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href='/nlab/show/diff/higher+category+theory'>higher category theory</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/manifold'>manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/differentiable+manifold'>differentiable manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tangential+structure'>tangential structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cobordism'>cobordism</a>, <a class='existingWikiWord' href='/nlab/show/diff/cobordism'>cobordism class</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/bordism+ring'>cobordism ring</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/submanifold'>submanifold</a>,</p> <p><a class='existingWikiWord' href='/nlab/show/diff/normal+bundle'>normal bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Pontryagin%27s+theorem'>Pontrjagin's theorem</a> (<a class='existingWikiWord' href='/nlab/show/diff/equivariant+Pontrjagin+theorem'>equivariant</a>, <a class='existingWikiWord' href='/nlab/show/diff/twisted+Pontrjagin+theorem'>twisted</a>):</p> <p><math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mo>↔</mo></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{\leftrightarrow}</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/cohomotopy'>Cohomotopy</a></p> <p><math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>↔</mo></mrow><annotation encoding='application/x-tex'>\leftrightarrow</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/cobordism'>cobordism classes</a> of <a class='existingWikiWord' href='/nlab/show/diff/normal+framing'>normally framed submanifolds</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Thom%27s+theorem'>Thom's theorem</a>:</p> <p><math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mo>↔</mo></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{\leftrightarrow}</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+class'>homotopy classes</a> of maps to <a class='existingWikiWord' href='/nlab/show/diff/Thom+space'>Thom space</a> <a class='existingWikiWord' href='/nlab/show/diff/MO'>MO</a></p> <p><math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>↔</mo></mrow><annotation encoding='application/x-tex'>\leftrightarrow</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/cobordism'>cobordism classes</a> of <a class='existingWikiWord' href='/nlab/show/diff/oriented+submanifold'>normally oriented submanifolds</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/vector+bundle'>vector bundle</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Thom+space'>Thom space</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Thom+isomorphism'>Thom isomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Thom+spectrum'>Thom spectrum</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Pontrjagin-Thom+collapse+map'>Pontryagin-Thom collapse construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cobordism+cohomology+theory'>cobordism cohomology theory</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/MU'>complex cobordism cohomology theory</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/orientation+in+generalized+cohomology'>orientation in generalized cohomology</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/genus'>genus</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29-category+of+cobordisms'>(∞,n)-category of cobordisms</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/cobordism+hypothesis'>cobordism hypothesis</a></p> </li> </ul> <p><strong>flavors of <a class='existingWikiWord' href='/nlab/show/diff/bordism+homology+theory'>bordism homology theories</a>/<a class='existingWikiWord' href='/nlab/show/diff/cobordism+cohomology+theory'>cobordism cohomology theories</a>, their <a class='existingWikiWord' href='/nlab/show/diff/Brown+representability+theorem'>representing</a> <a class='existingWikiWord' href='/nlab/show/diff/Thom+spectrum'>Thom spectra</a> and <a class='existingWikiWord' href='/nlab/show/diff/bordism+ring'>cobordism rings</a></strong>:</p> <p><a class='existingWikiWord' href='/nlab/show/diff/bordism+homology+theory'>bordism theory</a><math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thickmathspace' /></mrow><annotation encoding='application/x-tex'>\;</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/Thom+spectrum'>M(B,f)</a> (<a class='existingWikiWord' href='/nlab/show/diff/B-bordism'>B-bordism</a>):</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stable+cohomotopy'>MFr</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/MO'>MO</a>, <a class='existingWikiWord' href='/nlab/show/diff/MSO'>MSO</a>, <a class='existingWikiWord' href='/nlab/show/diff/MSpin'>MSpin</a>, <a class='existingWikiWord' href='/nlab/show/diff/String+bordism'>MString</a>, …</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/MU'>MU</a>, <a class='existingWikiWord' href='/nlab/show/diff/MSU'>MSU</a>, …</p> <p><a class='existingWikiWord' href='/nlab/show/diff/Ravenel%27s+spectrum'>MΩΩSU(n)</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/MP'>MP</a>, <a class='existingWikiWord' href='/nlab/show/diff/MR+cohomology+theory'>MR</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/MSpin%E1%B6%9C'>MSpin<sup><i>c</i></sup></a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/MSp'>MSp</a></p> </li> </ul> <p>relative bordism theories:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/MOFr'>MOFr</a>, <a class='existingWikiWord' href='/nlab/show/diff/MUFr'>MUFr</a>, <a class='existingWikiWord' href='/nlab/show/diff/MSUFr'>MSUFr</a></li> </ul> <p><a class='existingWikiWord' href='/nlab/show/diff/equivariant+bordism+homology+theory'>equivariant bordism theory</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivariant+MFr'>equivariant MFr</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivariant+bordism+homology+theory'>equivariant MO</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivariant+complex+cobordism+cohomology+theory'>equivariant MU</a></p> </li> </ul> <p><a class='existingWikiWord' href='/nlab/show/diff/global+equivariant+bordism+homology+theory'>global equivariant bordism theory</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/global+equivariant+bordism+homology+theory'>global equivariant mO</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/global+equivariant+bordism+homology+theory'>global equivariant mU</a></p> </li> </ul> <p>algebraic:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/algebraic+cobordism'>algebraic cobordism</a></li> </ul> </div> </div> </div> <blockquote> <p>under construction</p> </blockquote> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#BoundaryMorphism'>Boundary morphism to <math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>MFr</mi></mrow><annotation encoding='application/x-tex'>MFr</annotation></semantics></math></a></li><li><a href='#relation_to__and_'>Relation to <math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi><mi mathvariant='normal'>O</mi></mrow><annotation encoding='application/x-tex'>M \mathrm{O}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi><mi>Fr</mi></mrow><annotation encoding='application/x-tex'>M Fr</annotation></semantics></math></a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p><a class='existingWikiWord' href='/nlab/show/diff/cobordism+theory'>cobordism theory</a> for unoriented manifolds with stably <a class='existingWikiWord' href='/nlab/show/diff/framed+manifold'>framed</a> <a class='existingWikiWord' href='/nlab/show/diff/manifold+with+boundary'>boundaries</a>, thus unifying <a class='existingWikiWord' href='/nlab/show/diff/MO'>MO</a> with <a class='existingWikiWord' href='/nlab/show/diff/stable+cohomotopy'>MFr</a>.</p> <h2 id='definition'>Definition</h2> <p>Consider the <a class='existingWikiWord' href='/nlab/show/diff/cofiber+sequence'>cofiber sequence</a> of the unit morphism of the <a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>ring spectrum</a> <a class='existingWikiWord' href='/nlab/show/diff/MO'>MO</a></p> <div class='maruku-equation' id='eq:AsUnitCofiber'><span class='maruku-eq-number'>(1)</span><math class='maruku-mathml' display='block' id='mathml_a5c14c721361eed247328a827fec20262d788c78_9' xmlns='http://www.w3.org/1998/Math/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'>O</mi></mrow></msup></mrow></mover></mtd> <mtd><mi>M</mi><mi mathvariant='normal'>O</mi></mtd></mtr> <mtr><mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo></mtd> <mtd><msup><mrow /> <mrow><msub><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'>O</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{O}} }{ \longrightarrow } & M \mathrm{O} \\ \big\downarrow & {}^{{}_{(po)}} & \big\downarrow \\ \ast &\longrightarrow& M \mathrm{O}/ \mathbb{S} } </annotation></semantics></math></div> <p>The <a class='existingWikiWord' href='/nlab/show/diff/cofiber+sequence'>homotopy cofiber</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a5c14c721361eed247328a827fec20262d788c78_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi><mo stretchy='false'>(</mo><mi mathvariant='normal'>O</mi><mo>,</mo><mi>fr</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><mi>M</mi><mi mathvariant='normal'>O</mi><mo stretchy='false'>/</mo><mi>𝕊</mi><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> M(\mathrm{O},fr) \;\coloneqq\; M \mathrm{O} / \mathbb{S} \,, </annotation></semantics></math></div> <p>has <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group+of+a+spectrum'>stable homotopy groups</a> the <a class='existingWikiWord' href='/nlab/show/diff/bordism+ring'>cobordism ring</a> of unoriented bordisms with <a class='existingWikiWord' href='/nlab/show/diff/stable+tangent+bundle'>stably</a> <a class='existingWikiWord' href='/nlab/show/diff/framed+manifold'>framed</a> <a class='existingWikiWord' href='/nlab/show/diff/manifold+with+boundary'>boundaries</a></p> <div class='maruku-equation' id='eq:OFrCobordismRing'><span class='maruku-eq-number'>(2)</span><math class='maruku-mathml' display='block' id='mathml_a5c14c721361eed247328a827fec20262d788c78_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Ω</mi> <mo>•</mo> <mrow><mi mathvariant='normal'>O</mi><mo>,</mo><mi>fr</mi></mrow></msubsup><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><msub><mi>π</mi> <mo>•</mo></msub><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>M</mi><mi mathvariant='normal'>O</mi><mo stretchy='false'>/</mo><mi>𝕊</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'> \Omega^{\mathrm{O},fr}_{\bullet} \;\coloneqq\; \pi_{\bullet} \big( M\mathrm{O}/\mathbb{S} \big) </annotation></semantics></math></div> <h2 id='properties'>Properties</h2> <h3 id='BoundaryMorphism'>Boundary morphism to <math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_12' xmlns='http://www.w3.org/1998/Math/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'>(1)</a> makes it manifest that there is a <a class='existingWikiWord' href='/nlab/show/diff/cohomology+operation'>cohomology operation</a> to <a class='existingWikiWord' href='/nlab/show/diff/stable+cohomotopy'>MFr</a> of the form</p> <div class='maruku-equation' id='eq:BoundaryCohomologyOperation'><span class='maruku-eq-number'>(3)</span><math class='maruku-mathml' display='block' id='mathml_a5c14c721361eed247328a827fec20262d788c78_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>M</mi><mo stretchy='false'>(</mo><mi mathvariant='normal'>O</mi><mo>,</mo><mi>fr</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>=</mo></mtd> <mtd><mi>M</mi><mi mathvariant='normal'>O</mi><mo stretchy='false'>/</mo><mi>𝕊</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>∂</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover></mtd> <mtd><mi>Σ</mi><mi>𝕊</mi></mtd> <mtd><mo>=</mo><mspace width='thickmathspace' /><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'>O</mi><mo>,</mo><mi>fr</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd> <mtd /> <mtd><mo>⟶</mo></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' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ M(\mathrm{O},fr) \;= & M \mathrm{O}/\mathbb{S} & \overset{ \;\;\; \partial \;\;\; }{\longrightarrow} & \Sigma \mathbb{S} & =\; \Sigma Mfr \\ \pi_{2d+2}\big( M(\mathrm{O},fr) \big) && \longrightarrow && \pi_{2d+1}\big( Mfr \big) } \,. </annotation></semantics></math></div> <p>Namely, <math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_14' xmlns='http://www.w3.org/1998/Math/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/diff/cofiber+sequence'>homotopy cofiber</a>-sequence starting with <math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mn>1</mn> <mrow><mi>M</mi><mi mathvariant='normal'>O</mi></mrow></msup></mrow><annotation encoding='application/x-tex'>1^{M \mathrm{O}}</annotation></semantics></math>. In terms of the <a class='existingWikiWord' href='/nlab/show/diff/pasting+law+for+pullbacks'>pasting law</a>:</p> <div class='maruku-equation' id='eq:BoundaryOperationViaPastingLaw'><span class='maruku-eq-number'>(4)</span><math class='maruku-mathml' display='block' id='mathml_a5c14c721361eed247328a827fec20262d788c78_16' xmlns='http://www.w3.org/1998/Math/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'>O</mi></mrow></msup></mrow></mover></mtd> <mtd><mi>M</mi><mi mathvariant='normal'>O</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><msub><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><msub><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'>O</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{O}} }{ \longrightarrow } & M \mathrm{O} & \longrightarrow & \ast \\ \big\downarrow & {}^{{}_{(po)}} & \big\downarrow & {}^{{}_{(po)}} & \big\downarrow \\ \ast & \longrightarrow & M \mathrm{O}/ \mathbb{S} & \underset{ \partial }{ \longrightarrow } & \Sigma \mathbb{S} } </annotation></semantics></math></div> <h3 id='relation_to__and_'>Relation to <math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi><mi mathvariant='normal'>O</mi></mrow><annotation encoding='application/x-tex'>M \mathrm{O}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi><mi>Fr</mi></mrow><annotation encoding='application/x-tex'>M Fr</annotation></semantics></math></h3> <div class='num_prop' id='PositiveDegreeFramedBordismTrivialInUnorientedBordism'> <h6 id='proposition'>Proposition</h6> <p>The unit morphism of <a class='existingWikiWord' href='/nlab/show/diff/MO'>MO</a> is trivial on <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group+of+a+spectrum'>stable homotopy groups</a> in <a class='existingWikiWord' href='/nlab/show/diff/positive+number'>positive</a> degree:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a5c14c721361eed247328a827fec20262d788c78_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>π</mi> <mrow><mi>n</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><mover><mo>⟶</mo><mrow><msup><mn>1</mn> <mrow><mi>M</mi><mi mathvariant='normal'>O</mi></mrow></msup></mrow></mover></mtd> <mtd><msub><mi>π</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>M</mi><mi mathvariant='normal'>O</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd></mtr> <mtr><mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><msup><mrow /> <mpadded width='0'><mo>=</mo></mpadded></msup></mtd> <mtd /> <mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><msup><mrow /> <mpadded width='0'><mo>=</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msubsup><mi>Ω</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow> <mi>fr</mi></msubsup></mtd> <mtd><munder><mo>⟶</mo><mi>i</mi></munder></mtd> <mtd><msubsup><mi>Ω</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow> <mi mathvariant='normal'>O</mi></msubsup></mtd></mtr></mtable></mrow><mphantom><mi>AAAAAAA</mi></mphantom><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'> \array{ \pi_{n+1} \big( \mathbb{S} \big) & \overset{ 1^{M \mathrm{O}} }{\longrightarrow} & \pi_{n + 1} \big( M \mathrm{O} \big) \\ \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} \\ \Omega^{fr}_{n + 1} &\underset{i}{\longrightarrow}& \Omega^{\mathrm{O}}_{n + 1} } \phantom{AAAAAAA} n \in \mathbb{N} </annotation></semantics></math></div></div> <p>(<a href='#Stong68'>Stong 68, p. 102-103</a>)</p> <div class='num_prop' id='AShortExactSequenceOfOFrBordismRings'> <h6 id='proposition_2'>Proposition</h6> <p>In <a class='existingWikiWord' href='/nlab/show/diff/positive+number'>positive</a> degree, the underling <a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian groups</a> of the <a class='existingWikiWord' href='/nlab/show/diff/bordism+ring'>bordism rings</a> for <a class='existingWikiWord' href='/nlab/show/diff/MO'>MO</a>, <a class='existingWikiWord' href='/nlab/show/diff/stable+cohomotopy'>MFr</a> and <math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>MOFr</mi></mrow><annotation encoding='application/x-tex'>MOFr</annotation></semantics></math> <a class='maruku-eqref' href='#eq:OFrCobordismRing'>(2)</a> sit in <a class='existingWikiWord' href='/nlab/show/diff/exact+sequence'>short exact sequences</a> of this form:</p> <div class='maruku-equation' id='eq:ShortExactSequenceOfUFrBordismRings'><span class='maruku-eq-number'>(5)</span><math class='maruku-mathml' display='block' id='mathml_a5c14c721361eed247328a827fec20262d788c78_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>→</mo><msubsup><mi>Ω</mi> <mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow> <mi mathvariant='normal'>O</mi></msubsup><mover><mo>⟶</mo><mi>i</mi></mover><msubsup><mi>Ω</mi> <mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow> <mrow><mi mathvariant='normal'>O</mi><mo>,</mo><mi>fr</mi></mrow></msubsup><mover><mo>⟶</mo><mo>∂</mo></mover><msubsup><mi>Ω</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow> <mi>fr</mi></msubsup><mo>→</mo><mn>0</mn><mspace width='thinmathspace' /><mo>,</mo><mphantom><mi>AAAA</mi></mphantom><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> 0 \to \Omega^{\mathrm{O}}_{n+2} \overset{i}{\longrightarrow} \Omega^{\mathrm{O},fr}_{n+2} \overset{\partial}{ \longrightarrow } \Omega^{fr}_{n+1} \to 0 \,, \phantom{AAAA} n \in \mathbb{N} \,, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> is the evident inclusion, while <math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∂</mo></mrow><annotation encoding='application/x-tex'>\partial</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a> homomorphism from <a href='#BoundaryMorphism'>above</a>.</p> </div> <p>(<a href='#Stong68'>Stong 68, p. 102-103</a>)</p> <div class='proof'> <h6 id='proof'>Proof</h6> <p>We have the <a class='existingWikiWord' href='/nlab/show/diff/long+exact+sequence+of+homotopy+groups'>long exact sequence of homotopy groups</a> obtained from the <a class='existingWikiWord' href='/nlab/show/diff/cofiber+sequence'>cofiber sequence</a> <math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝕊</mi><mover><mo>⟶</mo><mrow><msup><mn>1</mn> <mrow><mi>M</mi><mi mathvariant='normal'>O</mi></mrow></msup></mrow></mover><mi>M</mi><mi mathvariant='normal'>O</mi><mo>→</mo><mi>M</mi><mi mathvariant='normal'>O</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{O}}}{\longrightarrow} M \mathrm{O} \to M \mathrm{O}/\mathbb{S} \overset{\partial}{\to} \Sigma \mathbb{S}</annotation></semantics></math> <a class='maruku-eqref' href='#eq:BoundaryOperationViaPastingLaw'>(4)</a>, the relevant part of which looks as follows:</p> <div class='maruku-equation' id='eq:SToMULongExactSequenceOfHomotopyGroups'><span class='maruku-eq-number'>(6)</span><math class='maruku-mathml' display='block' id='mathml_a5c14c721361eed247328a827fec20262d788c78_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>π</mi> <mrow><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></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mn>1</mn> <mrow><mi>M</mi><mi mathvariant='normal'>O</mi></mrow></msup></mrow></mover></mtd> <mtd><msub><mi>π</mi> <mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow></msub><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>M</mi><mi mathvariant='normal'>O</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow /></mover></mtd> <mtd><msub><mi>π</mi> <mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow></msub><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>M</mi><mi mathvariant='normal'>O</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><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><msub><mi>π</mi> <mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msub><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>M</mi><mi mathvariant='normal'>O</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd></mtr> <mtr><mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><msup><mrow /> <mpadded width='0'><mo>=</mo></mpadded></msup></mtd> <mtd /> <mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><msup><mrow /> <mpadded width='0'><mo>=</mo></mpadded></msup></mtd> <mtd /> <mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><msup><mrow /> <mpadded width='0'><mo>=</mo></mpadded></msup></mtd> <mtd /> <mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><msup><mrow /> <mpadded width='0'><mo>=</mo></mpadded></msup></mtd> <mtd /> <mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><msup><mrow /> <mpadded width='0'><mo>=</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msubsup><mi>Ω</mi> <mrow><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><mi>d</mi><mo>+</mo><mn>2</mn></mrow> <mi mathvariant='normal'>O</mi></msubsup></mtd> <mtd><munder><mo>⟶</mo><mi>i</mi></munder></mtd> <mtd><msubsup><mi>Ω</mi> <mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow> <mrow><mo stretchy='false'>(</mo><mi mathvariant='normal'>O</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><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><msubsup><mi>Ω</mi> <mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow> <mi mathvariant='normal'>O</mi></msubsup></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ \pi_{d+2} \big( \mathbb{S} \big) & \overset{ 1^{M\mathrm{O}} }{ \longrightarrow } & \pi_{d+2} \big( M\mathrm{O} \big) & \overset{ }{\longrightarrow} & \pi_{d+2} \big( M\mathrm{O}/\mathbb{S} \big) & \overset{ \partial }{\longrightarrow} & \pi_{d+1}\big(\mathbb{S}\big) &\longrightarrow& \pi_{d+1}\big(M\mathrm{O}\big) \\ \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} \\ \Omega^{fr}_{d+2} & \underset{ \color{green} 0 }{ \longrightarrow } & \Omega^{\mathrm{O}}_{d+2} & \underset{ i }{\longrightarrow} & \Omega^{(\mathrm{O},fr)}_{d+2} & \underset{ \partial }{\longrightarrow} & \Omega^{fr}_{d + 1} & \underset{ \color{green} 0 }{\longrightarrow} & \Omega^{\mathrm{O}}_{d+1} } </annotation></semantics></math></div> <p>Here the two outermost morphisms shown are <a class='existingWikiWord' href='/nlab/show/diff/zero+morphism'>zero morphisms</a>, by Prop. <a class='maruku-ref' href='#PositiveDegreeFramedBordismTrivialInUnorientedBordism'>1</a>, and hence the claim follows.</p> </div> <h2 id='related_concepts'>Related concepts</h2> <p><strong>flavors of <a class='existingWikiWord' href='/nlab/show/diff/bordism+homology+theory'>bordism homology theories</a>/<a class='existingWikiWord' href='/nlab/show/diff/cobordism+cohomology+theory'>cobordism cohomology theories</a>, their <a class='existingWikiWord' href='/nlab/show/diff/Brown+representability+theorem'>representing</a> <a class='existingWikiWord' href='/nlab/show/diff/Thom+spectrum'>Thom spectra</a> and <a class='existingWikiWord' href='/nlab/show/diff/bordism+ring'>cobordism rings</a></strong>:</p> <p><a class='existingWikiWord' href='/nlab/show/diff/bordism+homology+theory'>bordism theory</a><math class='maruku-mathml' display='inline' id='mathml_a5c14c721361eed247328a827fec20262d788c78_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thickmathspace' /></mrow><annotation encoding='application/x-tex'>\;</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/Thom+spectrum'>M(B,f)</a> (<a class='existingWikiWord' href='/nlab/show/diff/B-bordism'>B-bordism</a>):</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stable+cohomotopy'>MFr</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/MO'>MO</a>, <a class='existingWikiWord' href='/nlab/show/diff/MSO'>MSO</a>, <a class='existingWikiWord' href='/nlab/show/diff/MSpin'>MSpin</a>, <a class='existingWikiWord' href='/nlab/show/diff/String+bordism'>MString</a>, …</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/MU'>MU</a>, <a class='existingWikiWord' href='/nlab/show/diff/MSU'>MSU</a>, …</p> <p><a class='existingWikiWord' href='/nlab/show/diff/Ravenel%27s+spectrum'>MΩΩSU(n)</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/MP'>MP</a>, <a class='existingWikiWord' href='/nlab/show/diff/MR+cohomology+theory'>MR</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/MSpin%E1%B6%9C'>MSpin<sup><i>c</i></sup></a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/MSp'>MSp</a></p> </li> </ul> <p>relative bordism theories:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/MOFr'>MOFr</a>, <a class='existingWikiWord' href='/nlab/show/diff/MUFr'>MUFr</a>, <a class='existingWikiWord' href='/nlab/show/diff/MSUFr'>MSUFr</a></li> </ul> <p><a class='existingWikiWord' href='/nlab/show/diff/equivariant+bordism+homology+theory'>equivariant bordism theory</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivariant+MFr'>equivariant MFr</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivariant+bordism+homology+theory'>equivariant MO</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivariant+complex+cobordism+cohomology+theory'>equivariant MU</a></p> </li> </ul> <p><a class='existingWikiWord' href='/nlab/show/diff/global+equivariant+bordism+homology+theory'>global equivariant bordism theory</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/global+equivariant+bordism+homology+theory'>global equivariant mO</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/global+equivariant+bordism+homology+theory'>global equivariant mU</a></p> </li> </ul> <p>algebraic:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/algebraic+cobordism'>algebraic cobordism</a></li> </ul> <h2 id='references'>References</h2> <ul> <li id='Stong68'><a class='existingWikiWord' href='/nlab/show/diff/Robert+Stong'>Robert Stong</a>, p. 102-106 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><ins class='diffins'> </ins><ins class='diffins'><p>Analogous discussion for <a class='existingWikiWord' href='/nlab/show/diff/MO'>MO</a>-bordism with <a class='existingWikiWord' href='/nlab/show/diff/MSO'>MSO</a>-boundaries:</p></ins><ins class='diffins'> </ins><ins class='diffins'><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></ins> </div> <div class="revisedby"> <p> Last revised on January 18, 2021 at 15:32:55. 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