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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Boardman homomorphism </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/6960/#Item_7" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" 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href="/nlab/show/Introduction+to+Stable+Homotopy+Theory">Introduction</a></em></p> <h1 id="ingredients">Ingredients</h1> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h1 id="contents">Contents</h1> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+derivator">stable derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category+of+spectra">stable (∞,1)-category of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/stable+homotopy+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="cohomology">Cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>, <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>, <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</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/Spin%5Ec+structure">Spin^c 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> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </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='#examples'>Examples</a></li> <ul> <li><a href='#in_ordinary_cohomology'>In ordinary cohomology</a></li> <li><a href='#ForComplexOrientedCohomologyTheories'>In complex oriented cohomology</a></li> <li><a href='#in_topological_modular_forms'>In topological modular forms</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>Given any <a class="existingWikiWord" href="/nlab/show/homotopy+commutative+ring+spectrum">homotopy commutative ring spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E, \mu, e)</annotation></semantics></math>, then the <em>Boardman homomorphism</em> is the <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> from <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups">stable homotopy groups</a> (hence from <a class="existingWikiWord" href="/nlab/show/stable+homotopy+homology+theory">stable homotopy homology theory</a>) to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a> groups that is induced by <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> with the unit map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo lspace="verythinmathspace">:</mo><mi>𝕊</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">e \colon \mathbb{S} \longrightarrow E</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo>∧</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>E</mi><mo>∧</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_\bullet(-) \simeq \pi_\bullet(\mathbb{S} \wedge (-)) \longrightarrow \pi_\bullet(E \wedge (-)) = E_\bullet(-) \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">E = H \mathbb{Z}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a> for <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a>, then this reduces to the <a class="existingWikiWord" href="/nlab/show/Hurewicz+homomorphism">Hurewicz homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(-) \to H_\bullet(-)</annotation></semantics></math>.</p> <p>Dually, there is the Boardman homomorphism from <a class="existingWikiWord" href="/nlab/show/stable+cohomotopy">stable cohomotopy</a> to <a class="existingWikiWord" href="/nlab/show/generalized+cohomology">generalized cohomology</a> induced under forming <a class="existingWikiWord" href="/nlab/show/mapping+spectra">mapping spectra</a> into the unit map of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>π</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>𝕊</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>E</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi^\bullet(-) \simeq \pi_\bullet([(-),\mathbb{S}]) \longrightarrow \pi_\bullet([(-),E]) = E^\bullet(-) \,. </annotation></semantics></math></div> <p>Unifying these two cases, there is the bivariant Boardman homomorphism (<a href="#Adams74">Adams 74, p. 58</a>)</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><mi>X</mi><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">]</mo> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∧</mo><mi>𝕊</mi><msub><mo stretchy="false">]</mo> <mo>•</mo></msub><mo>⟶</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∧</mo><mi>E</mi><msub><mo stretchy="false">]</mo> <mo>•</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [X, Y]_\bullet \;\simeq\; [X, Y \wedge \mathbb{S}]_\bullet \longrightarrow [X,Y \wedge E]_\bullet \,. </annotation></semantics></math></div> <p>Since <a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a>/<a class="existingWikiWord" href="/nlab/show/generalized+cohomology">generalized cohomology</a> is typically more tractable than <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>/<a class="existingWikiWord" href="/nlab/show/cohomotopy">cohomotopy</a> (in particular when <a class="existingWikiWord" href="/nlab/show/homology+spectra+split">homology spectra split</a>), the Boardman homomorphism is often used to partially reduce computations of the latter in terms of computations of the former.</p> <p>One example is the computation of the homotopy groups of <a class="existingWikiWord" href="/nlab/show/MU">MU</a> via the <a class="existingWikiWord" href="/nlab/show/homology+of+MU">homology of MU</a> (<a class="existingWikiWord" href="/nlab/show/Quillen%27s+theorem+on+MU">Quillen's theorem on MU</a>), see <a href="#ForComplexOrientedCohomologyTheories">below</a>.</p> <h2 id="examples">Examples</h2> <h3 id="in_ordinary_cohomology">In ordinary cohomology</h3> <p>Consider the <a class="existingWikiWord" href="/nlab/show/unit">unit</a> morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi><mo>⟶</mo><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \mathbb{S} \longrightarrow H \mathbb{Z} </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a> to the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a> of the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>. For any <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>/<a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> postcomposition with this morphism induces <a class="existingWikiWord" href="/nlab/show/Boardman+homomorphisms">Boardman homomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a> (in fact of <a class="existingWikiWord" href="/nlab/show/commutative+rings">commutative rings</a>)</p> <div class="maruku-equation" id="eq:BoardmandCohomotopyToOrdinaryCohomology"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>b</mi> <mi>n</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>π</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> b^n \;\colon\; \pi^n(X) \longrightarrow H^n(X, \mathbb{Z}) </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/stable+cohomotopy">stable cohomotopy</a> of <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> in degree <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> to its <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> in degree <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>.</p> <div class="num_prop" id="BoundsOnCoKernelOfBoardmandFromStableCohomotopyToOrdinaryCohomology"> <h6 id="proposition">Proposition</h6> <p><strong>(bounds on (<a class="existingWikiWord" href="/nlab/show/cokernel">co-</a>)<a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of <a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman homomorphism</a> from <a class="existingWikiWord" href="/nlab/show/stable+cohomotopy">stable cohomotopy</a> to <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a>)</strong></p> <p>If <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> is a <a class="existingWikiWord" href="/nlab/show/CW-spectrum">CW-spectrum</a> which</p> <ol> <li> <p>is <a class="existingWikiWord" href="/nlab/show/n-connected+object+of+an+%28infinity%2C1%29-topos">(m-1)-connected</a></p> </li> <li> <p>of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">d \in \mathbb{N}</annotation></semantics></math></p> </li> </ol> <p>then</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of the <a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>b</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">b^n</annotation></semantics></math> <a class="maruku-eqref" href="#eq:BoardmandCohomotopyToOrdinaryCohomology">(1)</a> for</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>≤</mo><mi>n</mi><mo>≤</mo><mi>d</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex"> m \leq n\leq d -1 </annotation></semantics></math></div> <p>is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>ρ</mi><mo>¯</mo></mover> <mrow><mi>d</mi><mo>−</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\overline{\rho}_{d-n}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/torsion+subgroup">torsion group</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>ρ</mi><mo>¯</mo></mover> <mrow><mi>d</mi><mo>−</mo><mi>n</mi></mrow></msub><mi>ker</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mn>0</mn></mrow><annotation encoding="application/x-tex"> \overline{\rho}_{d-n} ker(b^n) \;\simeq\; 0 </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a> of the <a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>b</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">b^n</annotation></semantics></math> <a class="maruku-eqref" href="#eq:BoardmandCohomotopyToOrdinaryCohomology">(1)</a> for</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>≤</mo><mi>n</mi><mo>≤</mo><mi>d</mi><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex"> m \leq n \leq d - 2 </annotation></semantics></math></div> <p>is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>ρ</mi><mo>¯</mo></mover> <mrow><mi>d</mi><mo>−</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\overline{\rho}_{d-n-1}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/torsion+subgroup">torsion group</a>:</p> <div class="maruku-equation" id="eq:TorsionEstimateCokernel"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>ρ</mi><mo>¯</mo></mover> <mrow><mi>d</mi><mo>−</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mi>coker</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mn>0</mn></mrow><annotation encoding="application/x-tex"> \overline{\rho}_{d-n-1} coker(b^n) \;\simeq\; 0 </annotation></semantics></math></div></li> </ol> <p>where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>ρ</mi><mo>¯</mo></mover> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>i</mi><mo>≤</mo><mn>1</mn></mtd></mtr> <mtr><mtd><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></munderover><mi>exp</mi><mrow><mo>(</mo><msub><mi>π</mi> <mi>j</mi></msub><mrow><mo>(</mo><mi>𝕊</mi><mo>)</mo></mrow><mo>)</mo></mrow></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex"> \overline{\rho}_{i} \;\coloneqq\; \left\{ \array{ 1 &\vert& i\leq 1 \\ \underoverset{j = 1}{i}{\prod} exp\left( \pi_j\left( \mathbb{S}\right) \right) &\vert& \text{otherwise} } \right. </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/multiplication">product</a> of the <a class="existingWikiWord" href="/nlab/show/exponent+of+a+group">exponents</a> of the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a> in <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo><mi>i</mi></mrow><annotation encoding="application/x-tex">\leq i</annotation></semantics></math>.</p> </div> <p>(<a href="#Arlettaz04">Arlettaz 04, theorem 1.2</a>)</p> <div class="num_example" id="ExampleForEstimatesOfTorsionOfCokernelOfBeta"> <h6 id="example">Example</h6> <p><strong>(estimates for torsion of cokernel of Boardman homomorphism)</strong></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></p> <ul> <li> <p>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>d</mi><mo>=</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">d = 6</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply+connected+topological+space">simply connected</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\pi_2(X) \neq 0</annotation></semantics></math></p> </li> </ul> <p>then Prop. <a class="maruku-ref" href="#BoundsOnCoKernelOfBoardmandFromStableCohomotopyToOrdinaryCohomology"></a> asserts that the <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a> of the <a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman homomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>β</mi> <mn>4</mn></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>𝕊</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \beta^4 \;\colon\; \mathbb{S}^4(X) \longrightarrow H^4( X, \mathbb{Z} ) </annotation></semantics></math></div> <p>in</p> <ul> <li>degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">n = 4</annotation></semantics></math></li> </ul> <p>is <a class="existingWikiWord" href="/nlab/show/torsion+subgroup">2-torsion</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>coker</mi><mo stretchy="false">(</mo><msup><mi>β</mi> <mn>4</mn></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 2 coker(\beta^4) \;=\; 0 \,. </annotation></semantics></math></div> <p>This is because in this case <a class="maruku-eqref" href="#eq:TorsionEstimateCokernel">(2)</a> gives that the relevant torsion degree is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mover><mi>ρ</mi><mo>¯</mo></mover> <mrow><mi>d</mi><mo>−</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>=</mo><msub><mover><mi>ρ</mi><mo>¯</mo></mover> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>2</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \overline{\rho}_{d-n-1} & = \overline{\rho}_{1} \\ & = \exp( \pi_1(\mathbb{S}) ) \\ & = \exp( \mathbb{Z}/2 ) \\ & = 2 \end{aligned} \,. </annotation></semantics></math></div> <p>Similarly, if instead the manifold has dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">d = 7</annotation></semantics></math> but sticking to degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">n = 4</annotation></semantics></math>, then the estimate is that the cokernel is <a class="existingWikiWord" href="/nlab/show/torsion+subgroup">4-torsion</a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>4</mn><mi>coker</mi><mo stretchy="false">(</mo><msup><mi>β</mi> <mn>4</mn></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 4 coker(\beta^4) \;=\; 0 \,. </annotation></semantics></math></div> <p>since then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mover><mi>ρ</mi><mo>¯</mo></mover> <mrow><mi>d</mi><mo>−</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>=</mo><msub><mover><mi>ρ</mi><mo>¯</mo></mover> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>exp</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>2</mn><mo>⋅</mo><mn>2</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>4</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \overline{\rho}_{d-n-1} & = \overline{\rho}_{2} \\ & = \exp( \pi_1(\mathbb{S}) ) \cdot \exp( \pi_2(\mathbb{S}) ) \\ & = \exp( \mathbb{Z}/2 ) \cdot \exp( \mathbb{Z}/2 ) \\ & = 2 \cdot 2 \\ & = 4 \end{aligned} \,. </annotation></semantics></math></div> <p>Next for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">d = 8</annotation></semantics></math> we</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mover><mi>ρ</mi><mo>¯</mo></mover> <mrow><mi>d</mi><mo>−</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>=</mo><msub><mover><mi>ρ</mi><mo>¯</mo></mover> <mn>3</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>exp</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>exp</mi><mo stretchy="false">(</mo><msub><mi>π</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo><mo>⋅</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>24</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>2</mn><mo>⋅</mo><mn>2</mn><mo>⋅</mo><mn>6</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mn>24</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \overline{\rho}_{d-n-1} & = \overline{\rho}_{3} \\ & = \exp( \pi_1(\mathbb{S}) ) \cdot \exp( \pi_2(\mathbb{S}) ) \cdot \exp( \pi_3(\mathbb{S}) ) \\ & = \exp( \mathbb{Z}/2 ) \cdot \exp( \mathbb{Z}/2 ) \cdot \exp( \mathbb{Z}/{24} ) \\ & = 2 \cdot 2 \cdot 6 \\ & = 24 \end{aligned} \,. </annotation></semantics></math></div></div> <div class="num_prop" id="BoardmanIsoOn7SphereMod2I"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman</a> <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> on <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a> mod <a class="existingWikiWord" href="/nlab/show/binary+icosahedral+group">binary icosahedral group</a>)</strong></p> <p>Consider the <a class="existingWikiWord" href="/nlab/show/binary+icosahedral+group">binary icosahedral group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>I</mi></mrow><annotation encoding="application/x-tex">2 I</annotation></semantics></math> and its <a class="existingWikiWord" href="/nlab/show/action">action</a> on the <a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a> induced via the identification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><mo>≃</mo><mi>S</mi><mo stretchy="false">(</mo><mi>ℍ</mi><mo>×</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^7 \simeq S(\mathbb{H} \times \mathbb{H})</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/diagonal+action">diagonal</a> of the canonical action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>I</mi></mrow><annotation encoding="application/x-tex">2I</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">\mathbb{H}</annotation></semantics></math> induced via it being a <a class="existingWikiWord" href="/nlab/show/finite+subgroup+of+SU%282%29">finite subgroup of SU(2)</a>.</p> <p>On the <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><mo stretchy="false">/</mo><mn>2</mn><mi>I</mi></mrow><annotation encoding="application/x-tex">S^7/2 I</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman homomorphism</a> in degree 4 is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>𝕊</mi> <mn>4</mn></msup><mrow><mo>(</mo><msup><mi>S</mi> <mn>7</mn></msup><mo stretchy="false">/</mo><mn>2</mn><mi>I</mi><mo>)</mo></mrow><munderover><mo>⟶</mo><mo>≃</mo><mi>β</mi></munderover><msup><mi>H</mi> <mn>4</mn></msup><mrow><mo>(</mo><msup><mi>S</mi> <mn>7</mn></msup><mo stretchy="false">/</mo><mn>2</mn><mi>I</mi><mo>,</mo><mi>ℤ</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathbb{S}^4\left( S^7/2I \right) \underoverset{\simeq}{\beta}{\longrightarrow} H^4\left( S^7/2I , \mathbb{Z} \right) </annotation></semantics></math></div> <p>from <a class="existingWikiWord" href="/nlab/show/stable+cohomotopy">stable cohomotopy</a> in degree 4 to <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> in degree 4.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>In terms of the <a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a> for <a class="existingWikiWord" href="/nlab/show/stable+cohomotopy">stable cohomotopy</a> it is sufficient to see that the two differentials</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>4</mn></msup><mrow><mo>(</mo><msup><mi>S</mi> <mn>7</mn></msup><mo stretchy="false">/</mo><mn>2</mn><mi>I</mi><mo>,</mo><msubsup><mi>π</mi> <mi>s</mi> <mn>0</mn></msubsup><mo>=</mo><msubsup><mi>π</mi> <mn>0</mn> <mi>s</mi></msubsup><mo>=</mo><mi>ℤ</mi><mo>)</mo></mrow><mover><mo>⟶</mo><mrow><msub><mi>d</mi> <mn>3</mn></msub></mrow></mover><msup><mi>H</mi> <mn>6</mn></msup><mrow><mo>(</mo><msup><mi>S</mi> <mn>7</mn></msup><mo stretchy="false">/</mo><mn>2</mn><mi>I</mi><mo>,</mo><msubsup><mi>π</mi> <mi>s</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msubsup><mi>π</mi> <mn>1</mn> <mi>s</mi></msubsup><mo>=</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> H^4\left( S^7/2I, \pi^0_s = \pi^s_0 =\mathbb{Z} \right) \overset{d_3}{\longrightarrow} H^6\left( S^7/2I, \pi^{-1}_s = \pi^s_{1} =\mathbb{Z}/2 \right) </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><msup><mi>H</mi> <mn>4</mn></msup><mrow><mo>(</mo><msup><mi>S</mi> <mn>7</mn></msup><mo stretchy="false">/</mo><mn>2</mn><mi>I</mi><mo>,</mo><msubsup><mi>π</mi> <mi>s</mi> <mn>0</mn></msubsup><mo>=</mo><msubsup><mi>π</mi> <mn>0</mn> <mi>s</mi></msubsup><mo>=</mo><mi>ℤ</mi><mo>)</mo></mrow><mover><mo>⟶</mo><mrow><msub><mi>d</mi> <mn>3</mn></msub></mrow></mover><msup><mi>H</mi> <mn>7</mn></msup><mrow><mo>(</mo><msup><mi>S</mi> <mn>7</mn></msup><mo stretchy="false">/</mo><mn>2</mn><mi>I</mi><mo>,</mo><msubsup><mi>π</mi> <mi>s</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow></msubsup><mo>=</mo><msubsup><mi>π</mi> <mn>2</mn> <mi>s</mi></msubsup><mo>=</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> H^4\left( S^7/2I, \pi^0_s = \pi^s_0 =\mathbb{Z} \right) \overset{d_3}{\longrightarrow} H^7\left( S^7/2I, \pi^{-2}_s = \pi^s_{2} =\mathbb{Z}/2 \right) </annotation></semantics></math></div> <p>both vanish (all higher differentials on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msubsup><mi>π</mi> <mi>s</mi> <mn>0</mn></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^4(-,\pi^0_s)</annotation></semantics></math> vanish simply for dimensional reasons as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">S^7</annotation></semantics></math> is of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 7, while there are no differentials into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msubsup><mi>π</mi> <mi>s</mi> <mn>0</mn></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^4(-,\pi^0_s)</annotation></semantics></math> simply because the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a> is <a class="existingWikiWord" href="/nlab/show/connective+spectrum">connective</a>, so that the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a> vanish in <a class="existingWikiWord" href="/nlab/show/negative+number">negative</a> degree).</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">d_2</annotation></semantics></math> to vanish, it is sufficient that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>6</mn></msup><mrow><mo>(</mo><msup><mi>S</mi> <mn>7</mn></msup><mo stretchy="false">/</mo><mn>2</mn><mi>I</mi><mo>,</mo><msubsup><mi>π</mi> <mi>s</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msubsup><mi>π</mi> <mn>1</mn> <mi>s</mi></msubsup><mo>=</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mn>0</mn></mrow><annotation encoding="application/x-tex"> H^6\left( S^7/2I, \pi^{-1}_s = \pi^s_{1} =\mathbb{Z}/2 \right) \;\simeq\; 0 </annotation></semantics></math></div> <p>We now first show that this is the case:</p> <p>First, by the <a class="existingWikiWord" href="/nlab/show/Gysin+sequence">Gysin sequence</a> for the <a class="existingWikiWord" href="/nlab/show/spherical+fibration">spherical 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><msup><mi>S</mi> <mn>7</mn></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>S</mi> <mn>7</mn></msup><mo stretchy="false">/</mo><mi>SI</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi><mo stretchy="false">(</mo><mn>2</mn><mi>I</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^7 &\longrightarrow& S^7/SI \\ && \downarrow \\ && B (2 I) } </annotation></semantics></math></div> <p>we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>6</mn></msup><mrow><mo>(</mo><msup><mi>S</mi> <mn>7</mn></msup><mo stretchy="false">/</mo><mn>2</mn><mi>I</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mn>6</mn></msup><mrow><mo>(</mo><mi>B</mi><mo stretchy="false">(</mo><mn>2</mn><mi>I</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> H^6\left( S^7/2I, \, \mathbb{Z}/2 \right) \;\simeq\; H^6\left( B(2I),\, \mathbb{Z}/2 \right) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mn>2</mn><mi>I</mi><mo stretchy="false">)</mo><mo>≃</mo><mo>*</mo><mo>⫽</mo><mo stretchy="false">(</mo><mn>2</mn><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B (2 I) \simeq \ast \sslash (2I)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>I</mi></mrow><annotation encoding="application/x-tex">2I</annotation></semantics></math> (see e.g. at <a class="existingWikiWord" href="/nlab/show/infinity-action">infinity-action</a>).</p> <p>Moreover, by the <a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a> (<a href="universal+coefficient+theorem#OrdinaryStatementInTopology">this Prop.</a>) we have a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>H</mi> <mn>5</mn></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mo stretchy="false">(</mo><mn>2</mn><mi>I</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>⟶</mo><msup><mi>H</mi> <mn>6</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mo stretchy="false">(</mo><mn>2</mn><mi>I</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>⟶</mo><msub><mi>Hom</mi> <mi>Ab</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>H</mi> <mn>6</mn></msub><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">(</mo><mn>2</mn><mi>I</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>→</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \to Ext^1(H_{5}\big(B(2I), \mathbb{Z}), \mathbb{Z}/2\big) \longrightarrow H^6\big(B(2I), \mathbb{Z}/2\big) \longrightarrow Hom_{Ab}\big( H_6( B(2I), \mathbb{Z}) , \mathbb{Z}/2 \big) \to 0 \,. </annotation></semantics></math></div> <p>This means that it is sufficient to see that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>5</mn></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mo stretchy="false">(</mo><mn>2</mn><mi>I</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn><mphantom><mi>AAA</mi></mphantom><msub><mi>H</mi> <mn>6</mn></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mo stretchy="false">(</mo><mn>2</mn><mi>I</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> H_{5}\big(B(2I), \mathbb{Z}) \simeq 0 \phantom{AAA} H_{6}\big(B(2I), \mathbb{Z}) \simeq 0 </annotation></semantics></math></div> <p>But for every <a class="existingWikiWord" href="/nlab/show/finite+subgroup+of+SU%282%29">finite subgroup of SU(2)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>ADE</mi></msub><mo>⊂</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_{ADE} \subset SU(2)</annotation></semantics></math> we have (by <a href="https://ncatlab.org/nlab/show/finite+rotation+group#GroupCohomologyOfFiniteSubgroupsOfSU2">this Prop.</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mn>5</mn></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mo stretchy="false">(</mo><mn>2</mn><mi>I</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>G</mi> <mi>ADE</mi> <mi>ab</mi></msubsup><mphantom><mi>AAA</mi></mphantom><msub><mi>H</mi> <mn>6</mn></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mo stretchy="false">(</mo><mn>2</mn><mi>I</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> H_{5}\big(B(2I), \mathbb{Z}) \simeq G^{ab}_{ADE} \phantom{AAA} H_{6}\big(B(2I), \mathbb{Z}) \simeq 0 </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>G</mi> <mi>ADE</mi> <mi>ab</mi></msubsup></mrow><annotation encoding="application/x-tex">G^{ab}_{ADE}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/abelianization">abelianization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>ADE</mi></msub></mrow><annotation encoding="application/x-tex">G_{ADE}</annotation></semantics></math>. Specifically for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>ADE</mi></msub><mo>=</mo><mn>2</mn><mi>I</mi></mrow><annotation encoding="application/x-tex">G_{ADE} = 2I</annotation></semantics></math> this does vanish: the <a class="existingWikiWord" href="/nlab/show/binary+icosahedral+group">binary icosahedral group</a> is a <a class="existingWikiWord" href="/nlab/show/perfect+group">perfect group</a> (<a href="icosahedral+group#2IIsPerfect">this Prop.</a>).</p> <p>This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">d_2</annotation></semantics></math> vanishes on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mi>π</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^4(-, \pi^0)</annotation></semantics></math>.</p> <p>Now by a standard argument, the AHSS-differentials between ordinary cohomology groups are stable <a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a>, and thus, if non-trivial, must be the <a class="existingWikiWord" href="/nlab/show/Steenrod+operations">Steenrod operations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">Sq^n</annotation></semantics></math> (e.g. <a href="https://mathgroove.wordpress.com/2016/11/30/filling-the-details-5-two-words-about-the-atiyah-hirzebruch-spectral-sequence/">here</a>, but let’s add a more canonical reference).</p> <p>This means first of all that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">d_2</annotation></semantics></math> is not trivial then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>2</mn></msub><mo>=</mo><msup><mi>Sq</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">d_2 = Sq^2</annotation></semantics></math>. But since that vanishes on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mi>π</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^4(-,\pi^0)</annotation></semantics></math> by the above argument, and on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>7</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mi>π</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^7(-,\pi^2)</annotation></semantics></math> for dimension reasons, so that the relevant entries pass as ordinary cohomology groups to the third page of the <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a>, it follows similarly that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>3</mn></msub><mo>=</mo><msup><mi>Sq</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">d_3 = Sq^3</annotation></semantics></math>.</p> <p>But by the <a class="existingWikiWord" href="/nlab/show/Adem+relation">Adem relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mn>3</mn></msup><mo>=</mo><msup><mi>Sq</mi> <mn>1</mn></msup><mo>∘</mo><msup><mi>Sq</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">Sq^3 = Sq^1 \circ Sq^2</annotation></semantics></math>, the vanishing of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sq</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Sq_2</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mi>π</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^4(-,\pi^0)</annotation></semantics></math> then also implies the vanishing of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">d_3</annotation></semantics></math> on this entry.</p> </div> <h3 id="ForComplexOrientedCohomologyTheories">In complex oriented cohomology</h3> <p>Used for <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theories">complex oriented cohomology theories</a> and proof of <a class="existingWikiWord" href="/nlab/show/Quillen%27s+theorem+on+MU">Quillen's theorem on MU</a> via the <a class="existingWikiWord" href="/nlab/show/homology+of+MU">homology of MU</a> (…)</p> <p>(<a href="#Adams74">Adams 74, pages 60-62</a>, <a href="#Lurie10">Lurie 10, lecture 7</a>)</p> <h3 id="in_topological_modular_forms">In topological modular forms</h3> <p>See <em><a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism+in+tmf">Boardman homomorphism in tmf</a></em></p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hopf+degree+theorem">Hopf degree theorem</a></li> </ul> <h2 id="references">References</h2> <p>According to <a href="#Hunton95">Hunton 95</a>, the concept was introduced, in print, in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Michael+Boardman">John Michael Boardman</a>, <em>The eightfold way to BP-operations</em>, in: <em>Current trends in algebraic topology</em>, pp. 187–226, Canadian Mathematical Society Proceedings, 2, Part 1. Providence 1982 (<a href="https://bookstore.ams.org/cmsams-2">ISBN:978-0-8218-6003-8</a>)</li> </ul> <p>Further discussion:</p> <ul> <li id="Adams74"> <p><a class="existingWikiWord" href="/nlab/show/John+Frank+Adams">John Frank Adams</a>, part II, section 6 of: <em><a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalised+homology">Stable homotopy and generalised homology</a></em> (1974)</p> </li> <li id="Hunton95"> <p>John Hunton, <em>The Boardman homomorphism</em>, Contemporary Mathematics 181, 251-251, 1995 (<a href="https://books.google.ae/books?hl=en&lr=&id=hcYaCAAAQBAJ&oi=fnd&pg=PA251&dq=John+Hunton,+The+Boardman+homomorphism&ots=_ZI5SEcU28&sig=cK1Xd7AZwZP7WeTiy4dGv5v9BJk&redir_esc=y#v=onepage&q=John%20Hunton%2C%20The%20Boardman%20homomorphism&f=false">GoogleBooks</a>)</p> </li> <li id="Kochmann96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochmann">Stanley Kochmann</a>, section 4.3 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> <li id="Arlettaz04"> <p><a class="existingWikiWord" href="/nlab/show/Dominique+Arlettaz">Dominique Arlettaz</a>, <em>The generalized Boardman homomorphisms</em>, Central European Journal of Mathematics March 2004, Volume 2, Issue 1, pp 50-56 (<a href="https://doi.org/10.2478/BF02475949">doi:10.2478/BF02475949</a>)</p> </li> <li id="Lurie10"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, lecture 7 of <em><a class="existingWikiWord" href="/nlab/show/Chromatic+Homotopy+Theory">Chromatic Homotopy Theory</a></em>, 2010, (<a href="http://www.math.harvard.edu/~lurie/252xnotes/Lecture7.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Akhil+Mathew">Akhil Mathew</a>, <em>Torsion exponents in stable homotopy and the Hurewicz homomorphism</em>, Algebr. Geom. Topol. 16 (2016) 1025-1041 (<a href="https://arxiv.org/abs/1501.07561">arXiv:1501.07561</a>)</p> </li> <li> <p>Hadi Zare, <em>On the image of the unstable Boardman map</em> (<a href="https://arxiv.org/abs/1806.07079">arXiv:1806.07079</a>)</p> </li> </ul> <p>On the <a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman homomorphism</a> (generalized <a class="existingWikiWord" href="/nlab/show/Hurewicz+homomorphism">Hurewicz homomorphism</a>) to <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mark+Behrens">Mark Behrens</a>, <a class="existingWikiWord" href="/nlab/show/Mark+Mahowald">Mark Mahowald</a>, J.D. Quigley, <em>The 2-primary Hurewicz image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>tmf</mi></mrow><annotation encoding="application/x-tex">tmf</annotation></semantics></math></em> (<a href="https://arxiv.org/abs/2011.08956">arXiv:2011.08956</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 9, 2021 at 11:30:13. 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