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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='#representing_spectrum'>Representing spectrum</a></li> <li><a href='#relation_to__and_'>Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MSU</mi></mrow><annotation encoding="application/x-tex">MSU</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MFr</mi></mrow><annotation encoding="application/x-tex">MFr</annotation></semantics></math></a></li> <li><a href='#relation_to_todd_classes_and_the_invariant'>Relation to Todd classes and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>ℝ</mi></msub></mrow><annotation encoding="application/x-tex">e_{\mathbb{R}}</annotation></semantics></math>-invariant</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>In joint generalization of the <a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theories">cobordism cohomology theories</a> <a class="existingWikiWord" href="/nlab/show/MSU">MSU</a> and <a class="existingWikiWord" href="/nlab/show/MFr">MFr</a> of <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a> <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>-manifolds and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fr</mi></mrow><annotation encoding="application/x-tex">Fr</annotation></semantics></math>-manifolds, respectively, an <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>SU</mi><mo>,</mo><mi>fr</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(SU,fr)</annotation></semantics></math>-manifold</em> (<a href="#ConnerFloyd66">Conner-Floyd 66, Section 16, p. 103 onwards</a>) is a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/manifold+with+boundary">manifold with boundary</a> equipped with <a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>-<a class="existingWikiWord" href="/nlab/show/tangential+structure">tangential structure</a> on its <a class="existingWikiWord" href="/nlab/show/stable+tangent+bundle">stable tangent bundle</a> and equipped with a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivialization</a> (stable <a class="existingWikiWord" href="/nlab/show/framed+manifold">framing</a>) of that over the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a>.</p> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/bordism+classes">bordism classes</a> form a <a class="existingWikiWord" href="/nlab/show/bordism+ring">bordism ring</a> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>•</mo> <mrow><mi>SU</mi><mo>,</mo><mi>fr</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{SU,fr}_\bullet</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="representing_spectrum">Representing spectrum</h3> <p>In generalization to how <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mrow><mn>2</mn><mi>k</mi></mrow> <mi>SU</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{SU}_{2k}</annotation></semantics></math> is represented by <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of <a class="existingWikiWord" href="/nlab/show/maps">maps</a> into the <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> <a class="existingWikiWord" href="/nlab/show/MSU">MSU</a>, so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mrow><mn>2</mn><mi>k</mi></mrow> <mrow><mi>SU</mi><mo>,</mo><mi>fr</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{SU,fr}_{2k}</annotation></semantics></math> is represented by maps into the <a class="existingWikiWord" href="/nlab/show/quotient+spaces">quotient spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>MSU</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msub><mo stretchy="false">/</mo><msup><mi>S</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex">MSU_{2k}/S^{2k}</annotation></semantics></math> (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo>=</mo><mi>Th</mi><mo stretchy="false">(</mo><msup><mi>ℂ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>→</mo><mi>Th</mi><mo stretchy="false">(</mo><msup><mi>ℂ</mi> <mi>k</mi></msup><msub><mo>×</mo> <mrow><mi>U</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mi>E</mi><mi>U</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>MU</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">S^{2k} = Th(\mathbb{C}^{k}) \to Th( \mathbb{C}^k \times_{U(k)} E U(k) ) = MU_{2k}</annotation></semantics></math> the canonical inclusion):</p> <div class="maruku-equation" id="eq:InTermsOfHomotopyGroupsOfQuotientedThomSpace"><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> <mrow><mo stretchy="false">(</mo><mi>SU</mi><mo>,</mo><mi>fr</mi><mo stretchy="false">)</mo></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>2</mn><mi>k</mi></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>MSU</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msub><mo stretchy="false">/</mo><msup><mi>S</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mtext>for any</mtext><mspace width="thickmathspace"></mspace><mn>2</mn><mi>k</mi><mo>≥</mo><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>2</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^{(SU,fr)}_\bullet \;=\; \pi_{\bullet + 2k} \big( MSU_{2k}/S^{2k} \big) \,, \;\;\;\;\; \text{for any} \; 2k \geq \bullet + 2 \,. </annotation></semantics></math></div> <p>(<a href="#ConnerFloyd66">Conner-Floyd 66, p. 103</a>)</p> <h3 id="relation_to__and_">Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MSU</mi></mrow><annotation encoding="application/x-tex">MSU</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MFr</mi></mrow><annotation encoding="application/x-tex">MFr</annotation></semantics></math></h3> <p>In every eighth degree, the <a class="existingWikiWord" href="/nlab/show/bordism+rings">bordism rings</a> for <a class="existingWikiWord" href="/nlab/show/MSU">MSU</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MSUFr</mi></mrow><annotation encoding="application/x-tex">MSUFr</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/MFr">MFr</a> sit in a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of the form (<a href="#ConnerFloyd66">Conner-Floyd 66, p. 104</a>):</p> <div class="maruku-equation" id="eq:ShortExactSequenceOfUFrBordismRings"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msubsup><mi>Ω</mi> <mrow><mn>8</mn><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>4</mn></mrow> <mi>SU</mi></msubsup><mover><mo>⟶</mo><mi>i</mi></mover><msubsup><mi>Ω</mi> <mrow><mn>8</mn><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>4</mn></mrow> <mrow><mi>SU</mi><mo>,</mo><mi>fr</mi></mrow></msubsup><mover><mo>⟶</mo><mo>∂</mo></mover><msubsup><mi>Ω</mi> <mrow><mn>8</mn><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>3</mn></mrow> <mi>fr</mi></msubsup><mo>→</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> 0 \to \Omega^{SU}_{8\bullet+4} \overset{i}{ \longrightarrow } \Omega^{SU,fr}_{8\bullet+4} \overset{\partial}{ \longrightarrow } \Omega^{fr}_{8\bullet + 3} \to 0 \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is the evident inclusion, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo></mrow><annotation encoding="application/x-tex">\partial</annotation></semantics></math> is restriction to the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a>.</p> <p>In particular, this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo></mrow><annotation encoding="application/x-tex">\partial</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/surjective+function">surjective</a>, hence that every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fr</mi></mrow><annotation encoding="application/x-tex">Fr</annotation></semantics></math>-manifold of dimension <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>3</mn></mrow><annotation encoding="application/x-tex">8k + 3</annotation></semantics></math> is the boundary of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>fr</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U,fr)</annotation></semantics></math>-manifold.</p> <h3 id="relation_to_todd_classes_and_the_invariant">Relation to Todd classes and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>ℝ</mi></msub></mrow><annotation encoding="application/x-tex">e_{\mathbb{R}}</annotation></semantics></math>-invariant</h3> <p>In refinement of how the complex <a class="existingWikiWord" href="/nlab/show/e-invariant+is+the+Todd+class+of+cobounding+%28U%2Cfr%29-manifolds">e-invariant is the Todd class of cobounding (U,fr)-manifolds</a> we have for <a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>-structure instead of <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a>-structure and in dimensions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>8</mn><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">8\bullet + 4</annotation></semantics></math>:</p> <p>Since on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>8</mn><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(8 \bullet + 4)</annotation></semantics></math>-dimensional <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>-manifolds the <a class="existingWikiWord" href="/nlab/show/Todd+class">Todd class</a> is divisible by 2 <a href="#ConnerFloyd66">Conner-Floyd 66, Prop. 16.4</a>, we have (<a href="#ConnerFloyd66">Conner-Floyd 66, p. 104</a>) the following <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of <a class="existingWikiWord" href="/nlab/show/MSUFr">MSUFr</a>-<a class="existingWikiWord" href="/nlab/show/bordism+rings">bordism rings</a>:</p> <div class="maruku-equation" id="eq:HalfToddClassesOnShortExactSequenceOfSUFrBordismRings"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>0</mn><mo>→</mo></mtd> <mtd><msubsup><mi>Ω</mi> <mrow><mn>8</mn><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>4</mn></mrow> <mi>SU</mi></msubsup></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><msubsup><mi>Ω</mi> <mrow><mn>8</mn><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>4</mn></mrow> <mrow><mi>SU</mi><mo>,</mo><mi>fr</mi></mrow></msubsup></mtd> <mtd><mover><mo>⟶</mo><mo>∂</mo></mover></mtd> <mtd><msubsup><mi>Ω</mi> <mrow><mn>8</mn><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>3</mn></mrow> <mi>fr</mi></msubsup></mtd> <mtd><mo>≃</mo></mtd> <mtd><msubsup><mi>π</mi> <mrow><mn>8</mn><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>3</mn></mrow> <mi>s</mi></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mpadded width="0"><mi>Td</mi></mpadded></mrow></msup></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mpadded width="0"><mi>Td</mi></mpadded></mrow></msup></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mrow></mrow></msup></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mrow><msub><mi>e</mi> <mi>ℝ</mi></msub></mrow></msup></mtd></mtr> <mtr><mtd><mn>0</mn><mo>→</mo></mtd> <mtd><mi>ℤ</mi></mtd> <mtd><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover></mtd> <mtd><mi>ℚ</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover></mtd> <mtd><mi>ℚ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>ℚ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ 0 \to & \Omega^{SU}_{8\bullet+4} & \overset{i}{\longrightarrow} & \Omega^{SU,fr}_{8\bullet+4} & \overset{\partial}{ \longrightarrow } & \Omega^{fr}_{8\bullet + 3} & \simeq & \pi^s_{8\bullet+3} \\ & \big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}} && \big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}} && \big\downarrow{}^{} && \big\downarrow{}^{e_{\mathbb{R}}} \\ 0 \to & \mathbb{Z} &\overset{\;\;\;\;\;}{\hookrightarrow}& \mathbb{Q} &\overset{\;\;\;\;}{\longrightarrow}& \mathbb{Q}/\mathbb{Z} &=& \mathbb{Q}/\mathbb{Z} } \,. </annotation></semantics></math></div> <p>This produces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>ℝ</mi></msub></mrow><annotation encoding="application/x-tex">e_{\mathbb{R}}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/Adams+e-invariant">Adams e-invariant</a> with respect to <a class="existingWikiWord" href="/nlab/show/KO">KO</a>-theory instead of <a class="existingWikiWord" href="/nlab/show/KU">KU</a> (<a href="e-invariant#Adams66">Adams 66, p. 39</a>), which, 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>3</mn></mrow><annotation encoding="application/x-tex">8k + 3</annotation></semantics></math>, is indeed half of the e-invariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>ℂ</mi></msub></mrow><annotation encoding="application/x-tex">e_{\mathbb{C}}</annotation></semantics></math> for <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> (by <a href="e-invariant#Adams66">Adams 66, Prop. 7.14</a>).</p> <p>In fact, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k = 0</annotation></semantics></math> we have:</p> <div class="num_prop" id="AdamseRInvariantDetectsThirdStableHomotopyGroupOfSpheres"> <h6 id="proposition">Proposition</h6> <p><strong>(<a href="e-invariant#Adams66">Adams 66, Example 7.17 and p. 46</a>)</strong></p> <p>In degree 3, the <a class="existingWikiWord" href="/nlab/show/KO">KO</a>-theoretic <a class="existingWikiWord" href="/nlab/show/e-invariant">e-invariant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>ℝ</mi></msub></mrow><annotation encoding="application/x-tex">e_{\mathbb{R}}</annotation></semantics></math> takes the value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>24</mn></mfrac></mstyle><mo>]</mo></mrow><mo>∈</mo><mi>ℚ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\left[\tfrac{1}{24}\right] \in \mathbb{Q}/\mathbb{Z}</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/quaternionic+Hopf+fibration">quaternionic Hopf fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><mover><mo>⟶</mo><mrow><msub><mi>h</mi> <mi>ℍ</mi></msub></mrow></mover><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4</annotation></semantics></math> and hence reflects the full <a class="existingWikiWord" href="/nlab/show/third+stable+homotopy+group+of+spheres">third stable homotopy group of spheres</a>:</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>π</mi> <mn>3</mn> <mi>s</mi></msubsup></mtd> <mtd><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⟶</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><mo>≃</mo><mrow><msub><mi>e</mi> <mi>ℝ</mi></msub></mrow></munderover></mtd> <mtd><mi>ℤ</mi><mo stretchy="false">/</mo><mn>24</mn></mtd> <mtd><mo>⊂</mo></mtd> <mtd><mi>ℚ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msub><mi>h</mi> <mi>ℍ</mi></msub><mo stretchy="false">]</mo></mtd> <mtd></mtd> <mtd><mo>↦</mo></mtd> <mtd></mtd> <mtd><mrow><mo>[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>24</mn></mfrac></mstyle><mo>]</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \pi^s_3 & \underoverset{ \simeq }{ e_{\mathbb{R}} }{ \;\;\longrightarrow\;\; } & \mathbb{Z}/24 & \subset & \mathbb{Q}/\mathbb{Z} \\ [h_{\mathbb{H}}] &&\mapsto&& \left[\tfrac{1}{24}\right] } </annotation></semantics></math></div> <p>while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>ℂ</mi></msub></mrow><annotation encoding="application/x-tex">e_{\mathbb{C}}</annotation></semantics></math> sees only “half” of it (by <a href="e-invariant#Adams66">Adams 66, Prop. 7.14</a>).</p> </div> <h2 id="related_concepts">Related concepts</h2> <div> <p><strong>flavors of <a class="existingWikiWord" href="/nlab/show/bordism+homology+theories">bordism homology theories</a>/<a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theories">cobordism cohomology theories</a>, their <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">representing</a> <a class="existingWikiWord" href="/nlab/show/Thom+spectra">Thom spectra</a> and <a class="existingWikiWord" href="/nlab/show/cobordism+rings">cobordism rings</a></strong>:</p> <p><a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/M%28B%2Cf%29">M(B,f)</a> (<a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/MFr">MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MO">MO</a>, <a class="existingWikiWord" href="/nlab/show/MSO">MSO</a>, <a class="existingWikiWord" href="/nlab/show/MSpin">MSpin</a>, <a class="existingWikiWord" href="/nlab/show/MString">MString</a>, …</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MU">MU</a>, <a class="existingWikiWord" href="/nlab/show/MSU">MSU</a>, …</p> <p><a class="existingWikiWord" href="/nlab/show/Ravenel%27s+spectrum">MΩΩSU(n)</a></p> <p><a class="existingWikiWord" href="/nlab/show/MP-theory">MP</a>, <a class="existingWikiWord" href="/nlab/show/MR-theory">MR</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSpin%5Ec">MSpin<sup><i>c</i></sup></a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSp">MSp</a></p> </li> </ul> <p>relative bordism theories:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/MOFr">MOFr</a>, <a class="existingWikiWord" href="/nlab/show/MUFr">MUFr</a>, <a class="existingWikiWord" href="/nlab/show/MSUFr">MSUFr</a></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bordism+homology+theory">equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MFr">equivariant MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MO">equivariant MO</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MU">equivariant MU</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+bordism+homology+theory">global equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mO">global equivariant mO</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mU">global equivariant mU</a></p> </li> </ul> <p>algebraic:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+cobordism">algebraic cobordism</a></li> </ul> </div> <h2 id="references">References</h2> <p>The concept of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>SU</mi><mo>,</mo><mi>fr</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(SU,fr)</annotation></semantics></math>-bordism theory and its relation to the <a class="existingWikiWord" href="/nlab/show/e-invariant">e-invariant</a> originates with:</p> <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 16 from p. 103 onwards, in: <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> </body></html> </div> <div class="revisedby"> <p> Created on December 18, 2020 at 10:54:42. See the <a href="/nlab/history/MSUFr" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/MSUFr" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/12062/#Item_1">Discuss</a> <a href="/nlab/show/MSUFr/cite" style="color: black">Cite</a> <a href="/nlab/print/MSUFr" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/MSUFr" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>