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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='#RepresentingSpectrum'>Representing spectrum</a></li> <li><a href='#BoundaryMorphism'>Boundary morphism to <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='#RelationToMUAndFr'>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> 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_einvariant'>Relation to Todd classes and the e-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/MU">MU</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>U</mi></mrow><annotation encoding="application/x-tex">U</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, a <em><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</em> (<a href="#ConnerFloyd66">Conner-Floyd 66, Section 16</a>, <a href="#ConnerSmith69">Conner-Smith 69, Sections 6, 13</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/unitary+group">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 mathvariant="normal">U</mi><mo>,</mo><mi>fr</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{\mathrm{U},fr}_\bullet</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="RepresentingSpectrum">Representing spectrum</h3> <p>In generalization to how the complex <a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a> <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>U</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^U_{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/MU">MU</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 mathvariant="normal">U</mi><mo>,</mo><mi>fr</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{\mathrm{U},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>MU</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">MU_{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 mathvariant="normal">U</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mi>E</mi><mi mathvariant="normal">U</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>M</mi><msub><mi mathvariant="normal">U</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_{\mathrm{U}(k)} E \mathrm{U}(k) ) = M \mathrm{U}_{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 mathvariant="normal">U</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>MU</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^{(\mathrm{U},fr)}_\bullet \;=\; \pi_{\bullet + 2k} \big( MU_{2k}/S^{2k} \big) \,, \;\;\;\;\; \text{for any} \; 2k \geq \bullet + 2 \,. </annotation></semantics></math></div> <p>(<a href="#ConnerFloyd66">Conner-Floyd 66, p. 97</a>)</p> <p>Hence the representing spectrum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mi mathvariant="normal">U</mi><mo>,</mo><mi>fr</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M(\mathrm{U},fr)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a> of the <a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a> unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>1</mn> <mrow><mi>M</mi><mi mathvariant="normal">U</mi></mrow></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝕊</mi><mo>⟶</mo><mi>M</mi><mi mathvariant="normal">U</mi></mrow><annotation encoding="application/x-tex">1^{M\mathrm{U}} \;\colon\; \mathbb{S} \longrightarrow M \mathrm{U}</annotation></semantics></math> out of the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a> (<a href="#ConnerSmith69">Conner-Smith 69, p. 156 (41 of 106)</a>, <a href="#Smith71">Smith 71</a>) which deserves to be denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mi mathvariant="normal">U</mi><mo>,</mo><mi>fr</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>M</mi><mi mathvariant="normal">U</mi><mo stretchy="false">/</mo><mi>𝕊</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> M(\mathrm{U},fr) \;\coloneqq\; M \mathrm{U} / \mathbb{S} \,, </annotation></semantics></math></div> <p>but which in notation common around the <a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a> would be “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mover><mrow><mi>M</mi><mi mathvariant="normal">U</mi></mrow><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\Sigma \overline {M \mathrm{U}}</annotation></semantics></math>” (as in <a href="Adams+spectral+sequence#Adams74">Adams 74, theorem 15.1 page 319</a>) or just “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mi>M</mi><mi mathvariant="normal">U</mi></mrow><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{ M \mathrm{U} }</annotation></semantics></math>” (e.g. <a href="Adams+spectral+sequence#Hopkins99">Hopkins 99, Cor. 5.3</a>):</p> <div class="maruku-equation" id="eq:AsUnitCofiber"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝕊</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mn>1</mn> <mrow><mi>M</mi><mi mathvariant="normal">U</mi></mrow></msup></mrow></mover></mtd> <mtd><mi>M</mi><mi mathvariant="normal">U</mi></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msup></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>M</mi><mi mathvariant="normal">U</mi><mo stretchy="false">/</mo><mi>𝕊</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{S} & \overset{ 1^{M\mathrm{U}} }{ \longrightarrow } & M \mathrm{U} \\ \big\downarrow & {}^{{}_{(po)}} & \big\downarrow \\ \ast &\longrightarrow& M \mathrm{U}/ \mathbb{S} } </annotation></semantics></math></div> <p>So in terms of <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups">stable homotopy groups</a> of this spectrum we have the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">U</mi><mo>,</mo><mi>fr</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathrm{U},fr)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> <div class="maruku-equation" id="eq:UFrCobordismRing"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>•</mo> <mrow><mi mathvariant="normal">U</mi><mo>,</mo><mi>fr</mi></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>M</mi><mi mathvariant="normal">U</mi><mo stretchy="false">/</mo><mi>𝕊</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^{\mathrm{U},fr}_{\bullet} \;\coloneqq\; \pi_{\bullet} \big( M\mathrm{U}/\mathbb{S} \big) </annotation></semantics></math></div> <h3 id="BoundaryMorphism">Boundary morphism to <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>The realization <a class="maruku-eqref" href="#eq:AsUnitCofiber">(2)</a> makes it manifest (this is left implicit in <a href="#ConnerFloyd66">Conner-Floyd 66, p. 99</a>) that there is a <a class="existingWikiWord" href="/nlab/show/cohomology+operation">cohomology operation</a> to <a class="existingWikiWord" href="/nlab/show/MFr">MFr</a> of the form</p> <div class="maruku-equation" id="eq:BoundaryCohomologyOperation"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>M</mi><mo stretchy="false">(</mo><mi mathvariant="normal">U</mi><mo>,</mo><mi>fr</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo></mtd> <mtd><mi>M</mi><mi mathvariant="normal">U</mi><mo stretchy="false">/</mo><mi>𝕊</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∂</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover></mtd> <mtd><mi>Σ</mi><mi>𝕊</mi></mtd> <mtd><mo>=</mo><mspace width="thickmathspace"></mspace><mi>Σ</mi><mi>Mfr</mi></mtd></mtr> <mtr><mtd><msub><mi>π</mi> <mrow><mn>2</mn><mi>d</mi><mo>+</mo><mn>2</mn></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>M</mi><mo stretchy="false">(</mo><mi mathvariant="normal">U</mi><mo>,</mo><mi>fr</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><msub><mi>π</mi> <mrow><mn>2</mn><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Mfr</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ M(\mathrm{U},fr) \;= & M \mathrm{U}/\mathbb{S} & \overset{ \;\;\; \partial \;\;\; }{\longrightarrow} & \Sigma \mathbb{S} & =\; \Sigma Mfr \\ \pi_{2d+2}\big( M(\mathrm{U},fr) \big) && \longrightarrow && \pi_{2d+1}\big( Mfr \big) } \,. </annotation></semantics></math></div> <p>Namely, <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 the second next step in the long <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a>-sequence starting with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>1</mn> <mrow><mi>M</mi><mi mathvariant="normal">U</mi></mrow></msup></mrow><annotation encoding="application/x-tex">1^{M \mathrm{U}}</annotation></semantics></math>. In terms of the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>:</p> <div class="maruku-equation" id="eq:BoundaryOperationViaPastingLaw"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝕊</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mn>1</mn> <mrow><mi>M</mi><mi mathvariant="normal">U</mi></mrow></msup></mrow></mover></mtd> <mtd><mi>M</mi><mi mathvariant="normal">U</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msup></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msup></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>M</mi><mi mathvariant="normal">U</mi><mo stretchy="false">/</mo><mi>𝕊</mi></mtd> <mtd><munder><mo>⟶</mo><mo>∂</mo></munder></mtd> <mtd><mi>Σ</mi><mi>𝕊</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{S} & \overset{ 1^{M\mathrm{U}} }{ \longrightarrow } & M \mathrm{U} & \longrightarrow & \ast \\ \big\downarrow & {}^{{}_{(po)}} & \big\downarrow & {}^{{}_{(po)}} & \big\downarrow \\ \ast & \longrightarrow & M \mathrm{U}/ \mathbb{S} & \underset{ \partial }{ \longrightarrow } & \Sigma \mathbb{S} } </annotation></semantics></math></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The implicit idea in <a href="#ConnerFloyd66">Conner-Floyd 66, p. 99</a> must be to see <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> <a class="maruku-eqref" href="#eq:BoundaryCohomologyOperation">(4)</a> in terms of forming actual <a class="existingWikiWord" href="/nlab/show/boundaries">boundaries</a> of representative <a class="existingWikiWord" href="/nlab/show/manifolds+with+boundaries">manifolds with boundaries</a> under a version of the <a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a>. This is certainly plausible, but not proven either, as they “forego the tedious details” on <a href="#ConnerFloyd66">p. 97</a>. NB: for the purpose on p. 99 they might just <em>define</em> <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> in this geometric way, but then the claim wrapping around p. 100-101 needs proof. This claim however is immediate from the abstract homotopy theory, namely it is just the continuation yet one step further along the long cofiber sequence – this is Prop. <a class="maruku-ref" href="#AShortExactSequenceOfUFrBordismRings"></a> below.</p> </div> <p><br /></p> <h3 id="RelationToMUAndFr">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> 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> <div class="num_prop" id="AShortExactSequenceOfUFrBordismRings"> <h6 id="proposition">Proposition</h6> <p>In <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> degree, the underlying <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> of the <a class="existingWikiWord" href="/nlab/show/bordism+rings">bordism rings</a> for <a class="existingWikiWord" href="/nlab/show/MU">MU</a>, <a class="existingWikiWord" href="/nlab/show/MFr">MFr</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MUFr</mi></mrow><annotation encoding="application/x-tex">MUFr</annotation></semantics></math> <a class="maruku-eqref" href="#eq:UFrCobordismRing">(3)</a> sit in <a class="existingWikiWord" href="/nlab/show/short+exact+sequences">short exact sequences</a> of this form:</p> <div class="maruku-equation" id="eq:ShortExactSequenceOfUFrBordismRings"><span class="maruku-eq-number">(6)</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>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow> <mi mathvariant="normal">U</mi></msubsup><mover><mo>⟶</mo><mi>i</mi></mover><msubsup><mi>Ω</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow> <mrow><mi mathvariant="normal">U</mi><mo>,</mo><mi>fr</mi></mrow></msubsup><mover><mo>⟶</mo><mo>∂</mo></mover><msubsup><mi>Ω</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow> <mi>fr</mi></msubsup><mo>→</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo><mphantom><mi>AAAA</mi></mphantom><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> 0 \to \Omega^{\mathrm{U}}_{2n + 2} \overset{i}{\longrightarrow} \Omega^{\mathrm{U},fr}_{2n + 2} \overset{\partial}{ \longrightarrow } \Omega^{fr}_{2n + 1} \to 0 \,, \phantom{AAAA} n \in \mathbb{N} \,, </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 the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> homomorphism from <a href="#BoundaryMorphism">above</a>.</p> </div> <p>This is stated without comment in <a href="#ConnerFloyd66">Conner-Floyd 66, p. 99</a>. The beginning of an argument appears inside the proof of <a href="#ConnerFloyd66">CF66, Thm. 16.2 (p. 100)</a>, attributed there to <a class="existingWikiWord" href="/nlab/show/Peter+Landweber">Peter Landweber</a> (see Remark <a class="maruku-ref" href="#BoundaryOperationFromFrBordToUFrBordIsSurjective"></a> below). The idea for how to complete the argument is a little more explicit in <a href="#Stong68">Stong 68, p. 102</a>.</p> <p>The following is the complete and quick proof using the formulation <a class="maruku-eqref" href="#eq:BoundaryOperationViaPastingLaw">(5)</a> via abstract homotopy <a href="#BoundaryMorphism">above</a>:</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>We have the <a class="existingWikiWord" href="/nlab/show/long+exact+sequence+of+homotopy+groups">long exact sequence of homotopy groups</a> (<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+generalized+cohomology">long exact sequence in generalized cohomology</a> on <a class="existingWikiWord" href="/nlab/show/spheres">spheres</a>) obtained from the <a class="existingWikiWord" href="/nlab/show/cofiber+sequence">cofiber sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi><mover><mo>⟶</mo><mrow><msup><mn>1</mn> <mrow><mi>M</mi><mi mathvariant="normal">U</mi></mrow></msup></mrow></mover><mi>M</mi><mi mathvariant="normal">U</mi><mo>→</mo><mi>M</mi><mi mathvariant="normal">U</mi><mo stretchy="false">/</mo><mi>𝕊</mi><mover><mo>→</mo><mo>∂</mo></mover><mi>Σ</mi><mi>𝕊</mi></mrow><annotation encoding="application/x-tex">\mathbb{S} \overset{1^{M\mathrm{U}}}{\longrightarrow} M \mathrm{U} \to M \mathrm{U}/\mathbb{S} \overset{\partial}{\to} \Sigma \mathbb{S}</annotation></semantics></math> <a class="maruku-eqref" href="#eq:BoundaryOperationViaPastingLaw">(5)</a>, the relevant part of which looks as follows:</p> <div class="maruku-equation" id="eq:SToMULongExactSequenceOfHomotopyGroups"><span class="maruku-eq-number">(7)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mover><mover><mrow><msub><mi>π</mi> <mrow><mn>2</mn><mi>d</mi><mo>+</mo><mn>2</mn></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝕊</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>⏞</mo></mover><mpadded width="0" lspace="-50%width"><mstyle mathcolor="darkblue"><mi>pure</mi><mspace width="thickmathspace"></mspace><mi>torsion</mi></mstyle></mpadded></mover></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mn>1</mn> <mrow><mi>M</mi><mi mathvariant="normal">U</mi></mrow></msup></mrow></mover></mtd> <mtd><mover><mover><mrow><msub><mi>π</mi> <mrow><mn>2</mn><mi>d</mi><mo>+</mo><mn>2</mn></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>M</mi><mi mathvariant="normal">U</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>⏞</mo></mover><mpadded width="0" lspace="-50%width"><mstyle mathcolor="darkblue"><mi>free</mi><mspace width="thickmathspace"></mspace><mi>abelian</mi></mstyle></mpadded></mover></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msub><mi>π</mi> <mrow><mn>2</mn><mi>d</mi><mo>+</mo><mn>2</mn></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>M</mi><mi mathvariant="normal">U</mi><mo stretchy="false">/</mo><mi>𝕊</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mover><mo>⟶</mo><mo>∂</mo></mover></mtd> <mtd><msub><mi>π</mi> <mrow><mn>2</mn><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>𝕊</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mover><mover><mrow><msub><mi>π</mi> <mrow><mn>2</mn><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>M</mi><mi mathvariant="normal">U</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>⏞</mo></mover><mstyle mathcolor="darkblue"><mi>trivial</mi></mstyle></mover></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msubsup><mi>Ω</mi> <mrow><mn>2</mn><mi>d</mi><mo>+</mo><mn>2</mn></mrow> <mi>fr</mi></msubsup></mtd> <mtd><munder><mo>⟶</mo><mstyle mathcolor="green"><mn>0</mn></mstyle></munder></mtd> <mtd><msubsup><mi>Ω</mi> <mrow><mn>2</mn><mi>d</mi><mo>+</mo><mn>2</mn></mrow> <mi mathvariant="normal">U</mi></msubsup></mtd> <mtd><munder><mo>⟶</mo><mi>i</mi></munder></mtd> <mtd><msubsup><mi>Ω</mi> <mrow><mn>2</mn><mi>d</mi><mo>+</mo><mn>2</mn></mrow> <mrow><mo stretchy="false">(</mo><mi mathvariant="normal">U</mi><mo>,</mo><mi>fr</mi><mo stretchy="false">)</mo></mrow></msubsup></mtd> <mtd><munder><mo>⟶</mo><mo>∂</mo></munder></mtd> <mtd><msubsup><mi>Ω</mi> <mrow><mn>2</mn><mi>d</mi><mo>+</mo><mn>1</mn></mrow> <mi>fr</mi></msubsup></mtd> <mtd><munder><mo>⟶</mo><mstyle mathcolor="green"><mn>0</mn></mstyle></munder></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \overset{ \mathclap{ \color{darkblue} pure \; torsion } }{ \overbrace{ \pi_{2d+2} \big( \mathbb{S} \big) } } & \overset{ 1^{M\mathrm{U}} }{ \longrightarrow } & \overset{ \mathclap{ \color{darkblue} free \; abelian } }{ \overbrace{ \pi_{2d+2} \big( M\mathrm{U} \big) } } & \overset{ }{\longrightarrow} & \pi_{2d+2} \big( M\mathrm{U}/\mathbb{S} \big) & \overset{ \partial }{\longrightarrow} & \pi_{2d+1}\big(\mathbb{S}\big) &\longrightarrow& \overset{ \color{darkblue} trivial }{ \overbrace{ \pi_{2d+1}\big(M\mathrm{U}\big) } } \\ \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} \\ \Omega^{fr}_{2d+2} & \underset{ \color{green} 0 }{ \longrightarrow } & \Omega^{\mathrm{U}}_{2d+2} & \underset{ i }{\longrightarrow} & \Omega^{(\mathrm{U},fr)}_{2d+2} & \underset{ \partial }{\longrightarrow} & \Omega^{fr}_{2d + 1} & \underset{ \color{green} 0 }{\longrightarrow} & 0 } </annotation></semantics></math></div> <p>Observing now that the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups">stable homotopy groups</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi mathvariant="normal">U</mi></mrow><annotation encoding="application/x-tex">M\mathrm{U}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/free+abelian+groups">free abelian groups</a> concentrated in <a class="existingWikiWord" href="/nlab/show/even+number">even</a> degrees (by <a href="MU#RelationToCobordismRing">this theorem</a> at <em><a class="existingWikiWord" href="/nlab/show/MU">MU</a></em>) it follows that:</p> <ol> <li> <p>the rightmost morphism shown in <a class="maruku-eqref" href="#eq:SToMULongExactSequenceOfHomotopyGroups">(7)</a> is the <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero morphism</a> since its <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> is <a class="existingWikiWord" href="/nlab/show/zero+object">zero</a>;</p> </li> <li> <p>the leftmost morphism shown in <a class="maruku-eqref" href="#eq:SToMULongExactSequenceOfHomotopyGroups">(7)</a> is the <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero morphism</a>, since the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a> are all pure <a class="existingWikiWord" href="/nlab/show/torsion+groups">torsion groups</a> in <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> degrees (by the <a class="existingWikiWord" href="/nlab/show/Serre+finiteness+theorem">Serre finiteness theorem</a>), and the only morphism from a torsion group to a <a class="existingWikiWord" href="/nlab/show/free+abelian+group">free abelian group</a> is the zero morphism.</p> </li> </ol> </div> <div class="num_remark" id="BoundaryOperationFromFrBordToUFrBordIsSurjective"> <h6 id="remark_2">Remark</h6> <p>Here is an explicit construction of a <a class="existingWikiWord" href="/nlab/show/lift">lift</a> through the boundary map in <a class="maruku-eqref" href="#eq:ShortExactSequenceOfUFrBordismRings">(6)</a>, depending on a choice of trivialization in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mrow><mn>2</mn><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>M</mi><mi mathvariant="normal">U</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\pi_{2d-1}\big( M\mathrm{U}\big)</annotation></semantics></math>:</p> <p>(The following is essentially a streamlined account of the construction in the first half of the proof of <a href="#ConnerFloyd66">Conner-Floyd 66, Thm. 16.2</a>; for more and a more abstract perspective see at <em><a class="existingWikiWord" href="/nlab/show/d-invariant">d-invariant</a></em> the section <em><a href="d-invariant#TrivializationsOfThedInvariant">Trivializations of the d-invariant</a></em>):</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">[</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><msup><mi>S</mi> <mrow><mn>2</mn><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></msup><mover><mo>⟶</mo><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mstyle mathcolor="green"><mi>c</mi></mstyle></mrow></mover><msup><mi>Σ</mi> <mn>∞</mn></msup><msup><mi>S</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msubsup><mi>π</mi> <mrow><mn>2</mn><mi>d</mi><mo>−</mo><mn>1</mn></mrow> <mi>s</mi></msubsup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Ω</mi> <mrow><mn>2</mn><mi>d</mi><mo>−</mo><mn>1</mn></mrow> <mi>fr</mi></msubsup></mrow><annotation encoding="application/x-tex"> \big[ \Sigma^\infty S^{2(n+d)-1} \overset{ \Sigma^{\infty} {\color{green} c } }{\longrightarrow} \Sigma^\infty S^{2n} \big] \;\in\; \pi^s_{2d-1} \;\simeq\; \Omega^{fr}_{2d-1} </annotation></semantics></math></div> <p>be a given class in <a class="existingWikiWord" href="/nlab/show/stable+Cohomotopy">stable Cohomotopy</a>, hence in the <a class="existingWikiWord" href="/nlab/show/MFr">MFr</a>-<a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a>, under the <a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+isomorphism">PT isomorphism</a>. (We write this as the <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> of a class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathcolor="green"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">{\color{green} c}</annotation></semantics></math> in unstable <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a> just for emphasis that we can.)</p> <p>Consider then following <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+diagram">homotopy</a> <a class="existingWikiWord" href="/nlab/show/pasting+diagram">pasting diagram</a>:</p> <div style="margin: -20px 0px 20px 10px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/LiftingFromUBordismToUFrBordism.jpg" width="600px" /> <figcaption style="text-align: center">From <a href="https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory">SS21</a></figcaption> </figure> </div> <p>Here all squares are <a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a>-squares, they arise as follows (beware that we say “square” for any <em>single</em> cell and “rectangle” for the pasting composite of any adjacent <em>pair</em> of cells):</p> <ul> <li> <p>The top square witnesses the <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">C_c</annotation></semantics></math>, defined thereby;</p> </li> <li> <p>the left rectangle is a homotopy pushout by definition of <a class="existingWikiWord" href="/nlab/show/suspension">suspension</a>,</p> </li> <li> <p>hence the bottom left square is so by the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>.</p> </li> <li> <p>Again via the first point, a dashed morphism exists as shown, witnessing the fact that the pullback of the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><msup><mn>1</mn> <mrow><mi>M</mi><mi mathvariant="normal">U</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma^{2n}(1^{M\mathrm{U}})</annotation></semantics></math> to an odd-dimensional square vanishes (as does every <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>-class);</p> </li> <li> <p>with this, the bottom left rectangle exists, and it is a homotopy pushout by definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi mathvariant="normal">U</mi><mo stretchy="false">/</mo><mi>𝕊</mi></mrow><annotation encoding="application/x-tex">M \mathrm{U}/\mathbb{S}</annotation></semantics></math> <a class="maruku-eqref" href="#eq:BoundaryOperationViaPastingLaw">(5)</a>;</p> </li> <li> <p>hence the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathcolor="magenta"><msup><mi>M</mi> <mrow><mn>2</mn><mi>d</mi></mrow></msup></mstyle></mrow><annotation encoding="application/x-tex">\color{magenta} M^{2 d}</annotation></semantics></math> exists, and the resulting bottom middle square is a homotopy pushout, again by the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>.</p> </li> <li> <p>Now the bottom right rectangle is defined to be a homotopy pushout, and thus looks as shown by the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>;</p> </li> <li> <p>therefore the bottom right square exists, and it is a homotopy pushout by the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>.</p> </li> </ul> <p>But now the commuting three morphism in the very bottom and right part of the diagram shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathcolor="magenta"><msup><mi>M</mi> <mrow><mn>2</mn><mi>d</mi></mrow></msup></mstyle><mo>∈</mo><msub><mi>π</mi> <mrow><mn>2</mn><mi>d</mi></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>M</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>fr</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">{\color{magenta}M^{2d}} \in \pi_{2d}\big( M(U,fr) \big)</annotation></semantics></math> is a lift of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathcolor="green"><mi>c</mi></mstyle><mo>∈</mo><msub><mi>π</mi> <mrow><mn>2</mn><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>M</mi><mi>Fr</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">{\color{green} c} \in \pi_{2d-1}(M Fr)</annotation></semantics></math> through <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>.</p> </div> <h3 id="relation_to_todd_classes_and_the_einvariant">Relation to Todd classes and the e-invariant</h3> <div class="num_prop" id="EInvariantIsToddClassOnCoboundingUFrManifold"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/e-invariant+is+Todd+class+of+cobounding+%28U%2Cfr%29-manifold">e-invariant is Todd class of cobounding (U,fr)-manifold</a>)</strong></p> <p>Evaluation of the <a class="existingWikiWord" href="/nlab/show/Todd+class">Todd class</a> on <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>-manifolds yields <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a> which are <a class="existingWikiWord" href="/nlab/show/integers">integers</a> on actual <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-manifolds. It follows with the <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> <a class="maruku-eqref" href="#eq:ShortExactSequenceOfUFrBordismRings">(6)</a> that assigning to <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 the Todd class of any of their cobounding <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>-manifolds yields a well-defined element in <a class="existingWikiWord" href="/nlab/show/Q%2FZ">Q/Z</a>.</p> <p>Under the <a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+isomorphism">Pontrjagin-Thom isomorphism</a> between the <a class="existingWikiWord" href="/nlab/show/framed+bordism+ring">framed bordism ring</a> and the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+group+of+spheres">stable homotopy group of spheres</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>π</mi> <mo>•</mo> <mi>s</mi></msubsup></mrow><annotation encoding="application/x-tex">\pi^s_\bullet</annotation></semantics></math>, this assignment coincides with the <a class="existingWikiWord" href="/nlab/show/Adams+e-invariant">Adams e-invariant</a> in <a href="Adams+e-invariant#TheEInvariantAsAnElementOfQModZ">its Q/Z-incarnation</a>:</p> <div class="maruku-equation" id="eq:ToddClassesOnShortExactSequenceOfUFrBordismRings"><span class="maruku-eq-number">(8)</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><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow> <mi mathvariant="normal">U</mi></msubsup></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><msubsup><mi>Ω</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow> <mrow><mi mathvariant="normal">U</mi><mo>,</mo><mi>fr</mi></mrow></msubsup></mtd> <mtd><mover><mo>⟶</mo><mo>∂</mo></mover></mtd> <mtd><msubsup><mi>Ω</mi> <mo>•</mo> <mi>fr</mi></msubsup></mtd> <mtd><mo>≃</mo></mtd> <mtd><msubsup><mi>π</mi> <mo>•</mo> <mi>s</mi></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mi>Td</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mi>Td</mi></mpadded></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> <mi>e</mi></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^{\mathrm{U}}_{\bullet+1} & \overset{i}{\longrightarrow} & \Omega^{\mathrm{U},fr}_{\bullet+1} & \overset{\partial}{ \longrightarrow } & \Omega^{fr}_\bullet & \simeq & \pi^s_\bullet \\ & \big\downarrow{}^{\mathrlap{Td}} && \big\downarrow{}^{\mathrlap{Td}} && \big\downarrow{}^{} && \big\downarrow{}^{e} \\ 0 \to & \mathbb{Z} &\overset{\;\;\;\;\;}{\hookrightarrow}& \mathbb{Q} &\overset{\;\;\;\;}{\longrightarrow}& \mathbb{Q}/\mathbb{Z} &=& \mathbb{Q}/\mathbb{Z} } \,, </annotation></semantics></math></div></div> <p>(<a href="#ConnerFloyd66">Conner-Floyd 66, Theorem 16.2</a>)</p> <p>The first step in the proof of <a class="maruku-eqref" href="#eq:ToddClassesOnShortExactSequenceOfUFrBordismRings">(8)</a> is the observation (<a href="#ConnerFloyd66">Conner-Floyd 66, p. 100-101</a>) that the representing map <a class="maruku-eqref" href="#eq:InTermsOfHomotopyGroupsOfQuotientedThomSpace">(1)</a> for 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mrow><mn>2</mn><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex">M^{2k}</annotation></semantics></math> cobounding a <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 represented by a map <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 given by the following <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+diagram">homotopy</a> <a class="existingWikiWord" href="/nlab/show/pasting+diagram">pasting diagram</a> (see also at <em><a class="existingWikiWord" href="/nlab/show/Hopf+invariant">Hopf invariant</a> – <a href="Hopf+invariant#InGeneralizedCohomology">In generalized cohomology</a></em>):</p> <div style="margin: -20px 0px 20px 20px"> <figure style="margin: 0 0 0 0"> <img src="/nlab/files/CoboundingUFrManifoldDiagrammatically.jpg" width="470px" alt="homotopy pasting diagram exhibiting cobounding UFr-manifolds" /> <figcaption style="text-align: center">from <a href="https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory">SS21</a></figcaption> </figure> </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>U</mi><mo>,</mo><mi>fr</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U,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"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Conner">Pierre Conner</a>, <a class="existingWikiWord" href="/nlab/show/Edwin+Floyd">Edwin Floyd</a>, Section 16 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>)</p> </li> <li id="ConnerSmith69"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Conner">Pierre Conner</a>, <a class="existingWikiWord" href="/nlab/show/Larry+Smith">Larry Smith</a>, Section 6 of: <em>On the complex bordism of finite complexes</em>, Publications Mathématiques de l’IHÉS, Tome 37 (1969) , pp. 117-221 (<a href="http://www.numdam.org/item/?id=PMIHES_1969__37__117_0">numdam:PMIHES_1969__37__117_0</a>)</p> </li> </ul> <p>Analogous discussion for <a class="existingWikiWord" href="/nlab/show/MO">MO</a>-bordism with <a class="existingWikiWord" href="/nlab/show/MSO">MSO</a>-boundaries:</p> <ul> <li>G. E. Mitchell, <em>Bordism of Manifolds with Oriented Boundaries</em>, Proceedings of the American Mathematical Society Vol. 47, No. 1 (Jan., 1975), pp. 208-214 (<a href="https://doi.org/10.2307/2040234">doi:10.2307/2040234</a>)</li> </ul> <p>Analogous discussion for <a class="existingWikiWord" href="/nlab/show/MOFr">MOFr</a> is in</p> <ul> <li id="Stong68"><a class="existingWikiWord" href="/nlab/show/Robert+Stong">Robert Stong</a>, p. 102 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>)</li> </ul> <p>See also</p> <ul> <li id="Smith71"><a class="existingWikiWord" href="/nlab/show/Larry+Smith">Larry Smith</a>, <em>On characteristic numbers of almost complex manifolds with framed boundaries</em>, Topology Volume 10, Issue 3, August 1971, Pages 237-256 (<a href="https://doi.org/10.1016/0040-9383(71)90008-5">doi:10.1016/0040-9383(71)90008-5</a>)</li> </ul> <p>Generalization to <a class="existingWikiWord" href="/nlab/show/manifolds+with+corners">manifolds with corners</a> and relation to the <a class="existingWikiWord" href="/nlab/show/f-invariant">f-invariant</a>:</p> <ul> <li>Gerd Laures, <em>On cobordism of manifolds with corners</em>, Trans. Amer. Math. Soc. 352 (2000) (<a href="https://doi.org/10.1090/S0002-9947-00-02676-3">doi:10.1090/S0002-9947-00-02676-3</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 18, 2021 at 15:32:33. 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