> under the canonical morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mn>4</mn> <mi>SU</mi></msubsup><mo>→</mo><msubsup><mi>Ω</mi> <mn>4</mn> <mi mathvariant="normal">U</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{SU}_4 \to \Omega^{\mathrm{U}}_4</annotation></semantics></math> <a class="maruku-eqref" href="#eq:HomomorphismsOfCobordismRings">(1)</a> is non-trivial.</p> <p>In fact, the canonical morphism is an <a class="existingWikiWord" href="/nlab/show/injection">injection</a> in this degree</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>Ω</mi> <mn>4</mn> <mi>SU</mi></msubsup></mtd> <mtd><mo>↪</mo></mtd> <mtd><msubsup><mi>Ω</mi> <mn>4</mn> <mi mathvariant="normal">U</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Omega^{SU}_4 &\hookrightarrow& \Omega^{\mathrm{U}}_4 \\ [K3] &\mapsto& [K3] \,. } </annotation></semantics></math></div></div> <p>(This is vaguely indicated in <a href="#Novikov86">Novikov 86, p. 216 (218 of 321)</a>.)</p> <p> <div class='proof'> <h6>Proof</h6> <p>By Prop. <a class="maruku-ref" href="#KernelOfMapFromSUBordismToUBordismIsTorsion"></a> the kernel of the map to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mn>4</mn> <mi mathvariant="normal">U</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{\mathrm{U}}_4</annotation></semantics></math> is torsion, but by Prop. <a class="maruku-ref" href="#K3SurfaceSpansSUBordismRingInDegree4"></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[K3]</annotation></semantics></math> represents a non-torsion element. Since it is in fact a non-torsion generator, the kernel vanishes (as also implied by Prop. <a class="maruku-ref" href="#TorsionInSUBordismConcentratedInDegrees1And2Mod8"></a>).</p> <p></p> </div> </p> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds">Calabi-Yau manifolds</a> generate the <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">away from 2</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">away from 2</a> is multiplicatively generated by <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds">Calabi-Yau manifolds</a>.</p> </div> <p>(<a href="#LLP17">LLP 17, Theorem 2.4</a>)</p> <div class="num_prop"> <h6 id="proposition_7">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds">Calabi-Yau manifolds</a> in complex dim <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">\leq 4</annotation></semantics></math> span the <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo>≤</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">deg \leq 8</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">away from 2</a>)</strong></p> <p>There are <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds">Calabi-Yau manifolds</a> of complex dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math> whose whose SU-bordism classes equal the generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>±</mo><msub><mi>y</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">\pm y_6</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>±</mo><msub><mi>y</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">\pm y_8</annotation></semantics></math> in Prop. <a class="maruku-ref" href="#SUBordismRingAwayFromTwo"></a>.</p> <p>Together with the <a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a> representing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>y</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">- y_4</annotation></semantics></math> (Prop. <a class="maruku-ref" href="#K3SurfaceSpansSUBordismRingInDegree4"></a>), this means that <a class="existingWikiWord" href="/nlab/show/CYs">CYs</a> <a class="existingWikiWord" href="/nlab/show/linear+space">span</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mrow><mo>≤</mo><mn>8</mn></mrow> <mi>SU</mi></msubsup><mo maxsize="1.2em" minsize="1.2em">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">]</mo></mrow><annotation encoding="application/x-tex">\Omega^{SU}_{\leq 8}\big[ \tfrac{1}{2}\big]</annotation></semantics></math>.</p> </div> <p>(<a href="#CLP19">CLP 19, Theorem 13.5</a>)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+manifold">generalized Calabi-Yau manifold</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="ConnerFloyd66"><a class="existingWikiWord" href="/nlab/show/Pierre+Conner">Pierre Conner</a>, <a class="existingWikiWord" href="/nlab/show/Edwin+Floyd">Edwin Floyd</a>, Section 5 of: <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>)</li> </ul> <p>On the SU-bordism ring structure away from 2:</p> <ul> <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 id="Stong68"> <p><a class="existingWikiWord" href="/nlab/show/Robert+Stong">Robert Stong</a>, Chapter X 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> </ul> <p>Survey:</p> <ul> <li id="Novikov86"><a class="existingWikiWord" href="/nlab/show/Sergei+Novikov">Sergei Novikov</a>, p. 218 in: <em>Topology I – General survey</em>, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 (<a href="https://link.springer.com/book/10.1007/978-3-662-10579-5">doi:10.1007/978-3-662-10579-5</a>, <a href="https://web.math.rochester.edu/people/faculty/doug/otherpapers/novikovsurv.pdf">pdf</a>)</li> </ul> <p>On its <a class="existingWikiWord" href="/nlab/show/torsion+subgroups">torsion subgroups</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Pierre+Conner">Pierre Conner</a>, <a class="existingWikiWord" href="/nlab/show/Edwin+Floyd">Edwin Floyd</a>, <em>Torsion in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi></mrow><annotation encoding="application/x-tex">SU</annotation></semantics></math>-bordism</em>, Memoirs of the American Mathematical Society 1966 (<a href="https://bookstore.ams.org/memo-1-60">ISBN:978-1-4704-0007-1</a>)</li> </ul> <p>Relation to <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds">Calabi-Yau manifolds</a>:</p> <ul> <li id="LLP17"><a class="existingWikiWord" href="/nlab/show/Ivan+Limonchenko">Ivan Limonchenko</a>, <a class="existingWikiWord" href="/nlab/show/Zhi+Lu">Zhi Lu</a>, <a class="existingWikiWord" href="/nlab/show/Taras+Panov">Taras Panov</a>, <em>Calabi-Yau hypersurfaces and SU-bordism</em>, Proceedings of the Steklov Institute of Mathematics 302 (2018), 270-278 (<a href="https://arxiv.org/abs/1712.07350">arXiv:1712.07350</a>)</li> </ul> <p>On the (failure of) the <a class="existingWikiWord" href="/nlab/show/Conner-Floyd+isomorphism">Conner-Floyd isomorphism</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MSU</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">MSU \to </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/KO">KO</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Serge+Ochanine">Serge Ochanine</a>, <em>Modules de SU-bordisme. Applications</em>, Bulletin de la Société Mathématique de France, Tome 115 (1987) , pp. 257-289 (<a href="https://doi.org/10.24033/bsmf.2078">doi:10.24033/bsmf.2078</a>)</li> </ul> <p>Survey:</p> <ul> <li id="CLP19"> <p>Georgy Chernykh, <a class="existingWikiWord" href="/nlab/show/Ivan+Limonchenko">Ivan Limonchenko</a>, <a class="existingWikiWord" href="/nlab/show/Taras+Panov">Taras Panov</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi></mrow><annotation encoding="application/x-tex">SU</annotation></semantics></math>-bordism: structure results and geometric representatives</em>, Russian Math. Surveys 74 (2019), no. 3, 461-524 (<a href="https://arxiv.org/abs/1903.07178">arXiv:1903.07178</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Taras+Panov">Taras Panov</a>, <em>A geometric view on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi></mrow><annotation encoding="application/x-tex">SU</annotation></semantics></math>-bordism</em>, talk at Moscow State University 2020 (<a href="https://www.math.princeton.edu/events/geometric-view-su-bordism-2020-09-17t170000">webpage</a>, <a href="http://higeom.math.msu.su/people/taras/talks/2019SPb-Panov.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/PanovSU-Bordism.pdf" title="pdf">pdf</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 18, 2021 at 15:09:19. 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