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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> cohomotopy charge map </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/10424/#Item_9" 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+Cobordism+and+Complex+Oriented+Cohomology">Introduction</a></em></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+transformation">coordinate transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atlas">atlas</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, ,<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite+dimensional+manifold">infinite dimensional manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Banach+manifold">Banach manifold</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+manifold">Hilbert manifold</a>, <a class="existingWikiWord" href="/nlab/show/ILH+manifold">ILH manifold</a>, <a class="existingWikiWord" href="/nlab/show/Frechet+manifold">Frechet manifold</a>, <a class="existingWikiWord" href="/nlab/show/convenient+manifold">convenient manifold</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Genera and invariants</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/signature+genus">signature genus</a>, <a class="existingWikiWord" href="/nlab/show/Kervaire+invariant">Kervaire invariant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-hat+genus">A-hat genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-manifolds">2-manifolds</a>/<a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of a surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+transversality+theorem">Thom's transversality theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galatius-Tillmann-Madsen-Weiss+theorem">Galatius-Tillmann-Madsen-Weiss theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometrization+conjecture">geometrization conjecture</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptization+conjecture">elliptization conjecture</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </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> <ul> <li><a href='#general'>General</a></li> <li><a href='#ForChargedPoints'>For charged points</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#characterization_of_cobordism_classes_by_their_cohomotopy_charge'>Characterization of cobordism classes by their Cohomotopy charge</a></li> <li><a href='#CharacterizationOfPointConfigurations'>Characterization of point configurations by their Cohomotopy charge</a></li> <ul> <li><a href='#on_euclidean_spaces_via_plain_cohomotopy'>On Euclidean spaces via plain Cohomotopy</a></li> <li><a href='#on_closed_manifolds_via_twisted_cohomotopy'>On closed manifolds via twisted Cohomotopy</a></li> </ul> </ul> <li><a href='#references'>References</a></li> <ul> <li><a href='#ReferencesPontrjaginThomConstruction'>Pontrjagin-Thom construction</a></li> <ul> <li><a href='#ReferencesPontrjaginConstruction'>Pontrjagin’s construction</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#TwistedEquivariantPontrjaginConstruction'>Twisted/equivariant generalizations</a></li> <li><a href='#InNegativeCodimension'>In negative codimension</a></li> </ul> <li><a href='#thoms_construction'>Thom’s construction</a></li> <li><a href='#lashofs_construction'>Lashof’s construction</a></li> </ul> <li><a href='#for_point_configurations'>For point configurations</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <h3 id="general">General</h3> <p>The <em>Cohomotopy charge map</em> (terminology due to <a href="#SS22IntBranes">SS22, Rem. 2.6</a>) is the <a class="existingWikiWord" href="/nlab/show/function">function</a> that assigns to a <a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration</a> of <a class="existingWikiWord" href="/nlab/show/normally+framed+submanifolds">normally framed submanifolds</a> of <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> their total <a class="existingWikiWord" href="/nlab/show/charge">charge</a> as measured in <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>-<a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a>-<a class="existingWikiWord" href="/nlab/show/generalized+cohomology">cohomology theory</a>.</p> <p>Concretely, this is the function which assigns to a <a class="existingWikiWord" href="/nlab/show/normally+framed+submanifold">normally framed submanifold</a> its <strong>asymptotic normal distance function</strong>, namely the <a class="existingWikiWord" href="/nlab/show/distance">distance</a> from the <a class="existingWikiWord" href="/nlab/show/submanifold">submanifold</a> measured</p> <ol> <li> <p>in <a class="existingWikiWord" href="/nlab/show/direction+vector">direction</a> <a class="existingWikiWord" href="/nlab/show/orthogonality">perpendicular</a> to the submanifold, as encoded by the <a class="existingWikiWord" href="/nlab/show/normal+framing">normal framing</a>;</p> </li> <li> <p>asymptotically, regarding all points outside a <a class="existingWikiWord" href="/nlab/show/tubular+neighbourhood">tubular neighbourhood</a> as being <a class="existingWikiWord" href="/nlab/show/one-point+compactification">at infinity</a>.</p> </li> </ol> <center> <a href="https://arxiv.org/pdf/1909.12277.pdf#page=24"> <img src="https://ncatlab.org/nlab/files/CohomotopyChargeAsymptoticDistanceII.jpg" width="630" /> </a> </center> <blockquote> <p>graphics grabbed from <a href="#SatiSchreiber19">SS 19</a></p> </blockquote> <p>For general <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> this is known as the “<a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+collapse+construction">Pontrjagin-Thom collapse construction</a>”.</p> <h3 id="ForChargedPoints">For charged points</h3> <p>For maximal <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> inside an <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">manifold</a>, hence for 0-dimensional submanifolds, hence for <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configurations of points</a> and with all points regarded as equipped with positive normal framing, the Cohomotopy charge map is alternatively known as the “electric field map” (<a href="#Salvatore01">Salvatore 01</a> following <a href="#Segal73">Segal 73, Section 1</a>, see also <a href="#Knudsen18">Knudsen 18, p. 49</a>) or the “scanning map” (<a href="#Kallel98">Kallel 98</a>, <a href="#ManthorpeTillmann13">Manthorpe-Tillmann 13</a>):</p> <p>In maximal <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> <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>, the <em>Cohomotopy charge map</em> is thus the <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a></p> <div class="maruku-equation" id="eq:CohomotopyChargeMapOnEuclideanSpace"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Conf</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>ℝ</mi> <mi>D</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>⟶</mo><mi>cc</mi></mover><msup><mstyle mathvariant="bold"><mi>π</mi></mstyle> <mi>D</mi></msup><mo maxsize="1.8em" minsize="1.8em">(</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>ℝ</mi> <mi>D</mi></msup><msup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>cpt</mi></msup><mo maxsize="1.8em" minsize="1.8em">)</mo><mo>=</mo><msup><mi>Maps</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo><mspace width="negativethinmathspace"></mspace></mrow></msup><mo maxsize="1.8em" minsize="1.8em">(</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>ℝ</mi> <mi>D</mi></msup><msup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>cpt</mi></msup><mo>,</mo><msup><mi>S</mi> <mi>D</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><msup><mi>Ω</mi> <mi>D</mi></msup><msup><mi>S</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex"> Conf\big( \mathbb{R}^D \big) \overset{cc}{\longrightarrow} \mathbf{\pi}^D \Big( \big( \mathbb{R}^D \big)^{cpt} \Big) = Maps^{\ast/\!}\Big( \big(\mathbb{R}^D\big)^{cpt} , S^D\big) = \Omega^{D} S^D </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configuration space of points</a> in the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^D</annotation></semantics></math> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a> <a class="existingWikiWord" href="/nlab/show/cocycle+space">cocycle space</a> <a class="existingWikiWord" href="/nlab/show/vanishing+at+infinity">vanishing at infinity</a> on the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a>(which is equivalently the <a class="existingWikiWord" href="/nlab/show/space+of+maps">space of pointed maps</a> from the <a class="existingWikiWord" href="/nlab/show/one-point+compactification">one-point compactification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>D</mi></msup><mo>≃</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>ℝ</mi> <mi>D</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">S^D \simeq \big( \mathbb{R}^D \big)</annotation></semantics></math> to itself, and hence equivalently the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/iterated+based+loop+space">iterated based loop space</a> of the <a class="existingWikiWord" href="/nlab/show/n-sphere">D-sphere</a>), which sends a configuration of points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^D</annotation></semantics></math>, each regarded as carrying unit <a class="existingWikiWord" href="/nlab/show/charge">charge</a> to their total <a class="existingWikiWord" href="/nlab/show/charge">charge</a> as measured in <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a>-<a class="existingWikiWord" href="/nlab/show/generalized+cohomology">cohomology theory</a> (<a href="#Segal73">Segal 73, Section 3</a>).</p> <center> <a href="https://arxiv.org/pdf/1909.12277.pdf#page=10"> <img src="https://ncatlab.org/schreiber/files/CohomotopyChargeOfPoints.jpg" width="700" /> </a> </center> <blockquote> <p>graphics grabbed from <a href="#SatiSchreiber19">SS 19</a></p> </blockquote> <p>(See also at <em><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> – <a href="cobordism#RelationToCohomotopy">Relation to Cohomotopy</a></em>.)</p> <p>This has evident generalizations to other manifolds than just Euclidean spaces, to spaces of labeled configurations and to <a class="existingWikiWord" href="/nlab/show/equivariant+Cohomotopy">equivariant Cohomotopy</a>. The following graphics illustrates the Cohomotopy charge map on <a class="existingWikiWord" href="/nlab/show/G-space">G-space</a> <a class="existingWikiWord" href="/nlab/show/tori">tori</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G = \mathbb{Z}_2</annotation></semantics></math> with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+Cohomotopy">equivariant Cohomotopy</a>:</p> <center> <a href="https://arxiv.org/pdf/1909.12277.pdf#page=24"> <img src="https://ncatlab.org/schreiber/files/EquivariantCohomotopyTadpoleCancellationN.jpg" width="700" /> </a> </center> <blockquote> <p>graphics grabbed from <a href="#SatiSchreiber19">SS 19</a></p> </blockquote> <h2 id="properties">Properties</h2> <h3 id="characterization_of_cobordism_classes_by_their_cohomotopy_charge">Characterization of cobordism classes by their Cohomotopy charge</h3> <p>The unstable <a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+theorem">Pontrjagin-Thom theorem</a> states that Cohomotopy charge faithfully reflects <a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configurations</a> of <a class="existingWikiWord" href="/nlab/show/normally+framed+submanifolds">normally framed submanifolds</a> up to normally framed embedded <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>, hence that the <a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+collapse+construction">Pontrjagin-Thom collapse construction</a> induces a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> between <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of <a class="existingWikiWord" href="/nlab/show/normally+framed+submanifolds">normally framed submanifolds</a> and the <a class="existingWikiWord" href="/nlab/show/Cohomotopy+set">Cohomotopy set</a> in degree the respective <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mfrac linethickness="0"><mrow><mtext>normally framed submanifolds</mtext></mrow><mrow><mrow><mtext>in</mtext><mspace width="thickmathspace"></mspace><mi>X</mi><mspace width="thickmathspace"></mspace><mtext>of codimension</mtext><mspace width="thickmathspace"></mspace><mi>n</mi></mrow></mrow></mfrac><mo>}</mo></mrow><msub><mo maxsize="1.8em" minsize="1.8em">/</mo> <mrow><msub><mo>∼</mo> <mi>cobordism</mi></msub></mrow></msub><munderover><mo>⟶</mo><mo>≃</mo><mi>cc</mi></munderover><munder><mrow><msup><mi>π</mi> <mi>n</mi></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mpadded width="0" lspace="-50%width"><mstyle mathcolor="blue"><mfrac linethickness="0"><mrow><mtext>Cohomotopy</mtext></mrow><mrow><mtext>set</mtext></mrow></mfrac></mstyle></mpadded></munder></mrow><annotation encoding="application/x-tex"> \left\{ { { \text{normally framed submanifolds} } \atop { \text{in}\;X\;\text{of codimension}\; n } } \right\} \Big/_{\sim_{cobordism}} \underoverset{\simeq}{cc}{\longrightarrow} \underset{ \mathclap{ \color{blue} { \text{Cohomotopy} \atop \text{set} } } }{ \pi^n\big( X \big) } </annotation></semantics></math></div> <p>For more details see <a href="cohomotopy#RelationToCobordismGroup">here</a>.</p> <p>In good situations this <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> of <a class="existingWikiWord" href="/nlab/show/sets">sets</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> enhances to a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> of <a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration spaces</a>/<a class="existingWikiWord" href="/nlab/show/cocycle+spaces">cocycle spaces</a>. See <em><a href="#CharacterizationOfPointConfigurations">Characterization of point configurations by their Cohomotopy charge</a></em> below.</p> <p><br /></p> <h3 id="CharacterizationOfPointConfigurations">Characterization of point configurations by their Cohomotopy charge</h3> <p>In some situations the <a class="existingWikiWord" href="/nlab/show/Cohomotopy+charge+map">Cohomotopy charge map</a> is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> and hence exhibits, for all purposes of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, the <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a> <a class="existingWikiWord" href="/nlab/show/cocycle+space">cocycle space</a> of Cohomotopy charges as an equivalent reflection of the <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configuration space of points</a>.</p> <h4 id="on_euclidean_spaces_via_plain_cohomotopy">On Euclidean spaces via plain Cohomotopy</h4> <div class="num_prop" id="GroupCompletionOfConfigurationSpaceIsIteratedBasedLoopSpace"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a> on <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configuration space of points</a> is <a class="existingWikiWord" href="/nlab/show/iterated+based+loop+space">iterated based loop space</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/Cohomotopy+charge+map">Cohomotopy charge map</a> <a class="maruku-eqref" href="#eq:CohomotopyChargeMapOnEuclideanSpace">(1)</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Conf</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>ℝ</mi> <mi>D</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>⟶</mo><mi>cc</mi></mover><msup><mi>Ω</mi> <mi>D</mi></msup><msup><mi>S</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex"> Conf \big( \mathbb{R}^D \big) \overset{ cc }{\longrightarrow} \Omega^D S^D </annotation></semantics></math></div> <p>from the full unordered and unlabeled configuration space (<a href="configuration+space+of+points#eq:UnorderedUnlabeledConfigurationSpace">here</a>) of <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^D</annotation></semantics></math> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/iterated+based+loop+space">iterated based loop space</a> of the <a class="existingWikiWord" href="/nlab/show/n-sphere">D-sphere</a>, exhibits the <a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a> (<a href="configuration+space+of+points#eq:GroupCompletionOfConfigurationSpaceMonoid">here</a>) of the configuration space monoid</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><msub><mi>B</mi> <mrow><msub><mrow></mrow> <mo>⊔</mo></msub><mspace width="negativethinmathspace"></mspace></mrow></msub><mi>Conf</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>ℝ</mi> <mi>D</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>Ω</mi> <mi>D</mi></msup><msup><mi>S</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex"> \Omega B_{{}_{\sqcup}\!} Conf \big( \mathbb{R}^D \big) \overset{ \simeq }{\longrightarrow} \Omega^D S^D </annotation></semantics></math></div></div> <p>(<a href="#Segal73">Segal 73, Theorem 1</a>)</p> <div class="num_prop" id="CohomotopyChargeMapIsEquivalenceOnSPhereLabeledConfihgurationSpace"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Cohomotopy+charge+map">Cohomotopy charge map</a> is <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> on sphere-labeled <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configuration space of points</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>,</mo><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">D, k \in \mathbb{N}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k \geq 1</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/Cohomotopy+charge+map">Cohomotopy charge map</a> <a class="maruku-eqref" href="#eq:CohomotopyChargeMapOnEuclideanSpace">(1)</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Conf</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>ℝ</mi> <mi>D</mi></msup><mo>,</mo><msup><mi>S</mi> <mi>k</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><munderover><mo>⟶</mo><mo>≃</mo><mi>cc</mi></munderover><msup><mi>Ω</mi> <mi>D</mi></msup><msup><mi>S</mi> <mrow><mi>D</mi><mo>+</mo><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> Conf \big( \mathbb{R}^D, S^k \big) \underoverset{\simeq}{cc}{\longrightarrow} \Omega^D S^{D + k} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> from the configuration space (<a href="configuration+space+of+points#eq:UnorderedLabeledCOnfigurationSpace">here</a>) of unordered points with labels in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">S^k</annotation></semantics></math> and vanishing at the base point of the label space to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/iterated+loop+space">iterated loop space</a> of the <a class="existingWikiWord" href="/nlab/show/n-sphere">D+k-sphere</a>.</p> </div> <p>(<a href="#Segal73">Segal 73, Theorem 3</a>)</p> <h4 id="on_closed_manifolds_via_twisted_cohomotopy">On closed manifolds via twisted Cohomotopy</h4> <p>The May-Segal theorem <a class="maruku-ref" href="#ScanningMapEquivalenceOverCartesianSpace"></a> generalizes from <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> to <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> if at the same time one passes from plain <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a> to <a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a>, twisted, via the <a class="existingWikiWord" href="/nlab/show/J-homomorphism">J-homomorphism</a>, by the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>:</p> <div class="num_prop" id="ScanningMapEquivalenceOverClosedManifold"> <h6 id="proposition_3">Proposition</h6> <p>Let</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X^n</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth</a> <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">1 \leq k \in \mathbb{N}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>.</p> </li> </ol> <p>Then the <a class="existingWikiWord" href="/nlab/show/Cohomotopy+charge+map">Cohomotopy charge map</a> constitutes a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mrow><msub><mi>Maps</mi> <mrow><msub><mrow></mrow> <mrow><mo stretchy="false">/</mo><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><msup><mi>X</mi> <mi>n</mi></msup><mspace width="thickmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><msup><mi>S</mi> <mrow><msub><mstyle mathvariant="bold"><mi>n</mi></mstyle> <mi>def</mi></msub><mo>+</mo><msub><mstyle mathvariant="bold"><mi>k</mi></mstyle> <mi mathvariant="normal">triv</mi></msub></mrow></msup><mspace width="negativethinmathspace"></mspace><mo>⫽</mo><mspace width="negativethinmathspace"></mspace><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><mstyle mathcolor="blue"><mfrac linethickness="0"><mrow><mphantom><mi>a</mi></mphantom></mrow><mrow><mtext> J-twisted Cohomotopy space</mtext></mrow></mfrac></mstyle></munder><munderover><mo>⟵</mo><mo>≃</mo><mstyle mathcolor="blue"><mtext>Cohomotopy charge map</mtext></mstyle></munderover><munder><mrow><mi>Conf</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>S</mi> <mi>k</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mpadded width="0" lspace="-50%width"><mstyle mathcolor="blue"><mfrac linethickness="0"><mrow><mphantom><mi>a</mi></mphantom></mrow><mrow><mfrac linethickness="0"><mrow><mtext>configuration space</mtext></mrow><mrow><mtext>of points</mtext></mrow></mfrac></mrow></mfrac></mstyle></mpadded></munder></mrow><annotation encoding="application/x-tex"> \underset{ \color{blue} { \phantom{a} \atop \text{ J-twisted Cohomotopy space}} }{ Maps_{{}_{/B O(n)}} \Big( X^n \;,\; S^{ \mathbf{n}_{def} + \mathbf{k}_{\mathrm{triv}} } \!\sslash\! O(n) \Big) } \underoverset {\simeq} { \color{blue} \text{Cohomotopy charge map} } {\longleftarrow} \underset{ \mathclap{ \color{blue} { \phantom{a} \atop { \text{configuration space} \atop \text{of points} } } } }{ Conf \big( X^n, S^k \big) } </annotation></semantics></math></div> <p>between</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/J-homomorphism">J</a>-<a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted (n+k)-Cohomotopy</a> space of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X^n</annotation></semantics></math>, hence the <a class="existingWikiWord" href="/nlab/show/space+of+sections">space of sections</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n + k)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/spherical+fibration">spherical fibration</a> over <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> which is <a class="existingWikiWord" href="/nlab/show/associated+fiber+bundle">associated</a> via the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> by the <a class="existingWikiWord" href="/nlab/show/O%28n%29">O(n)</a>-<a class="existingWikiWord" href="/nlab/show/action">action</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msup><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>ℝ</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})</annotation></semantics></math></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configuration space of points</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X^n</annotation></semantics></math> with labels in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">S^k</annotation></semantics></math>.</p> </li> </ol> </div> <p>(<a href="#Boedigheimer87">Bödigheimer 87, Prop. 2</a>, following <a href="#McDuff75">McDuff 75</a>)</p> <div class="num_prop" id="ScanningMapEquivalenceOverClosedFramedManifold"> <h6 id="remark">Remark</h6> <p>In the special case that the <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X^n</annotation></semantics></math> in Prop. <a class="maruku-ref" href="#ScanningMapEquivalenceOverClosedManifold"></a> is <a class="existingWikiWord" href="/nlab/show/parallelizable+manifold">parallelizable</a>, hence that its <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> is <a class="existingWikiWord" href="/nlab/show/trivial+bundle">trivializable</a>, the statement of Prop. <a class="maruku-ref" href="#ScanningMapEquivalenceOverClosedManifold"></a> reduces to this:</p> <p>Let</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X^n</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/parallelizable+manifold">parallelizable</a> <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">1 \leq k \in \mathbb{N}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>.</p> </li> </ol> <p>Then the <a class="existingWikiWord" href="/nlab/show/Cohomotopy+charge+map">Cohomotopy charge map</a> constitutes a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mrow><mi>Maps</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><msup><mi>X</mi> <mi>n</mi></msup><mspace width="thickmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><msup><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msup><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><mstyle mathcolor="blue"><mfrac linethickness="0"><mrow><mphantom><mi>a</mi></mphantom></mrow><mrow><mtext> Cohomotopy space</mtext></mrow></mfrac></mstyle></munder><munderover><mo>⟵</mo><mo>≃</mo><mstyle mathcolor="blue"><mtext>Cohomotopy charge map</mtext></mstyle></munderover><munder><mrow><mi>Conf</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>X</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>S</mi> <mi>k</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mpadded width="0" lspace="-50%width"><mstyle mathcolor="blue"><mfrac linethickness="0"><mrow><mphantom><mi>a</mi></mphantom></mrow><mrow><mfrac linethickness="0"><mrow><mtext>configuration space</mtext></mrow><mrow><mtext>of points</mtext></mrow></mfrac></mrow></mfrac></mstyle></mpadded></munder></mrow><annotation encoding="application/x-tex"> \underset{ \color{blue} { \phantom{a} \atop \text{ Cohomotopy space}} }{ Maps \Big( X^n \;,\; S^{ n + k } \Big) } \underoverset {\simeq} { \color{blue} \text{Cohomotopy charge map} } {\longleftarrow} \underset{ \mathclap{ \color{blue} { \phantom{a} \atop { \text{configuration space} \atop \text{of points} } } } }{ Conf \big( X^n, S^k \big) } </annotation></semantics></math></div> <p>between</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+k)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a> space of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X^n</annotation></semantics></math>, hence the <a class="existingWikiWord" href="/nlab/show/space+of+maps">space of maps</a> from <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> to the <a class="existingWikiWord" href="/nlab/show/n-sphere">(n+k)-sphere</a></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configuration space of points</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X^n</annotation></semantics></math> with labels in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">S^k</annotation></semantics></math>.</p> </li> </ol> </div> <h2 id="references">References</h2> <div> <h3 id="ReferencesPontrjaginThomConstruction">Pontrjagin-Thom construction</h3> <h4 id="ReferencesPontrjaginConstruction">Pontrjagin’s construction</h4> <h5 id="general">General</h5> <p>The <em><a class="existingWikiWord" href="/nlab/show/Pontryagin+theorem">Pontryagin theorem</a></em>, i.e. the unstable and <a class="existingWikiWord" href="/nlab/show/normally+framed+submanifold">framed</a> version of the <a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a>, identifying <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of <a class="existingWikiWord" href="/nlab/show/normally+framed+submanifolds">normally framed submanifolds</a> with their <a class="existingWikiWord" href="/nlab/show/Cohomotopy+charge">Cohomotopy charge</a> in unstable <a class="existingWikiWord" href="/nlab/show/Karol+Borsuk">Borsuk</a>-<a class="existingWikiWord" href="/nlab/show/Edwin+Spanier">Spanier</a> <a class="existingWikiWord" href="/nlab/show/Cohomotopy+sets">Cohomotopy sets</a>, is due to:</p> <ul> <li id="Pontryagin38a"> <p><a class="existingWikiWord" href="/nlab/show/Lev+Pontrjagin">Lev Pontrjagin</a>, <em><a class="existingWikiWord" href="/nlab/show/Classification+of+continuous+maps+of+a+complex+into+a+sphere">Classification of continuous maps of a complex into a sphere</a></em>, <em>Communication I</em>, Doklady Akademii Nauk SSSR <strong>19</strong> 3 (1938) 147-149</p> </li> <li id="Pontryagin50"> <p><a class="existingWikiWord" href="/nlab/show/Lev+Pontryagin">Lev Pontryagin</a>, <em><a class="existingWikiWord" href="/nlab/show/Homotopy+classification+of+mappings+of+an+%28n%2B2%29-dimensional+sphere+on+an+n-dimensional+one">Homotopy classification of mappings of an (n+2)-dimensional sphere on an n-dimensional one</a></em>, Doklady Akad. Nauk SSSR (N.S.) 19 (1950), 957–959 (<a href="https://www.maths.ed.ac.uk/~v1ranick/papers/pont3.pdf">pdf</a>)</p> </li> </ul> <p>(both available in English translation in <a href="Revaz+Gamkrelidze#Gamkrelidze86">Gamkrelidze 86</a>),</p> <p>as presented more comprehensively in:</p> <ul> <li id="Pontryagin55"><a class="existingWikiWord" href="/nlab/show/Lev+Pontrjagin">Lev Pontrjagin</a>, <em><a class="existingWikiWord" href="/nlab/show/Smooth+manifolds+and+their+applications+in+Homotopy+theory">Smooth manifolds and their applications in Homotopy theory</a></em>, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (<a href="https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001">doi:10.1142/9789812772107_0001</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/pont001.pdf">pdf</a>)</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/Pontrjagin+theorem">Pontrjagin theorem</a> must have been known to Pontrjagin at least by 1936, when he announced the computation of the <a class="existingWikiWord" href="/nlab/show/second+stable+homotopy+group+of+spheres">second stem of homotopy groups of spheres</a>:</p> <ul> <li id="Pontrjagin36"><a class="existingWikiWord" href="/nlab/show/Lev+Pontrjagin">Lev Pontrjagin</a>, <em>Sur les transformations des sphères en sphères</em> (<a class="existingWikiWord" href="/nlab/files/PontrjaginSurLesTransformationDesSpheres.pdf" title="pdf">pdf</a>) in: <em>Comptes Rendus du Congrès International des Mathématiques – Oslo 1936</em> (<a href="https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1936.2/ICM1936.2.ocr.pdf">pdf</a>)</li> </ul> <p>Review:</p> <ul> <li id="FreedUhlenbeck91"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Karen+Uhlenbeck">Karen Uhlenbeck</a>, Appendix B of: <em>Instantons and Four-Manifolds</em>, Mathematical Sciences Research Institute Publications, Springer 1991 (<a href="https://link.springer.com/book/10.1007/978-1-4613-9703-8">doi:10.1007/978-1-4613-9703-8</a>)</p> </li> <li id="Bredon93"> <p><a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a>, chapter II.16 of: <em>Topology and Geometry</em>, Graduate Texts in Mathematics <strong>139</strong>, Springer (1993) [<a href="https://link.springer.com/book/10.1007/978-1-4757-6848-0">doi:10.1007/978-1-4757-6848-0</a>, <a href="http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf">pdf</a>]</p> </li> </ul> <div style="margin: -30px 0px 20px 10px"> <img src="/nlab/files/Bredon-FigII13-PontryaginThom.jpg" width="450px" /> </div> <ul> <li id="Kosinski93"> <p><a class="existingWikiWord" href="/nlab/show/Antoni+Kosinski">Antoni Kosinski</a>, chapter IX of: <em>Differential manifolds</em>, Academic Press (1993) [<a href="http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf">pdf</a>, <a href="https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/138/suppl/C">ISBN:978-0-12-421850-5</a>]</p> </li> <li id="Milnor97"> <p><a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, Chapter 7 of: <em>Topology from the differentiable viewpoint</em>, Princeton University Press, 1997. (<a href="https://press.princeton.edu/books/paperback/9780691048338/topology-from-the-differentiable-viewpoint">ISBN:9780691048338</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/milnortop.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mladen+Bestvina">Mladen Bestvina</a> (notes by <a class="existingWikiWord" href="/nlab/show/Adam+Keenan">Adam Keenan</a>), Chapter 16 in: <em>Differentiable Topology and Geometry</em>, 2002 (<a class="existingWikiWord" href="/nlab/files/BestvinaKeenanDifferentialTopology.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michel+Kervaire">Michel Kervaire</a>, <em>La méthode de Pontryagin pour la classification des applications sur une sphère</em>, in: E. Vesentini (ed.), <em>Topologia Differenziale</em>, CIME Summer Schools, vol. 26, Springer 2011 (<a href="https://doi.org/10.1007/978-3-642-10988-1_3">doi:10.1007/978-3-642-10988-1_3</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Rustam+Sadykov">Rustam Sadykov</a>, Section 1 of: <em>Elements of Surgery Theory</em>, 2013 (<a href="https://www.math.ksu.edu/~sadykov/Lecture%20Notes/Surgery%20Theory.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/SadykovSurgeryTheory.pdf" title="pdf">pdf</a>)</p> </li> <li id="Csepai20"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A1s+Cs%C3%A9pai">András Csépai</a>, <em>Stable Pontryagin-Thom construction for proper maps</em>, Period Math Hung 80, 259–268 (2020) (<a href="https://arxiv.org/abs/1905.07734">arXiv:1905.07734</a>, <a href="https://doi.org/10.1007/s10998-020-00327-0">doi:10.1007/s10998-020-00327-0</a>)</p> </li> </ul> <p>Discussion of the early history:</p> <ul> <li><a href="#Kosinski93">Kosinski 93, Section IX.9</a></li> </ul> <h5 id="TwistedEquivariantPontrjaginConstruction">Twisted/equivariant generalizations</h5> <p>The (fairly straightforward) generalization of the <a class="existingWikiWord" href="/nlab/show/Pontrjagin+theorem">Pontrjagin theorem</a> to the <a class="existingWikiWord" href="/nlab/show/twisted+Pontrjagin+theorem">twisted Pontrjagin theorem</a>, identifying <a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a> with <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of <a class="existingWikiWord" href="/nlab/show/normally+twisted-framed+submanifolds">normally twisted-framed submanifolds</a>, is made explicit in:</p> <ul> <li id="Cruickshank03"><a class="existingWikiWord" href="/nlab/show/James+Cruickshank">James Cruickshank</a>, Lemma 5.2 using Sec. 5.1 in: <em>Twisted homotopy theory and the geometric equivariant 1-stem</em>, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (<a href="https://doi.org/10.1016/S0166-8641(02)00183-9">doi:10.1016/S0166-8641(02)00183-9</a>)</li> </ul> <p>A general <a class="existingWikiWord" href="/nlab/show/equivariant+Pontrjagin+theorem">equivariant Pontrjagin theorem</a> – relating <a class="existingWikiWord" href="/nlab/show/equivariant+Cohomotopy">equivariant Cohomotopy</a> to normal equivariant framed submanifolds – remains elusive, but on <a class="existingWikiWord" href="/nlab/show/free+action">free</a> <a class="existingWikiWord" href="/nlab/show/G-manifolds">G-manifolds</a> it is again straightforward (and reduces to the <a class="existingWikiWord" href="/nlab/show/twisted+Pontrjagin+theorem">twisted Pontrjagin theorem</a> on the <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>), made explicit in:</p> <ul> <li id="Cruickshank99"><a class="existingWikiWord" href="/nlab/show/James+Cruickshank">James Cruickshank</a>, Thm. 5.0.6, Cor. 6.0.13 in: <em>Twisted Cobordism and its Relationship to Equivariant Homotopy Theory</em>, 1999 (<a href="http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape9/PQDD_0030/NQ46823.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Cruickshank99.pdf" title="pdf">pdf</a>)</li> </ul> <h5 id="InNegativeCodimension">In negative codimension</h5> <p>In <a class="existingWikiWord" href="/nlab/show/negative+number">negative</a> <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a>, the <a class="existingWikiWord" href="/nlab/show/Cohomotopy+charge+map">Cohomotopy charge map</a> from the <a class="existingWikiWord" href="/nlab/show/Pontrjagin+theorem">Pontrjagin theorem</a> gives the <a href="configuration+space+of+points#LoopSpacesOfSuspensions">May-Segal theorem</a>, now identifying <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a> <em><a class="existingWikiWord" href="/nlab/show/cocycle+spaces">cocycle spaces</a></em> with <a class="existingWikiWord" href="/nlab/show/configuration+spaces+of+points">configuration spaces of points</a>:</p> <ul> <li id="May72"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em>The geometry of iterated loop spaces</em>, Springer 1972 (<a href="https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf">pdf</a>)</p> </li> <li id="Segal73"> <p><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>Configuration-spaces and iterated loop-spaces</em>, Invent. Math. <strong>21</strong> (1973), 213–221. MR 0331377 (<a href="http://dodo.pdmi.ras.ru/~topology/books/segal.pdf">pdf</a>)</p> <p>c Generalization of these constructions and results is due to</p> </li> <li id="McDuff75"> <p><a class="existingWikiWord" href="/nlab/show/Dusa+McDuff">Dusa McDuff</a>, <em>Configuration spaces of positive and negative particles</em>, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (<a href="https://doi.org/10.1016/0040-9383(75)90038-5">doi:10.1016/0040-9383(75)90038-5</a>)</p> </li> <li id="Boedigheimer87"> <p><a class="existingWikiWord" href="/nlab/show/Carl-Friedrich+B%C3%B6digheimer">Carl-Friedrich Bödigheimer</a>, <em>Stable splittings of mapping spaces</em>, Algebraic topology. Springer 1987. 174-187 (<a href="http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BoedigheimerStableSplittings87.pdf" title="pdf">pdf</a>)</p> </li> </ul> <h4 id="thoms_construction">Thom’s construction</h4> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+theorem">Thom's theorem</a> i.e. the unstable and <em>oriented</em> version of the <a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a>, identifying <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of <a class="existingWikiWord" href="/nlab/show/normally+oriented+submanifolds">normally oriented submanifolds</a> with <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of <a class="existingWikiWord" href="/nlab/show/maps">maps</a> to the <a class="existingWikiWord" href="/nlab/show/universal+vector+bundle">universal</a> <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal</a> <a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M SO(n)</annotation></semantics></math>, is due to:</p> <ul> <li id="Thom54"><a class="existingWikiWord" href="/nlab/show/Ren%C3%A9+Thom">René Thom</a>, <em><a class="existingWikiWord" href="/nlab/show/Quelques+propri%C3%A9t%C3%A9s+globales+des+vari%C3%A9t%C3%A9s+diff%C3%A9rentiables">Quelques propriétés globales des variétés différentiables</a></em>, Comment. Math. Helv. 28, (1954). 17-86 (<a href="https://doi.org/10.1007/BF02566923">doi:10.1007/BF02566923</a>, <a href="https://eudml.org/doc/139072">dml:139072</a>, <a href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259">digiz:GDZPPN002056259</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf">pdf</a>)</li> </ul> <p>Textbook accounts:</p> <ul> <li id="Stong68"><a class="existingWikiWord" href="/nlab/show/Robert+Stong">Robert Stong</a>, <em>Notes on Cobordism theory</em>, 1968 (<a href="http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf">toc pdf</a>, <a href="http://press.princeton.edu/titles/6465.html">publisher page</a>)</li> </ul> <h4 id="lashofs_construction">Lashof’s construction</h4> <p>The joint generalization of <a href="#Pontryagin38a">Pontryagin 38a</a>, <a href="#Pontryagin55">55</a> (framing structure) and <a href="#Thom54">Thom 54</a> (orientation structure) to any family of <a class="existingWikiWord" href="/nlab/show/tangential+structures">tangential structures</a> (“<a class="existingWikiWord" href="/nlab/show/%28B%2Cf%29-structure">(B,f)-structure</a>”) is first made explicit in</p> <ul> <li id="Lashof63"><a class="existingWikiWord" href="/nlab/show/Richard+Lashof">Richard Lashof</a>, <em>Poincaré duality and cobordism</em>, Trans. AMS 109 (1963), 257-277 (<a href="https://doi.org/10.1090/S0002-9947-1963-0156357-4">doi:10.1090/S0002-9947-1963-0156357-4</a>)</li> </ul> <p>and the general statement that has come to be known as the <em><a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+isomorphism">Pontryagin-Thom isomorphism</a></em> (identifying the stable <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of normally <a class="existingWikiWord" href="/nlab/show/tangential+structure">(B,f)-structured</a> <a class="existingWikiWord" href="/nlab/show/submanifolds">submanifolds</a> with <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of <a class="existingWikiWord" href="/nlab/show/maps">maps</a> to the <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> <a class="existingWikiWord" href="/nlab/show/Mf">Mf</a>) is really due to <a href="#Lashof63">Lashof 63, Theorem C</a>.</p> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Theodor+Br%C3%B6cker">Theodor Bröcker</a>, <a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, Satz 3.1 & 4.9 in: <em>Kobordismentheorie</em>, Lecture Notes in Mathematics <strong>178</strong>, Springer (1970) &lbrack;<a href="https://link.springer.com/book/9783540053415">ISBN:9783540053415</a>&rbrack;</p> </li> <li id="Kochman96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochman">Stanley Kochman</a>, section 1.5 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="Rudyak98"> <p><a class="existingWikiWord" href="/nlab/show/Yuli+Rudyak">Yuli Rudyak</a>, <em>On Thom spectra, orientability and cobordism</em>, Springer Monographs in Mathematics (1998) [<a href="https://doi.org/10.1007/978-3-540-77751-9">doi:10.1007/978-3-540-77751-9</a>, <a href="http://www.maths.ed.ac.uk/~aar/papers/rudyakthom.pdf">pdf</a>]</p> </li> </ul> <p>Lecture notes:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Francis">John Francis</a>, <em>Topology of manifolds</em> course notes (2010) (<a href="http://math.northwestern.edu/~jnkf/classes/mflds/">web</a>), Lecture 3: <em>Thom’s theorem</em> (<a href="http://math.northwestern.edu/~jnkf/classes/mflds/3thom.pdf">pdf</a>), Lecture 4 <em>Transversality</em> (notes by I. Bobkova) (<a href="http://math.northwestern.edu/~jnkf/classes/mflds/4transversality.pdf">pdf</a>)</p> </li> <li id="Malkiewich11"> <p><a class="existingWikiWord" href="/nlab/show/Cary+Malkiewich">Cary Malkiewich</a>, Section 3 of: <em>Unoriented cobordism and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>O</mi></mrow><annotation encoding="application/x-tex">M O</annotation></semantics></math></em>, 2011 (<a href="http://math.uiuc.edu/~cmalkiew/cobordism.pdf">pdf</a>)</p> </li> <li> <p>Tom Weston, Part I of <em>An introduction to cobordism theory</em> (<a href="http://people.math.umass.edu/~weston/oldpapers/cobord.pdf">pdf</a>)</p> </li> </ul> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Manifold+Atlas">Manifold Atlas</a>, <em><a href="http://www.map.mpim-bonn.mpg.de/B-Bordism#The_Pontrjagin-Thom_isomorphism">The Pontrjagin-Thom isomorphism</a></em></li> </ul> </div> <h3 id="for_point_configurations">For point configurations</h3> <p>The theorem that, with due care, for <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">point configurations</a> the Cohomotopy charge map is in fact a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> between the <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configuration space of points</a> and the <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a> <a class="existingWikiWord" href="/nlab/show/cocycle+space">cocycle space</a> originates with</p> <ul> <li id="May72"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em>The geometry of iterated loop spaces</em>, Springer 1972 (<a href="https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf">pdf</a>)</p> </li> <li id="Segal73"> <p><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>Configuration-spaces and iterated loop-spaces</em>, Invent. Math. <strong>21</strong> (1973), 213–221. MR 0331377 (<a href="http://dodo.pdmi.ras.ru/~topology/books/segal.pdf">pdf</a>)</p> </li> <li id="McDuff75"> <p><a class="existingWikiWord" href="/nlab/show/Dusa+McDuff">Dusa McDuff</a>, <em>Configuration spaces of positive and negative particles</em>, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (<a href="https://doi.org/10.1016/0040-9383(75)90038-5">doi:10.1016/0040-9383(75)90038-5</a>)</p> </li> </ul> <p>with comprehensive review in</p> <ul> <li id="Boedigheimer87"><a class="existingWikiWord" href="/nlab/show/Carl-Friedrich+B%C3%B6digheimer">Carl-Friedrich Bödigheimer</a>, <em>Stable splittings of mapping spaces</em>, Algebraic topology. Springer 1987. 174-187 (<a href="http://www.math.uni-bonn.de/~cfb/PUBLICATIONS/stable-splittings-of-mapping-spaces.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BoedigheimerStableSplittings87.pdf" title="pdf">pdf</a>)</li> </ul> <p>See also:</p> <ul> <li id="Kallel98"> <p><a class="existingWikiWord" href="/nlab/show/Sadok+Kallel">Sadok Kallel</a>, <em>Spaces of particles on manifolds and Generalized Poincaré Dualities</em>, The Quarterly Journal of Mathematics, Volume 52, Issue 1, 1 March 2001 (<a href="https://doi.org/10.1093/qjmath/52.1.45">doi:10.1093/qjmath/52.1.45</a>)</p> </li> <li id="Salvatore01"> <p><a class="existingWikiWord" href="/nlab/show/Paolo+Salvatore">Paolo Salvatore</a>, <em>Configuration spaces with summable labels</em>, In: Aguadé J., Broto C., <a class="existingWikiWord" href="/nlab/show/Carles+Casacuberta">Carles Casacuberta</a> (eds.) <em>Cohomological Methods in Homotopy Theory</em>. Progress in Mathematics, vol 196. Birkhäuser, Basel, 2001 (<a href="https://arxiv.org/abs/math/9907073">arXiv:math/9907073</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Oscar+Randal-Williams">Oscar Randal-Williams</a>, section 10 of: <em>Embedded Cobordism Categories and Spaces of Manifolds</em>, Int. Math. Res. Not. IMRN 2011, no. 3, 572-608 (<a href="https://arxiv.org/abs/0912.2505">arXiv:0912.2505</a>)</p> </li> <li id="ManthorpeTillmann13"> <p>Richard Manthorpe, <a class="existingWikiWord" href="/nlab/show/Ulrike+Tillmann">Ulrike Tillmann</a>, <em>Tubular configurations: equivariant scanning and splitting</em>, Journal of the London Mathematical Society, Volume 90, Issue 3 (<a href="https://arxiv.org/abs/1307.5669">arxiv:1307.5669</a>, <a href="https://doi.org/10.1112/jlms/jdu050">doi:10.1112/jlms/jdu050</a>)</p> </li> <li id="Knudsen18"> <p><a class="existingWikiWord" href="/nlab/show/Ben+Knudsen">Ben Knudsen</a>, <em>Configuration spaces in algebraic topology</em> (<a href="https://arxiv.org/abs/1803.11165">arXiv:1803.11165</a>)</p> </li> <li id="SatiSchreiber19"> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Equivariant+Cohomotopy+implies+orientifold+tadpole+cancellation">Equivariant Cohomotopy implies orientifold tadpole cancellation</a></em> (<a href="https://arxiv.org/abs/1909.12277">arXiv:1909.12277</a>)</p> </li> <li id="SS22IntBranes"> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Differential+Cohomotopy+implies+intersecting+brane+observables">Differential Cohomotopy implies intersecting brane observables</a></em>, Adv. Theor. Math. Phys. <strong>26</strong> 4 (2022) [<a href="https://dx.doi.org/10.4310/ATMP.2022.v26.n4.a4">doi:10.4310/ATMP.2022.v26.n4.a4</a><a href="https://arxiv.org/abs/1912.10425">arXiv:1912.10425</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 18, 2024 at 12:40:40. 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