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href="/nlab/show/Introduction+to+Cobordism+and+Complex+Oriented+Cohomology">Introduction</a></em></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+transformation">coordinate transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atlas">atlas</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, ,<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite+dimensional+manifold">infinite dimensional manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Banach+manifold">Banach manifold</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+manifold">Hilbert manifold</a>, <a class="existingWikiWord" href="/nlab/show/ILH+manifold">ILH manifold</a>, <a class="existingWikiWord" href="/nlab/show/Frechet+manifold">Frechet manifold</a>, <a class="existingWikiWord" href="/nlab/show/convenient+manifold">convenient manifold</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</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> <ul> <li><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Genera and invariants</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/signature+genus">signature genus</a>, <a class="existingWikiWord" href="/nlab/show/Kervaire+invariant">Kervaire invariant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-hat+genus">A-hat genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-manifolds">2-manifolds</a>/<a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of a surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+transversality+theorem">Thom's transversality theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galatius-Tillmann-Madsen-Weiss+theorem">Galatius-Tillmann-Madsen-Weiss theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometrization+conjecture">geometrization conjecture</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptization+conjecture">elliptization conjecture</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#details'>Details</a></li> <ul> <li><a href='#InTermsOfBfStructures'>In terms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,f)</annotation></semantics></math>-structures</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Given a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, by a <em>tangential structure</em> one typically understands (e.g. <a href="GMWT09">GMWT 09, Sec. 5</a>) a <a class="existingWikiWord" href="/nlab/show/lift">lift</a> of the <a class="existingWikiWord" href="/nlab/show/classifying+map">classifying map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mo>⊢</mo></mover><mi>B</mi><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \overset{\vdash}{\longrightarrow} B GL(n)</annotation></semantics></math> of its <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> through any prescribed <a class="existingWikiWord" href="/nlab/show/map">map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>B</mi><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \overset{f}{\longrightarrow} B GL(n)</annotation></semantics></math> into the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a>, up to <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mfrac linethickness="0"><mrow><mi>tangential</mi></mrow><mrow><mi>structure</mi></mrow></mfrac></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo maxsize="1.2em" minsize="1.2em">↓</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>⊢</mo><mi>T</mi><mi>X</mi></mrow></munder></mtd> <mtd><mi>B</mi><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && B \\ & {}^{ \mathllap{ {tangential} \atop {structure} } }\nearrow & \big\downarrow^{ f } \\ X &\underset{ \vdash T X }{\longrightarrow}& B GL(n) } </annotation></semantics></math></div> <p>Since this is all considered (only) for <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> of topological spaces (e.g. via the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>) and there is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B GL(n) \simeq B O(n)</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> (the latter being the <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math>), authors typically consider the equivalent diagram over <a class="existingWikiWord" href="/nlab/show/BO%28n%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>B</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">B O(n)</annotation> </semantics> </math></a>.</p> <p>Beware that the same kind of lift but understood in <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable</a> <a class="existingWikiWord" href="/nlab/show/classifying+stacks">classifying stacks</a> instead of just <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> is a <em><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a></em> as commonly understood now (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">A = \mathbf{B}G</annotation></semantics></math>, the classifying stack/<a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>).</p> <h2 id="details">Details</h2> <h3 id="InTermsOfBfStructures">In terms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,f)</annotation></semantics></math>-structures</h3> <div class="num_defn" id="BfStructure"> <h6 id="definition">Definition</h6> <p>A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,f)</annotation></semantics></math>-structure</strong> is</p> <ol> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n\in \mathbb{N}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a> <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>n</mi></msub><mo>∈</mo><msubsup><mi>Top</mi> <mi>CW</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">B_n \in Top_{CW}^{\ast/}</annotation></semantics></math></p> </li> <li> <p>equipped with a pointed <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre fibration</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>B</mi> <mi>n</mi></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mi>n</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ B_n \\ \downarrow^{\mathrlap{f_n}} \\ B O(n) } </annotation></semantics></math></div> <p>to the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <a class="existingWikiWord" href="/nlab/show/BO%28n%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>B</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">B O(n)</annotation> </semantics> </math></a> (<a href="classifying+space#EOn">def.</a>);</p> </li> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>≤</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">n_1 \leq n_2</annotation></semantics></math> a pointed continuous function</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>⟶</mo><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\iota_{n_1, n_2} \;\colon\; B_{n_1} \longrightarrow B_{n_2}</annotation></semantics></math></p> <p>which is the identity for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">n_1 = n_2</annotation></semantics></math>;</p> </li> </ol> <p>such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>≤</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n_1 \leq n_2 \in \mathbb{N}</annotation></semantics></math> these <a class="existingWikiWord" href="/nlab/show/commuting+square">squares commute</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ι</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mrow></mover></mtd> <mtd><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><msub><mi>n</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ B_{n_1} &\overset{\iota_{n_1,n_2}}{\longrightarrow}& B_{n_2} \\ {}^{\mathllap{f_{n_1}}}\downarrow && \downarrow^{\mathrlap{f_{n_2}}} \\ B O(n_1) &\longrightarrow& B O(n_2) } \,, </annotation></semantics></math></div> <p>where the bottom map is the canonical one (<a href="classifying+space#InclusionOfBOnIntoBOnPlusOne">def.</a>).</p> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,f)</annotation></semantics></math>-structure is <strong>multiplicative</strong> if it is moreover equipped with a system of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><mo lspace="verythinmathspace">:</mo><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>×</mo><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><mo>→</mo><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mu_{n_1,n_2} \colon B_{n_1}\times B_{n_2} \to B_{n_1 + n_2}</annotation></semantics></math> which cover the canonical multiplication maps (<a href="classifying+space#WhitneySumMapOnClassifyingSpaces">def.</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>×</mo><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>μ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mrow></mover></mtd> <mtd><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>×</mo><msub><mi>f</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><msub><mi>n</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>×</mo><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ B_{n_1} \times B_{n_2} &\overset{\mu_{n_1, n_2}}{\longrightarrow}& B_{n_1 + n_2} \\ {}^{\mathllap{f_{n_1} \times f_{n_2}}}\downarrow && \downarrow^{\mathrlap{f_{n_1 + n_2}}} \\ B O(n_1) \times B O(n_2) &\longrightarrow& B O(n_1 + n_2) } </annotation></semantics></math></div> <p>and which satisfy the evident <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a> and <a class="existingWikiWord" href="/nlab/show/unitality">unitality</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mn>0</mn></msub><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">B_0 = \ast</annotation></semantics></math> the unit, and, finally, which commute with the maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi></mrow><annotation encoding="application/x-tex">\iota</annotation></semantics></math> in that all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>3</mn></msub><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n_1,n_2, n_3 \in \mathbb{N}</annotation></semantics></math> these squares commute:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>×</mo><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>id</mi><mo>×</mo><msub><mi>ι</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>3</mn></msub></mrow></msub></mrow></mover></mtd> <mtd><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>×</mo><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>3</mn></msub></mrow></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>3</mn></msub></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>ι</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>3</mn></msub></mrow></msub></mrow></munder></mtd> <mtd><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>3</mn></msub></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ B_{n_1} \times B_{n_2} &\overset{id \times \iota_{n_2,n_2+n_3}}{\longrightarrow}& B_{n_1} \times B_{n_2 + n_3} \\ {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2 + n_3}}} \\ B_{n_1 + n_2} &\underset{\iota_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}& B_{n_1 + n_2 + n_3} } </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>×</mo><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ι</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>3</mn></msub></mrow></msub><mo>×</mo><mi>id</mi></mrow></mover></mtd> <mtd><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>3</mn></msub></mrow></msub><mo>×</mo><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>ι</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>3</mn></msub></mrow></msub></mrow></munder></mtd> <mtd><msub><mi>B</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>3</mn></msub></mrow></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ B_{n_1} \times B_{n_2} &\overset{\iota_{n_1,n_1+n_3} \times id}{\longrightarrow}& B_{n_1+n_3} \times B_{n_2 } \\ {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1 + n_3 , n_2}}} \\ B_{n_1 + n_2} &\underset{\iota_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}& B_{n_1 + n_2 + n_3} } \,. </annotation></semantics></math></div> <p>Similarly, an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,f)</annotation></semantics></math>-structure</strong> is a compatible system</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><msub><mi>B</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>⟶</mo><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f_{2n} \colon B_{2n} \longrightarrow B O(2n) </annotation></semantics></math></div> <p>indexed only on the even natural numbers.</p> <p>Generally, an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">S^k</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,f)</annotation></semantics></math>-structure</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k \geq 1</annotation></semantics></math> is a compatible system</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><mi>k</mi><mi>n</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><msub><mi>B</mi> <mrow><mi>k</mi><mi>n</mi></mrow></msub><mo>⟶</mo><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>k</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f_{k n} \colon B_{ k n} \longrightarrow B O(k n) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, hence for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>n</mi><mo>∈</mo><mi>k</mi><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k n \in k \mathbb{N}</annotation></semantics></math>.</p> </div> <p>(<a href="#Lashof63">Lashof 63</a>, <a href="#Stong68">Stong 68, beginning of chapter II</a>, <a href="#Kochman96">Kochman 96, section 1.4</a>)</p> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a></em>.</p> <div class="num_example" id="ExamplesOfBfStructures"> <h6 id="example">Example</h6> <p>Examples of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,f)</annotation></semantics></math>-structures (def. <a class="maruku-ref" href="#BfStructure"></a>) include the following:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>n</mi></msub><mo>=</mo><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_n = B O(n)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">f_n = id</annotation></semantics></math> is <strong>orthogonal structure</strong> (or “no structure”);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>n</mi></msub><mo>=</mo><mi>E</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_n = E O(n)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">f_n</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>-projection is <strong><a class="existingWikiWord" href="/nlab/show/framing">framing</a>-structure</strong>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>n</mi></msub><mo>=</mo><mi>B</mi><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_n = B SO(n) = E O(n)/SO(n)</annotation></semantics></math> the classifying space <a class="existingWikiWord" href="/nlab/show/BSO%28n%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>B</mi> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">B S O(n)</annotation> </semantics> </math></a> of the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">f_n</annotation></semantics></math> the canonical projection is <strong><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> structure</strong>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>n</mi></msub><mo>=</mo><mi>B</mi><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_n = B Spin(n) = E O(n)/Spin(n)</annotation></semantics></math> the classifying space of the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">f_n</annotation></semantics></math> the canonical projection is <strong><a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a></strong>.</p> </li> </ol> <p>Examples of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,f)</annotation></semantics></math>-structures include</p> <ol> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>=</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_{2n} = B U(n) = E O(2n)/U(n)</annotation></semantics></math> the classifying space of the <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f_{2n}</annotation></semantics></math> the canonical projection is <strong><a class="existingWikiWord" href="/nlab/show/almost+complex+structure">almost complex structure</a></strong>.</li> </ol> </div> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, and given a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,f)</annotation></semantics></math>-structure as in def. <a class="maruku-ref" href="#BfStructure"></a>, then a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,f)</annotation></semantics></math>-structure on the manifold</strong> is an <a class="existingWikiWord" href="/nlab/show/equivalence+class">equivalence class</a> of the following structure:</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/embedding">embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>↪</mo><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">i_X \; \colon \; X \hookrightarrow \mathbb{R}^k</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/homotopy+class">homotopy class</a> of a <a class="existingWikiWord" href="/nlab/show/lift">lift</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>g</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat g</annotation></semantics></math> of the classifying map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>B</mi> <mi>n</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mover><mi>g</mi><mo stretchy="false">^</mo></mover></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mi>n</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && B_{n} \\ &{}^{\mathllap{\hat g}}\nearrow& \downarrow^{\mathrlap{f_n}} \\ X &\overset{g}{\longrightarrow}& B O(n) } \,. </annotation></semantics></math></div></li> </ol> <p>The equivalence relation on such structures is to be that generated by the relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>X</mi></msub><msub><mo stretchy="false">)</mo> <mn>1</mn></msub><mo>,</mo><msub><mover><mi>g</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∼</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>X</mi></msub><msub><mo stretchy="false">)</mo> <mo>,</mo></msub><msub><mover><mi>g</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((i_{X})_1, \hat g_1) \sim ((i_{X})_,\hat g_2)</annotation></semantics></math> if</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>k</mi> <mn>2</mn></msub><mo>≥</mo><msub><mi>k</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">k_2 \geq k_1</annotation></semantics></math></p> </li> <li> <p>the second inclusion factors through the first as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>X</mi></msub><msub><mo stretchy="false">)</mo> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>↪</mo><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>X</mi></msub><msub><mo stretchy="false">)</mo> <mn>1</mn></msub></mrow></mover><msup><mi>ℝ</mi> <mrow><msub><mi>k</mi> <mn>1</mn></msub></mrow></msup><mo>↪</mo><msup><mi>ℝ</mi> <mrow><msub><mi>k</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> (i_X)_2 \;\colon\; X \overset{(i_X)_1}{\hookrightarrow} \mathbb{R}^{k_1} \hookrightarrow \mathbb{R}^{k_2} </annotation></semantics></math></div></li> <li> <p>the lift of the classifying map factors accordingly (as homotopy classes)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>g</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mover><mi>g</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow></mover><msub><mi>B</mi> <mi>n</mi></msub><mo>⟶</mo><msub><mi>B</mi> <mi>n</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \hat g_2 \;\colon\; X \overset{\hat g_1}{\longrightarrow} B_{n} \longrightarrow B_{n} \,. </annotation></semantics></math></div></li> </ol> </div> <h2 id="examples">Examples</h2> <p>The tangential structures corresponding to lifts through the <a href="Whitehead+tower#OfTheOrthogonalGroup">Whitehead tower of the orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></p> </li> <li> <p>…</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/framing">framing</a></p> </li> </ul> <h2 id="references">References</h2> <p>The concept of tangential structure originates with <a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a>, originally under the name <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B,f)</annotation></semantics></math>-structures</em>:</p> <ul> <li id="Lashof63"> <p><a class="existingWikiWord" href="/nlab/show/Richard+Lashof">Richard Lashof</a>, <em>Poincaré duality and cobordism</em>, Trans. AMS 109 (1963), 257-277 (<a href="https://doi.org/10.1090/S0002-9947-1963-0156357-4">doi:10.1090/S0002-9947-1963-0156357-4</a>)</p> </li> <li id="Stong68"> <p><a class="existingWikiWord" href="/nlab/show/Robert+Stong">Robert Stong</a>, beginning of chapter II of: <em>Notes on Cobordism theory</em>, 1968,</p> <p>reprinted as: Princeton Legacy Library, Princeton University Press 2016 (<a href="http://press.princeton.edu/titles/6465.html">ISBN:9780691649016</a>, <a href="http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf">toc pdf</a>)</p> </li> <li id="Kochman96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochman">Stanley Kochman</a>, section 1.4 of <em><a class="existingWikiWord" href="/nlab/show/Bordism%2C+Stable+Homotopy+and+Adams+Spectral+Sequences">Bordism, Stable Homotopy and Adams Spectral Sequences</a></em>, AMS 1996</p> </li> </ul> <p>The terminology “tangential structure” became popular around</p> <ul> <li id="GMWT09"><a class="existingWikiWord" href="/nlab/show/S%C3%B8ren+Galatius">Søren Galatius</a>, <a class="existingWikiWord" href="/nlab/show/Ib+Madsen">Ib Madsen</a>, <a class="existingWikiWord" href="/nlab/show/Ulrike+Tillmann">Ulrike Tillmann</a>,<a class="existingWikiWord" href="/nlab/show/Michael+Weiss">Michael Weiss</a>, Section 5 of: <em>The homotopy type of the cobordism category</em>, Acta Math. 202 (2009), no. 2, 195–239 (<a href="http://arxiv.org/abs/math/0605249">arXiv:math/0605249</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 4, 2024 at 21:31:07. 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