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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> stable cohomotopy </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/8970/#Item_21" 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/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> <h4 id="spheres">Spheres</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unit+sphere">unit sphere</a>, <a class="existingWikiWord" href="/nlab/show/polar+coordinates">polar coordinates</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/round+sphere">round sphere</a>, <a class="existingWikiWord" href="/nlab/show/squashed+sphere">squashed sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hemisphere">hemisphere</a>, <a class="existingWikiWord" href="/nlab/show/equator">equator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stereographic+projection">stereographic projection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+spheres">homotopy groups of spheres</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+sphere">homotopy sphere</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+sphere">rational homotopy sphere</a>, <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+fibration">spherical fibration</a>, <a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a>, <a class="existingWikiWord" href="/nlab/show/stable+Cohomotopy+theory">stable Cohomotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology+sphere">homology sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+sphere">rational homotopy sphere</a>, <a class="existingWikiWord" href="/nlab/show/rational+n-sphere">rational n-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+sphere">motivic sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+actions+on+spheres">group actions on spheres</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+sphere">representation sphere</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+Cohomotopy">equivariant Cohomotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeb+sphere+theorem">Reeb sphere theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+packing">sphere packing</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Music+of+the+Spheres">Music of the Spheres</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/low+dimensional+topology">low dimensional</a> <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+projective+space">real projective space</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℝ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{R}P^1</annotation></semantics></math></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+projective+line">complex projective line</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{C}P^1</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+the+2-sphere">geometric quantization of the 2-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fuzzy+2-sphere">fuzzy 2-sphere</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fuzzy+3-sphere">fuzzy 3-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quaternionic+projective+line">quaternionic projective line</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℍ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{H}P^1</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fuzzy+4-sphere">fuzzy 4-sphere</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/5-sphere">5-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6-sphere">6-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/exotic+7-sphere">exotic 7-sphere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/8-sphere">8-sphere</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/octonionic+projective+line">octonionic projective line</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>𝕆</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\,\mathbb{O}P^1</annotation></semantics></math></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/13-sphere">13-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/15-sphere">15-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite-dimensional+sphere">infinite-dimensional sphere</a></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='#properties'>Properties</a></li> <ul> <li><a href='#AsAlgebraicKTheoryOverTheFieldWithOneElement'>As algebraic K-theory over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_1</annotation></semantics></math></a></li> <li><a href='#the_third_stable_framed_bordism_group'>The third stable framed bordism group</a></li> <li><a href='#kahnpriddy_theorem'>Kahn-Priddy theorem</a></li> <li><a href='#boardman_homomorphisms'>Boardman homomorphisms</a></li> <ul> <li><a href='#to_ordinary_cohomology'>To ordinary cohomology</a></li> <li><a href='#to_topological_modular_forms'>To topological modular forms</a></li> </ul> </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>The <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a> which is <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">represented</a> by the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a> is also called <em>stable cohomotopy</em>, as it is the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> version of <a class="existingWikiWord" href="/nlab/show/cohomotopy">cohomotopy</a>.</p> <p>Equivalently, it is the cohomological <a class="existingWikiWord" href="/nlab/show/duality">dual</a> concept to <a class="existingWikiWord" href="/nlab/show/stable+homotopy+homology+theory">stable homotopy homology theory</a>.</p> <p>By the <a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+theorem">Pontryagin-Thom theorem</a> this is equivalently <a class="existingWikiWord" href="/nlab/show/framed+manifold">framed</a> <a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a>.</p> <h2 id="properties">Properties</h2> <h3 id="AsAlgebraicKTheoryOverTheFieldWithOneElement">As algebraic K-theory over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_1</annotation></semantics></math></h3> <p>The following is known as the <em>Barratt-Priddy-Quillen theorem</em>:</p> <div class="num_prop" id="StableCohomotopyIsKTheoryOfFinSet"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/stable+cohomotopy">stable cohomotopy</a> is K-theory of <a class="existingWikiWord" href="/nlab/show/FinSet">FinSet</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/FinSet">FinSet</a> be a <a class="existingWikiWord" href="/nlab/show/skeleton">skeleton</a> of the category of <a class="existingWikiWord" href="/nlab/show/finite+sets">finite sets</a>, regarded as a <a class="existingWikiWord" href="/nlab/show/permutative+category">permutative category</a>. Then the <a class="existingWikiWord" href="/nlab/show/K-theory+of+a+permutative+category">K-theory of this permutative category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>FinSet</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>𝕊</mi></mrow><annotation encoding="application/x-tex"> K(FinSet) \;\simeq\; \mathbb{S} </annotation></semantics></math></div> <p>is represented by the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a>, hence is stable cohomotopy.</p> </div> <p>This is due to <a href="#BarrattPriddy72">Barratt-Priddy 72</a> reproved in <a href="#Segal74">Segal 74, Prop. 3.5</a>. See also <a href="#Priddy73">Priddy 73</a>, <a href="#Glasman13">Glasman 13</a>.</p> <div class="num_remark" id="StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement"> <h6 id="remark">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/stable+cohomotopy">stable cohomotopy</a> as <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a> over the <a class="existingWikiWord" href="/nlab/show/field+with+one+element">field with one element</a>)</strong></p> <p>Notice that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/field">field</a>, the <a class="existingWikiWord" href="/nlab/show/K-theory+of+a+permutative+category">K-theory of a permutative category</a> of its <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">F Mod</annotation></semantics></math> is its <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">K F</annotation></semantics></math> (see <a href="K-theory+of+a+permutative+category#OrdinaryAlgebraicKTheoryFromPermutativeCategoryOfProjectiveModules">this example</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mi>F</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>K</mi><mo stretchy="false">(</mo><mi>F</mi><mi>Mod</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K F \;\simeq\; K(F Mod) \,. </annotation></semantics></math></div> <p>Now (<a class="existingWikiWord" href="/nlab/show/pointed+sets">pointed</a>) <a class="existingWikiWord" href="/nlab/show/finite+sets">finite sets</a> may be regarded as the modules over the “<a class="existingWikiWord" href="/nlab/show/field+with+one+element">field with one element</a>” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_1</annotation></semantics></math> (see <a href="field+with+one+element#Modules">there</a>):</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> <mn>1</mn></msub><mi>Mod</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msup><mi>FinSet</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> \mathbb{F}_1 Mod \;=\; FinSet^{\ast/} </annotation></semantics></math></div> <p>If this is understood, example <a class="maruku-ref" href="#StableCohomotopyIsKTheoryOfFinSet"></a> says that <a class="existingWikiWord" href="/nlab/show/stable+cohomotopy">stable cohomotopy</a> is the algebraic K-theory of the <a class="existingWikiWord" href="/nlab/show/field+with+one+element">field with one element</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>K</mi><msub><mi>𝔽</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{S} \;\simeq\; K \mathbb{F}_1 \,. </annotation></semantics></math></div></div> <p>This perspective is highlighted in: <a href="#Deitmar06">Deitmar 06, p. 2</a>; <a href="#Guillot06">Guillot 06</a>; <a href="#Mahanta17">Mahanta 17</a>; <a href="#DundasGoodwillieMcCarthy13">Dundas, Goodwillie* McCarthy 13, II 1.2</a>; <a href="#MoravaSomeBackground">Morava</a>, <a href="#ConnesConsani16">Connes & Consani 16</a> and fully explicitly in <a href="#ChuLorscheidSanthanam10">Chu, Lorscheid & Santhanam 10, Thm. 5.9</a> and <a href="#BeardsleyNakamura24">Beardsley & Nakamura 2024, Cor. 2.25</a>. (<a href="#ChuLorscheidSanthanam10">Chu et al.</a> also generalize to <a class="existingWikiWord" href="/nlab/show/equivariant+stable+Cohomotopy">equivariant stable Cohomotopy</a> and <a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a>.)</p> <div> <table><thead><tr><th>(<a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant</a>) <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">cohomology</a></th><th><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">representing</a> <br /> <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></th><th><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> <br /> of the <a class="existingWikiWord" href="/nlab/show/point">point</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">cohomology</a> <br /> of <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">B G</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+ordinary+cohomology">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">HZ</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Borel+equivariant+cohomology">Borel equivariance</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>G</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KU">KU</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>KU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>R</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">KU_G(\ast) \simeq R_{\mathbb{C}}(G)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah-Segal+completion+theorem">Atiyah-Segal completion theorem</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>KU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mtext>compl.</mtext></mover><mover><mrow><msub><mi>KU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mo>≃</mo><mi>KU</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+complex+cobordism+cohomology+theory">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology">complex cobordism cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/MU">MU</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>MU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">MU_G(\ast)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/completion+theorem+for+complex+cobordism+cohomology">completion theorem for complex cobordism cohomology</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>MU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mtext>compl.</mtext></mover><mover><mrow><msub><mi>MU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mo>≃</mo><mi>MU</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+algebraic+K-theory">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">K \mathbb{F}_p</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>K</mi><msub><mi>𝔽</mi> <mi>p</mi></msub><msub><mo stretchy="false">)</mo> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>R</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Rector+completion+theorem">Rector completion theorem</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mrow><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mi>p</mi></msub><msub><mo stretchy="false">)</mo> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mtext>compl.</mtext></mover><mover><mrow><mo stretchy="false">(</mo><mi>K</mi><msub><mi>𝔽</mi> <mi>p</mi></msub><msub><mo stretchy="false">)</mo> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a></mtext></mover><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi>K</mi><msub><mi>𝔽</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G) </annotation></semantics></math> <br /></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+stable+cohomotopy">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/stable+cohomotopy">stable cohomotopy</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><msub><mi>𝔽</mi> <mn>1</mn></msub><mover><mo>≃</mo><mtext><a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a></mtext></mover></mrow><annotation encoding="application/x-tex">K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq} </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/equivariant+sphere+spectrum">S</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Burnside+ring">Burnside ring</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝕊</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>A</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{S}_G(\ast) \simeq A(G)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Segal-Carlsson+completion+theorem">Segal-Carlsson completion theorem</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a></mtext></mover><msub><mi>𝕊</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mtext>compl.</mtext></mover><mover><mrow><msub><mi>𝕊</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a></mtext></mover><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi>𝕊</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G) </annotation></semantics></math> <br /></td></tr> </tbody></table> </div> <h3 id="the_third_stable_framed_bordism_group">The third stable framed bordism group</h3> <p>The <a class="existingWikiWord" href="/nlab/show/third+stable+homotopy+group+of+spheres">third stable homotopy group of spheres</a> is the <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> of <a class="existingWikiWord" href="/nlab/show/order+of+a+group">order</a> 24:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>π</mi> <mn>3</mn> <mi>s</mi></msubsup></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>ℤ</mi><mo stretchy="false">/</mo><mn>24</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msub><mi>h</mi> <mi>ℍ</mi></msub><mo stretchy="false">]</mo></mtd> <mtd><mo>↔</mo></mtd> <mtd><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \pi_3^s &\simeq& \mathbb{Z}/24 \\ [h_{\mathbb{H}}] &\leftrightarrow& [1] } </annotation></semantics></math></div> <p>where the generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>∈</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>24</mn></mrow><annotation encoding="application/x-tex">[1] \in \mathbb{Z}/24</annotation></semantics></math> is represented by the <a class="existingWikiWord" href="/nlab/show/quaternionic+Hopf+fibration">quaternionic Hopf fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>7</mn></msup><mover><mo>⟶</mo><mrow><msub><mi>h</mi> <mi>ℍ</mi></msub></mrow></mover><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4</annotation></semantics></math>.</p> <p>Under the <a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+isomorphism">Pontrjagin-Thom isomorphism</a>, identifying the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a> with the <a class="existingWikiWord" href="/nlab/show/bordism+ring">bordism ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mo>•</mo> <mi>fr</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^{fr}_\bullet</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/stable+framing">stably framed</a> manifolds (see at <em><a class="existingWikiWord" href="/nlab/show/MFr">MFr</a></em>), this generator is represented by the <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a> (with its left-invariant framing induced from the identification with the <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <a class="existingWikiWord" href="/nlab/show/SU%282%29">SU(2)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Sp%281%29">Sp(1)</a> )</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>π</mi> <mn>3</mn> <mi>s</mi></msubsup></mtd> <mtd><mo>≃</mo></mtd> <mtd><msubsup><mi>Ω</mi> <mn>3</mn> <mi>fr</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msub><mi>h</mi> <mi>ℍ</mi></msub><mo stretchy="false">]</mo></mtd> <mtd><mo>↔</mo></mtd> <mtd><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>3</mn></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \pi_3^s & \simeq & \Omega_3^{fr} \\ [h_{\mathbb{H}}] & \leftrightarrow & [S^3] \,. } </annotation></semantics></math></div> <p>Moreover, the relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mn>4</mn><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>3</mn></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><mn>0</mn></mrow><annotation encoding="application/x-tex">2 4 [S^3] \,\simeq\, 0</annotation></semantics></math> is represented by the <a class="existingWikiWord" href="/nlab/show/complement">complement</a> of 24 <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a> inside <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <a class="existingWikiWord" href="/nlab/show/K3">K3</a>-manifold (<a href="https://mathoverflow.net/a/44885/381">MO:a/44885/381</a>, <a href="https://mathoverflow.net/a/218053/381">MO:a/218053/381</a>).</p> <h3 id="kahnpriddy_theorem">Kahn-Priddy theorem</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Kahn-Priddy+theorem">Kahn-Priddy theorem</a> characterizes a comparison map between stable cohomotopy and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the infinite <a class="existingWikiWord" href="/nlab/show/real+projective+space">real projective space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><msup><mi>P</mi> <mn>∞</mn></msup><mo>≃</mo><mi>B</mi><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{R}P^\infty \simeq B \mathbb{Z}/2</annotation></semantics></math>.</p> <h3 id="boardman_homomorphisms">Boardman homomorphisms</h3> <h4 id="to_ordinary_cohomology">To ordinary cohomology</h4> <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_2">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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(m-1)</annotation></semantics></math>-<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"><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> <h4 id="to_topological_modular_forms">To topological modular forms</h4> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi></mrow><annotation encoding="application/x-tex">\mathbb{S}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a> and <em><a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></em> for the <a class="existingWikiWord" href="/nlab/show/connective+spectrum">connective spectrum</a> of <a class="existingWikiWord" href="/nlab/show/topological+modular+forms">topological modular forms</a>.</p> <p>Since <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a> is an <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞</a><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, there is an essentially unique homomorphism of <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞</a><a class="existingWikiWord" href="/nlab/show/ring+spectra">ring spectra</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>tmf</mi></msub></mrow></mover><mi>tmf</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{S} \overset{e_{tmf}}{\longrightarrow} tmf \,. </annotation></semantics></math></div> <p>Regarded as a morphism of <a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a>-theories, this is called the <a class="existingWikiWord" href="/nlab/show/Hurewicz+homomorphism">Hurewicz homomorphism</a>, or rather the <a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman homomorphism</a> for <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></p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p><strong>(Boardman homomorphism in <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> is 6-connected)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism+in+tmf">Boardman homomorphism in tmf</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>tmf</mi></msub></mrow></mover><mi>tmf</mi></mrow><annotation encoding="application/x-tex"> \mathbb{S} \overset{e_{tmf}}{\longrightarrow} tmf </annotation></semantics></math></div> <p>induces an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> on <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups">stable homotopy groups</a> (hence from the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups+of+spheres">stable homotopy groups of spheres</a> to the stable homotopy groups of tmf), up to degree 6:</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> <mrow><mo>•</mo><mo>≤</mo><mn>6</mn></mrow></msub><mo stretchy="false">(</mo><mi>𝕊</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>π</mi> <mrow><mo>•</mo><mo>≤</mo><mn>6</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>tmf</mi></msub><mo stretchy="false">)</mo></mrow></munderover><msub><mi>π</mi> <mrow><mo>•</mo><mo>≤</mo><mn>6</mn></mrow></msub><mo stretchy="false">(</mo><mi>tmf</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_{\bullet \leq 6}(\mathbb{S}) \underoverset{\simeq}{\pi_{\bullet \leq 6}(e_{tmf})}{\longrightarrow} \pi_{\bullet\leq 6}(tmf) \,. </annotation></semantics></math></div></div> <p>(<a href="Boardman+homomorphism+in+tmf#Hopkins02">Hopkins 02, Prop. 4.6</a>, <a href="Boardman+homomorphism+in+tmf#DFHH14">DFHH 14, Ch. 13</a>)</p> <h2 id="related_concepts">Related concepts</h2> <div> <table><thead><tr><th>flavours of <br /><strong><a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a></strong> <br /><a class="existingWikiWord" href="/nlab/show/generalized+cohomology">cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> <br /> (<a class="existingWikiWord" href="/nlab/show/homotopy+theory">full</a> or <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>)</th><th><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> <br /> (<a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">full</a> or <a class="existingWikiWord" href="/nlab/show/rational+equivariant+homotopy+theory">rational</a>)</th></tr></thead><tbody><tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/non-abelian+cohomology">non-abelian cohomology</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a> <br /> (full or <a class="existingWikiWord" href="/nlab/show/rational+Cohomotopy">rational</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivariant+Cohomotopy">equivariant Cohomotopy</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></strong> <br /> (<strong><a class="existingWikiWord" href="/nlab/show/parametrized+homotopy+theory">full</a></strong> or <strong><a class="existingWikiWord" href="/nlab/show/rational+parameterized+stable+homotopy+theory">rational</a></strong>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a></td><td style="text-align: left;">twisted equivariant Cohomotopy</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">stable cohomology</a></strong> <br /> (<strong><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">full</a></strong> or <strong><a class="existingWikiWord" href="/nlab/show/rational+stable+homotopy+theory">rational</a></strong>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/stable+Cohomotopy">stable Cohomotopy</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivariant+stable+Cohomotopy">equivariant stable Cohomotopy</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+Cohomotopy">differential Cohomotopy</a></td><td style="text-align: left;">equivariant differential cohomotopy</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/persistent+homotopy">persistent cohomology</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/persistent+Cohomotopy">persistent Cohomotopy</a></td><td style="text-align: left;">persistent equivariant Cohomotopy</td></tr> </tbody></table> </div> <p><br /></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></p> </li> </ul> <p><br /></p> <div> <p><strong>flavors of <a class="existingWikiWord" href="/nlab/show/bordism+homology+theories">bordism homology theories</a>/<a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theories">cobordism cohomology theories</a>, their <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">representing</a> <a class="existingWikiWord" href="/nlab/show/Thom+spectra">Thom spectra</a> and <a class="existingWikiWord" href="/nlab/show/cobordism+rings">cobordism rings</a></strong>:</p> <p><a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/M%28B%2Cf%29">M(B,f)</a> (<a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/MFr">MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MO">MO</a>, <a class="existingWikiWord" href="/nlab/show/MSO">MSO</a>, <a class="existingWikiWord" href="/nlab/show/MSpin">MSpin</a>, <a class="existingWikiWord" href="/nlab/show/MString">MString</a>, …</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MU">MU</a>, <a class="existingWikiWord" href="/nlab/show/MSU">MSU</a>, …</p> <p><a class="existingWikiWord" href="/nlab/show/Ravenel%27s+spectrum">MΩΩSU(n)</a></p> <p><a class="existingWikiWord" href="/nlab/show/MP-theory">MP</a>, <a class="existingWikiWord" href="/nlab/show/MR-theory">MR</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSpin%5Ec">MSpin<sup><i>c</i></sup></a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSp">MSp</a></p> </li> </ul> <p>relative bordism theories:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/MOFr">MOFr</a>, <a class="existingWikiWord" href="/nlab/show/MUFr">MUFr</a>, <a class="existingWikiWord" href="/nlab/show/MSUFr">MSUFr</a></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bordism+homology+theory">equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MFr">equivariant MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MO">equivariant MO</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MU">equivariant MU</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+bordism+homology+theory">global equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mO">global equivariant mO</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mU">global equivariant mU</a></p> </li> </ul> <p>algebraic:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+cobordism">algebraic cobordism</a></li> </ul> </div> <h2 id="references">References</h2> <p>The concept of stable Cohomotopy as such:</p> <ul> <li id="Adams74"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, part III, section 6, p. 204 of: <em><a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalised+homology">Stable homotopy and generalised homology</a></em>, 1974</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Rognes">John Rognes</a>, p. 1 of: <em>The sphere spectrum</em>, 2004 (<a class="existingWikiWord" href="/nlab/files/RognesSphereSpectrum.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Discussion of stable Cohomotopy as <a class="existingWikiWord" href="/nlab/show/framed+manifold">framed</a> <a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a>:</p> <ul> <li id="Stong68"><a class="existingWikiWord" href="/nlab/show/Robert+Stong">Robert Stong</a>, Chapter IV, Example 1, p. 40 of <em>Notes on Cobordism theory</em>, Princeton University Press, 1968 (<a href="http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf">toc pdf</a>, <a href="http://press.princeton.edu/titles/6465.html">ISBN:9780691649016</a>)</li> </ul> <p>Discussion of stable Cohomotopy of <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a>:</p> <ul> <li> <p>C. T. Stretch, <em>Stable cohomotopy and cobordism of abelian groups</em>, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 90, Issue 2 September 1981, pp. 273-278 (<a href="https://doi.org/10.1017/S0305004100058734">doi:10.1017/S0305004100058734</a>)</p> </li> <li> <p>Ken-ichi Maruyama, <em><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>-invariants on the stable cohomotopy groups of Lie groups</em>, Osaka J. Math. Volume 25, Number 3 (1988), 581-589 (<a href="https://projecteuclid.org/euclid.ojm/1200780982">euclid:ojm/1200780982</a>)</p> </li> <li> <p>Sławomir Nowak, <em>Stable cohomotopy groups of compact spaces</em>, Fundamenta Mathematicae 180 (2003), 99-137 (<a href="https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/180/2">doi:10.4064/fm180-2-1</a>)</p> </li> </ul> <p>The identification of stable cohomotopy with the <a class="existingWikiWord" href="/nlab/show/K-theory+of+a+permutative+category">K-theory of the permutative category</a> of <a class="existingWikiWord" href="/nlab/show/finite+sets">finite sets</a> is due to</p> <ul> <li id="BarrattPriddy72"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Barratt">Michael Barratt</a>, <a class="existingWikiWord" href="/nlab/show/Stewart+Priddy">Stewart Priddy</a>, <em>On the homology of non-connected monoids and their associated groups</em>, Commentarii Mathematici Helvetici, <strong>47</strong> 1 (1972) 1–14 [<a href="https://link.springer.com/article/10.1007/BF02566785">doi:10.1007/BF02566785</a>, <a href="https://eudml.org/doc/139496">eudml:139496</a>]</p> </li> <li id="Segal74"> <p><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>Categories and cohomology theories</em>, Topology vol 13, pp. 293-312, 1974 (<a href="https://doi.org/10.1016/0040-9383(74)90022-6">doi:10.1016/0040-9383(74)90022-6</a>, <a href="http://ncatlab.org/nlab/files/SegalCategoriesAndCohomologyTheories.pdf">pdf</a>)</p> </li> </ul> <p>see also:</p> <ul> <li id="Priddy73"> <p><a class="existingWikiWord" href="/nlab/show/Stewart+Priddy">Stewart Priddy</a>, <em>Transfer, symmetric groups, and stable homotopy theory</em>, in <em>Higher K-Theories</em>, Springer, Berlin, Heidelberg, 1973. 244-255 (<a href="https://link.springer.com/content/pdf/10.1007/BFb0067060.pdf">pdf</a>)</p> </li> <li id="Glasman13"> <p><a class="existingWikiWord" href="/nlab/show/Saul+Glasman">Saul Glasman</a>, <em>The multiplicative Barratt-Priddy-Quillen theorem and beyond</em>, talk at <em>AMS Sectional Meeting</em> <strong>1095</strong> (2013) [<a href="https://www.ams.org/meetings/sectional/2199_program_ss7.html">webpage</a>, <a href="http://math.mit.edu/~sglasman/bpq-beamer.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pedro+Boavida+de+Brito">Pedro Boavida de Brito</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, Thm. 1.2 in: <em>Dendroidal spaces, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>-spaces and the special Barratt-Priddy-Quillen theorem</em>, Journal für die reine und angewandte Mathematik <strong>2020</strong> 760 (2020) 229-265 [<a href="https://arxiv.org/abs/1701.06459">arXiv:1701.06459</a>, <a href="https://doi.org/10.1515/crelle-2018-0002">doi:10.1515/crelle-2018-0002</a>]</p> </li> </ul> <p>The resulting interpretation of stable cohomotopy as <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a> over the <a class="existingWikiWord" href="/nlab/show/field+with+one+element">field with one element</a> is amplified in the following texts:</p> <ul> <li id="DundasGoodwillieMcCarthy13"> <p><a class="existingWikiWord" href="/nlab/show/Bj%C3%B8rn+Dundas">Bjørn Dundas</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Goodwillie">Thomas Goodwillie</a>, <a class="existingWikiWord" href="/nlab/show/Randy+McCarthy">Randy McCarthy</a>, chapter II, section 1.2 of <em>The local structure of algebraic K-theory</em>, Springer (2013) [<a href="https://doi.org/10.1007/978-1-4471-4393-2">doi:10.1007/978-1-4471-4393-2</a>]</p> </li> <li id="Deitmar06"> <p><a class="existingWikiWord" href="/nlab/show/Anton+Deitmar">Anton Deitmar</a>, <em>Remarks on zeta functions and K-theory over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_1</annotation></semantics></math></em> (<a href="https://arxiv.org/abs/math/0605429">arXiv:math/0605429</a>)</p> </li> <li id="Guillot06"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Guillot">Pierre Guillot</a>, <em>Adams operations in cohomotopy</em> (<a href="https://arxiv.org/abs/math/0612327">arXiv:0612327</a>)</p> </li> <li id="Mahanta17"> <p><a class="existingWikiWord" href="/nlab/show/Snigdhayan+Mahanta">Snigdhayan Mahanta</a>, <em>G-theory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_1</annotation></semantics></math>-algebras I: the equivariant Nishida problem</em>, J. Homotopy Relat. Struct. 12 (4), 901-930, 2017 (<a href="https://arxiv.org/abs/1110.6001">arXiv:1110.6001</a>)</p> </li> <li id="ChuLorscheidSanthanam10"> <p>Chenghao Chu, <a class="existingWikiWord" href="/nlab/show/Oliver+Lorscheid">Oliver Lorscheid</a>, Rekha Santhanam, <em>Sheaves and K-theory for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_1</annotation></semantics></math>-schemes</em>, Advances in Mathematics, <strong>229</strong> 4, (2012) 2239-2286 [<a href="https://arxiv.org/abs/1010.2896">arxiv:1010.2896</a>, <a href="https://doi.org/10.1016/j.aim.2011.12.023">doi:10.1016/j.aim.2011.12.023</a>]</p> </li> <li id="BeardsleyNakamura24"> <p><a class="existingWikiWord" href="/nlab/show/Jonathan+Beardsley">Jonathan Beardsley</a>, <a class="existingWikiWord" href="/nlab/show/So+Nakamura">So Nakamura</a>, <em>Projective Geometries and Simple Pointed Matroids as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_1</annotation></semantics></math>-modules</em> [<a href="https://arxiv.org/abs/2404.04730">arXiv:2404.04730</a>]</p> </li> </ul> <p>see also</p> <ul> <li id="MoravaSomeBackground"> <p><a class="existingWikiWord" href="/nlab/show/Jack+Morava">Jack Morava</a>, <em>Some background on Manin’s theorem <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∼</mo><mi>𝕊</mi></mrow><annotation encoding="application/x-tex">K(\mathbb{F}_1) \sim \mathbb{S}</annotation></semantics></math></em> (<a href="http://www.alainconnes.org/docs/Morava.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/MoravaSomeBackground.pdf" title="">MoravaSomeBackground.pdf</a>)</p> </li> <li id="ConnesConsani16"> <p><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <a class="existingWikiWord" href="/nlab/show/Caterina+Consani">Caterina Consani</a>, <em>Absolute algebra and Segal’s Gamma sets</em>, Journal of Number Theory 162 (2016): 518-551 (<a href="https://arxiv.org/abs/1502.05585">arXiv:1502.05585</a>)</p> </li> <li id="Berman18"> <p><a class="existingWikiWord" href="/nlab/show/John+D.+Berman">John D. Berman</a>, p. 92 of: <em>Categorified algebra and equivariant homotopy theory</em>, PhD thesis (2018) [<a href="https://arxiv.org/abs/1805.08745">arXiv:1805.08745</a>, <a href="http://www.people.virginia.edu/~jdb8pc/Thesis.pdf">pdf</a>]</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/Kahn-Priddy+theorem">Kahn-Priddy theorem</a> is due to</p> <ul> <li id="KahnPriddy72"><a class="existingWikiWord" href="/nlab/show/Daniel+Kahn">Daniel Kahn</a>, <a class="existingWikiWord" href="/nlab/show/Stewart+Priddy">Stewart Priddy</a>, <em>Applications of the transfer to stable homotopy theory</em>, Bull. Amer. Math. Soc. Volume 78, Number 6 (1972), 981-987 (<a href="https://projecteuclid.org/euclid.bams/1183534135">Euclid</a>)</li> </ul> <p>Discussion of stable Cohomotopy as <a class="existingWikiWord" href="/nlab/show/framed+manifold">framed</a> <a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a>:</p> <ul> <li id="ConnerFloyd66"><a class="existingWikiWord" href="/nlab/show/Pierre+Conner">Pierre Conner</a>, <a class="existingWikiWord" href="/nlab/show/Edwin+Floyd">Edwin Floyd</a>, Section 5 of: <em><a class="existingWikiWord" href="/nlab/show/The+Relation+of+Cobordism+to+K-Theories">The Relation of Cobordism to K-Theories</a></em>, Lecture Notes in Mathematics <strong>28</strong> Springer 1966 (<a href="https://link.springer.com/book/10.1007/BFb0071091">doi:10.1007/BFb0071091</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=216511">MR216511</a>)</li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/%CE%B2-rings">β-rings</a> is discussed in</p> <ul> <li> <p>E. Vallejo, <em>Polynomial operations from Burnside rings to representation functors</em>, J. Pure Appl. Algebra, 65 (1990), pp. 163–190.</p> </li> <li> <p>E. Vallejo, <em>Polynomial operations on stable cohomotopy</em>, Manuscripta Math., 67 (1990), pp. 345–365</p> </li> <li> <p>E. Vallejo, <em>The free β-ring on one generator, J. Pure Appl. Algebra, 86 (1993), pp. 95–108.</em></p> </li> <li> <p><a href="#Guillot06">Guillot 06</a></p> </li> </ul> <p>see also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jack+Morava">Jack Morava</a>, Rekha Santhanam, <em>Power operations and absolute geometry</em> (<a href="http://www.lemiller.net/media/slidesconf/AbsolutePower.pdf">pdf</a>)</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/Boardman+homomorphisms">Boardman homomorphisms</a> from stable cohomotopy is in</p> <ul> <li id="Arlettaz04"><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>)</li> </ul> <p>A lift of <a class="existingWikiWord" href="/nlab/show/Seiberg-Witten+invariants">Seiberg-Witten invariants</a> to classes in <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-<a class="existingWikiWord" href="/nlab/show/equivariant+stable+cohomotopy">equivariant stable cohomotopy</a> is discussed in</p> <ul> <li> <p><em>A stable cohomotopy refinement of Seiberg-Witten invariants: I</em> (<a href="http://arxiv.org/abs/math/0204340">arXiv:math/0204340</a>)</p> </li> <li> <p><em>A stable cohomotopy refinement of Seiberg-Witten invariants: II</em> (<a href="http://arxiv.org/abs/math/0204267">arXiv:math/0204267</a>)</p> </li> </ul> <p>On (<a class="existingWikiWord" href="/nlab/show/stable+cohomotopy">stable</a>) <a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic</a> <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a> of <a class="existingWikiWord" href="/nlab/show/schemes">schemes</a> (as <a class="existingWikiWord" href="/nlab/show/motivic+homotopy+theory">motivic homotopy classes</a> of maps into <a class="existingWikiWord" href="/nlab/show/motivic+sphere">motivic</a> <a class="existingWikiWord" href="/nlab/show/Tate+spheres">Tate spheres</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Aravind+Asok">Aravind Asok</a>, <a class="existingWikiWord" href="/nlab/show/Jean+Fasel">Jean Fasel</a>, <a class="existingWikiWord" href="/nlab/show/Mrinal+Kanti+Das">Mrinal Kanti Das</a>, <em>Euler class groups and motivic stable cohomotopy</em>, Journal of the EMS <strong>24</strong> 8 (2022) 2775–2822 [<a href="https://arxiv.org/abs/1601.05723">arXiv:1601.05723</a>, <a href="https://doi.org/10.4171/jems/1156">doi:10.4171/jems/1156</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Lerbet">Samuel Lerbet</a>, <em>Motivic stable cohomotopy and unimodular rows</em>, Advances in Mathematics <strong>436</strong> 109415 (2024) [<a href="https://arxiv.org/abs/2206.11688">arXiv:2206.11688</a>, <a href="https://doi.org/10.1016/j.aim.2023.109415">doi:10.1016/j.aim.2023.109415</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 8, 2024 at 18:18:32. 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