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value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #48 to #49: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='cohomology'>Cohomology</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycle</a>, <a class='existingWikiWord' href='/nlab/show/diff/coboundary'>coboundary</a>, <a class='existingWikiWord' href='/nlab/show/diff/coefficient'>coefficient</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homology'>homology</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/chain'>chain</a>, <a class='existingWikiWord' href='/nlab/show/diff/cycle'>cycle</a>, <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/characteristic+class'>characteristic class</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+characteristic+class'>universal characteristic class</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/secondary+characteristic+class'>secondary characteristic class</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+characteristic+class'>differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>fiber sequence</a>/<a class='existingWikiWord' href='/nlab/show/diff/long+exact+sequence+in+homology'>long exact sequence in cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+infinity-bundle'>fiber ∞-bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/principal+infinity-bundle'>principal ∞-bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/associated+infinity-bundle'>associated ∞-bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/twisted+infinity-bundle'>twisted ∞-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group+extension'>∞-group extension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/chain+homology+and+cohomology'>cochain cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ordinary+cohomology'>ordinary cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/singular+cohomology'>singular cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group+cohomology'>group cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/nonabelian+group+cohomology'>nonabelian group cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lie+group+cohomology'>Lie group cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Galois+cohomology'>Galois cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/groupoid+cohomology'>groupoid cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/nonabelian+groupoid+cohomology'>nonabelian groupoid cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/generalized+%28Eilenberg-Steenrod%29+cohomology'>generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cobordism+cohomology+theory'>cobordism cohomology theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/integral+cohomology'>integral cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/K-theory'>K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/elliptic+cohomology'>elliptic cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/tmf'>tmf</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+automorphic+form'>taf</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>abelian sheaf cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Deligne+cohomology'>Deligne cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/de+Rham+complex'>de Rham cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Dolbeault+cohomology'>Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%C3%A9tale+cohomology'>etale cohomology</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/group+of+units'>group of units</a>, <a class='existingWikiWord' href='/nlab/show/diff/Picard+group'>Picard group</a>, <a class='existingWikiWord' href='/nlab/show/diff/Brauer+group'>Brauer group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/crystalline+cohomology'>crystalline cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/syntomic+cohomology'>syntomic cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/motivic+cohomology'>motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohomology+of+operads'>cohomology of operads</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hochschild+cohomology'>Hochschild cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/cyclic+homology'>cyclic cohomology</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/string+topology'>string topology</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nonabelian+cohomology'>nonabelian cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+infinity-bundle'>principal ∞-bundle</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+principal+infinity-bundle'>universal principal ∞-bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/groupal+model+for+universal+principal+infinity-bundles'>groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+bundle'>principal bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/Atiyah+Lie+groupoid'>Atiyah Lie groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+2-bundle'>principal 2-bundle</a>/<a class='existingWikiWord' href='/nlab/show/diff/gerbe'>gerbe</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+constant+infinity-stack'>covering ∞-bundle</a>/<a class='existingWikiWord' href='/nlab/show/diff/local+system'>local system</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module+bundle'>(∞,1)-vector bundle</a> / <a class='existingWikiWord' href='/nlab/show/diff/n-vector+bundle'>(∞,n)-vector bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quantum+anomaly'>quantum anomaly</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/orientation'>orientation</a>, <a class='existingWikiWord' href='/nlab/show/diff/spin+structure'>Spin structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/spin%E1%B6%9C+structure'>Spin^c structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/string+structure'>String structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/Fivebrane+structure'>Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohomology+with+constant+coefficients'>cohomology with constant coefficients</a> / <a class='existingWikiWord' href='/nlab/show/diff/local+system'>with a local system of coefficients</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-Lie+algebra+cohomology'>∞-Lie algebra cohomology</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra+cohomology'>Lie algebra cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/nonabelian+Lie+algebra+cohomology'>nonabelian Lie algebra cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra+extension'>Lie algebra extensions</a>, <a class='existingWikiWord' href='/nlab/show/diff/Gelfand-Fuks+cohomology'>Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/%C4%8Cech+cohomology'>?ech cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hypercohomology'>hypercohomology</a></p> </li> </ul> <h3 id='variants'>Variants</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivariant+cohomology'>equivariant cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivariant+homotopy+theory'>equivariant homotopy theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Bredon+cohomology'>Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/twisted+cohomology'>twisted cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/twisted+bundle'>twisted bundle</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/twisted+K-theory'>twisted K-theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/twisted+spin+structure'>twisted spin structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/twisted+spin%E1%B6%9C+structure'>twisted spin^c structure</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/twisted+differential+c-structure'>twisted differential c-structures</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/differential+string+structure'>twisted differential string structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/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/diff/differential+cohomology'>differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+cobordism+cohomology'>differential cobordism cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Deligne+cohomology'>Deligne cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+K-theory'>differential K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+elliptic+cohomology'>differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/schreiber/show/diff/differential+cohomology+in+a+cohesive+topos' title='schreiber'>differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory'>Chern-Weil theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd'>∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relative+cohomology'>relative cohomology</a></p> </li> </ul> <h3 id='extra_structure'>Extra structure</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hodge+structure'>Hodge structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/orientation'>orientation</a>, <a class='existingWikiWord' href='/nlab/show/diff/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/diff/cohomology+operation'>cohomology operations</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cup+product'>cup product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connecting+homomorphism'>connecting homomorphism</a>, <a class='existingWikiWord' href='/nlab/show/diff/Bockstein+homomorphism'>Bockstein homomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+integration'>fiber integration</a>, <a class='existingWikiWord' href='/nlab/show/diff/transgression'>transgression</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohomology+localization'>cohomology localization</a></p> </li> </ul> <h3 id='theorems'>Theorems</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+coefficient+theorem'>universal coefficient theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/K%C3%BCnneth+theorem'>Künneth theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/de+Rham+theorem'>de Rham theorem</a>, <a class='existingWikiWord' href='/nlab/show/diff/Poincar%C3%A9+lemma'>Poincare lemma</a>, <a class='existingWikiWord' href='/nlab/show/diff/Stokes+theorem'>Stokes theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hodge+theory'>Hodge theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hodge+theorem'>Hodge theorem</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/nonabelian+Hodge+theory'>nonabelian Hodge theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/noncommutative+Hodge+structure'>noncommutative Hodge theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brown+representability+theorem'>Brown representability theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>hypercovering theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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> <h4 id='differential_geometry'>Differential geometry</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/synthetic+differential+geometry'>synthetic</a> <a class='existingWikiWord' href='/nlab/show/diff/differential+geometry'>differential geometry</a></strong></p> <p><strong>Introductions</strong></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+1'>from point-set topology to differentiable manifolds</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics'>geometry of physics</a>: <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+coordinate+systems'>coordinate systems</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+smooth+sets'>smooth spaces</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+manifolds+and+orbifolds'>manifolds</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+smooth+homotopy+types'>smooth homotopy types</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+supergeometry'>supergeometry</a></p> <p><strong>Differentials</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differentiation'>differentiation</a>, <a class='existingWikiWord' href='/nlab/show/diff/chain+rule'>chain rule</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differentiable+map'>differentiable function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+object'>infinitesimal space</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinitesimally+thickened+point'>infinitesimally thickened point</a>, <a class='existingWikiWord' href='/nlab/show/diff/amazing+right+adjoint'>amazing right adjoint</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/V-manifold'>V-manifolds</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differentiable+manifold'>differentiable manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/coordinate+system'>coordinate chart</a>, <a class='existingWikiWord' href='/nlab/show/diff/atlas'>atlas</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/smooth+structure'>smooth structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/exotic+smooth+structure'>exotic smooth structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/analytic+manifold'>analytic manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+manifold'>complex manifold</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/formal+smooth+manifold'>formal smooth manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/derived+smooth+manifold'>derived smooth manifold</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/smooth+set'>smooth space</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/diffeological+space'>diffeological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Fr%C3%B6licher+space'>Frölicher space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/manifold+structure+of+mapping+spaces'>manifold structure of mapping spaces</a></p> </li> </ul> <p><strong>Tangency</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/frame+bundle'>frame bundle</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/vector+field'>vector field</a>, <a class='existingWikiWord' href='/nlab/show/diff/multivector+field'>multivector field</a>, <a class='existingWikiWord' href='/nlab/show/diff/tangent+Lie+algebroid'>tangent Lie algebroid</a>;</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+forms+in+synthetic+differential+geometry'>differential forms</a>, <a class='existingWikiWord' href='/nlab/show/diff/de+Rham+complex'>de Rham complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dolbeault+complex'>Dolbeault complex</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pullback+of+a+differential+form'>pullback of differential forms</a>, <a class='existingWikiWord' href='/nlab/show/diff/invariant+differential+form'>invariant differential form</a>, <a class='existingWikiWord' href='/nlab/show/diff/Maurer-Cartan+form'>Maurer-Cartan form</a>, <a class='existingWikiWord' href='/nlab/show/diff/horizontal+differential+form'>horizontal differential form</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cogerm+differential+form'>cogerm differential form</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/integration+of+differential+forms'>integration of differential forms</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/local+diffeomorphism'>local diffeomorphism</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+%C3%A9tale+morphism'>formally étale morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/submersion'>submersion</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+smooth+morphism'>formally smooth morphism</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/immersion'>immersion</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+unramified+morphism'>formally unramified morphism</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/de+Rham+space'>de Rham space</a>, <a class='existingWikiWord' href='/nlab/show/diff/crystal'>crystal</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+disk+bundle'>infinitesimal disk bundle</a></p> </li> </ul> <p><strong>The magic algebraic facts</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras'>embedding of smooth manifolds into formal duals of R-algebras</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+Serre-Swan+theorem'>smooth Serre-Swan theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derivations+of+smooth+functions+are+vector+fields'>derivations of smooth functions are vector fields</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hadamard+lemma'>Hadamard lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Borel%27s+theorem'>Borel's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Boman%27s+theorem'>Boman's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitney+extension+theorem'>Whitney extension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Steenrod+approximation+theorem'>Steenrod-Wockel approximation theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitney+embedding+theorem'>Whitney embedding theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Poincar%C3%A9+lemma'>Poincare lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Stokes+theorem'>Stokes theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/de+Rham+theorem'>de Rham theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hochschild-Kostant-Rosenberg+theorem'>Hochschild-Kostant-Rosenberg theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology+diagram'>differential cohomology hexagon</a></p> </li> </ul> <p><strong>Axiomatics</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kock-Lawvere+axiom'>Kock-Lawvere axiom</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+topos'>smooth topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/super+smooth+topos'>super smooth topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/microlinear+space'>microlinear space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/integration+axiom'>integration axiom</a></p> </li> </ul> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/cohesive'>cohesion</a></strong></p> <ul> <li> <p>(<a class='existingWikiWord' href='/nlab/show/diff/shape+modality'>shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/flat+modality'>flat modality</a> <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/sharp+modality'>sharp modality</a>)</p> <p><math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='0em' rspace='thinmathspace'>esh</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete object</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+object'>codiscrete object</a>, <a class='existingWikiWord' href='/nlab/show/diff/concrete+object'>concrete object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/points-to-pieces+transform'>points-to-pieces transform</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohesive+%28infinity%2C1%29-topos+--+structures'>structures in cohesion</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/dR-shape+modality'>dR-shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/dR-flat+modality'>dR-flat modality</a></p> <p><math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='0em' rspace='thinmathspace'>esh</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding='application/x-tex'>\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+cohesive+%28infinity%2C1%29-topos'>infinitesimal cohesion</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/classical+modality'>classical modality</a></li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/tangent+cohesive+%28%E2%88%9E%2C1%29-topos'>tangent cohesion</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology+diagram'>differential cohomology diagram</a></li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/differential+cohesive+%28infinity%2C1%29-topos'>differential cohesion</a></strong></p> <ul> <li> <p>(<a class='existingWikiWord' href='/nlab/show/diff/reduction+modality'>reduction modality</a> <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+shape+modality'>infinitesimal shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+flat+modality'>infinitesimal flat modality</a>)</p> <p><math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\Re \dashv \Im \dashv \&)</annotation></semantics></math></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reduced+object'>reduced object</a>, <a class='existingWikiWord' href='/nlab/show/diff/coreduced+object'>coreduced object</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+smooth+object'>formally smooth object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/formally+%C3%A9tale+morphism'>formally étale map</a></p> </li> <li> <p><a href='cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion'>structures in differential cohesion</a></p> </li> </ul> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/super+formal+smooth+infinity-groupoid'>graded differential cohesion</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bosonic+modality'>fermionic modality</a> <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/bosonic+modality'>bosonic modality</a> <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/rheonomy+modality'>rheonomy modality</a></p> <p><math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/orbifold+cohomology'>singular cohesion</a></strong></p> <div class='maruku-equation' id='Diagram'><math class='maruku-mathml' display='block' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow /><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>⊥</mo></mtd> <mtd /> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant='normal'>R</mi><mspace width='negativethinmathspace' /><mspace width='negativethinmathspace' /><mi mathvariant='normal'>h</mi></mtd> <mtd><mover><mrow /><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow /><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>⊥</mo></mtd> <mtd /> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow /><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>cohesive</mi></mover></mtd> <mtd><mo lspace='0em' rspace='thinmathspace'>esh</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow /><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>⊥</mo></mtd> <mtd /> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow /><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div> <p id='models_2'><strong>Models</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Models+for+Smooth+Infinitesimal+Analysis'>Models for Smooth Infinitesimal Analysis</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/C%5E%E2%88%9E-ring'>smooth algebra</a> (<math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding='application/x-tex'>C^\infty</annotation></semantics></math>-ring)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+locus'>smooth locus</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Fermat+theory'>Fermat theory</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cahiers+topos'>Cahiers topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+infinity-groupoid'>smooth ∞-groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/formal+smooth+infinity-groupoid'>formal smooth ∞-groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/super+formal+smooth+infinity-groupoid'>super formal smooth ∞-groupoid</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/Lie+theory'>Lie theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-Lie+theory'>∞-Lie theory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra'>Lie algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>Lie n-algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+group'>Lie group</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lie+2-group'>Lie 2-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/smooth+infinity-groupoid'>smooth ∞-group</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/differential+equation'>differential equations</a>, <a class='existingWikiWord' href='/nlab/show/diff/variational+calculus'>variational calculus</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/D-geometry'>D-geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/D-module'>D-module</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/jet+bundle'>jet bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/variational+bicomplex'>variational bicomplex</a>, <a class='existingWikiWord' href='/nlab/show/diff/Euler-Lagrange+complex'>Euler-Lagrange complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euler-Lagrange+equation'>Euler-Lagrange equation</a>, <a class='existingWikiWord' href='/nlab/show/diff/De+Donder-Weyl-Hamilton+equation'>de Donder-Weyl formalism</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/phase+space'>phase space</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory'>Chern-Weil theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd'>∞-Chern-Weil theory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connection+on+a+bundle'>connection on a bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/connection+on+a+smooth+principal+infinity-bundle'>connection on an ∞-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology'>differential cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ordinary+differential+cohomology'>ordinary differential cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Deligne+cohomology'>Deligne complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+K-theory'>differential K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+cobordism+cohomology'>differential cobordism cohomology</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/parallel+transport'>parallel transport</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+parallel+transport'>higher parallel transport</a>, <a class='existingWikiWord' href='/nlab/show/diff/fiber+integration+in+differential+cohomology'>fiber integration in differential cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/holonomy'>holonomy</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+parallel+transport'>higher holonomy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/gauge+theory'>gauge theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+gauge+field'>higher gauge theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Wilson+loop'>Wilson line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Wilson+surface'>Wilson surface</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/Cartan+geometry'>Cartan geometry</a> (<a class='existingWikiWord' href='/nlab/show/diff/super-Cartan+geometry'>super</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+Cartan+geometry'>higher</a>)</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Klein+geometry'>Klein geometry</a>, (<a class='existingWikiWord' href='/nlab/show/diff/higher+Klein+geometry'>higher</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/G-structure'>G-structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/torsion+of+a+G-structure'>torsion of a G-structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euclidean+geometry'>Euclidean geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/hyperbolic+geometry'>hyperbolic geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/elliptic+geometry'>elliptic geometry</a></p> </li> <li> <p>(<a class='existingWikiWord' href='/nlab/show/diff/pseudo-Riemannian+metric'>pseudo</a>-)<a class='existingWikiWord' href='/nlab/show/diff/Riemannian+geometry'>Riemannian geometry</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/orthogonal+structure'>orthogonal structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/isometry'>isometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/Killing+vector+field'>Killing vector field</a>, <a class='existingWikiWord' href='/nlab/show/diff/Killing+spinor'>Killing spinor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetime</a>, <a class='existingWikiWord' href='/nlab/show/diff/super+spacetime'>super-spacetime</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/complex+geometry'>complex geometry</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symplectic+geometry'>symplectic geometry</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/conformal+geometry'>conformal geometry</a></p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a><ul><li><a href='#for_smooth_manifolds'>For smooth manifolds</a></li><li><a href='#for_algebraic_objects'>For algebraic objects</a></li><li><a href='#ForCohesiveHomotopyTypes'>For cohesive homotopy types</a></li></ul></li><li><a href='#examples'>Examples</a><ul><li><a href='#de_rham_cohomology_of_spheres'>de Rham cohomology of spheres</a></li></ul></li><li><a href='#properties'>Properties</a><ul><li><a href='#basic_theorems'>Basic theorems</a></li><li><a href='#RelaionToPLDeRhamComplex'>Relation to PL de Rham complex</a></li><li><a href='#relation_to_deligne_complex'>Relation to Deligne complex</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><ul><li><a href='#in_differential_geometry'>In differential geometry</a></li><li><a href='#in_algebraic_geometry'>In algebraic geometry</a></li></ul></li></ul></div> <h2 id='idea'>Idea</h2> <p>The <em>de Rham complex</em> (named after <a class='existingWikiWord' href='/nlab/show/diff/Georges+de+Rham'>Georges de Rham</a>) <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Omega^\bullet(X)</annotation></semantics></math> of a <a class='existingWikiWord' href='/nlab/show/diff/space'>space</a> <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/cochain+complex'>cochain complex</a> that in degree <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> has the <a class='existingWikiWord' href='/nlab/show/diff/differential+form'>differential form</a>s (which may mean: <a class='existingWikiWord' href='/nlab/show/diff/K%C3%A4hler+differential'>Kähler differential form</a>s) of degree <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>, and whose <a class='existingWikiWord' href='/nlab/show/diff/differential'>differential</a> is the <strong>de Rham differential</strong> or <strong>exterior derivative</strong>.</p> <p>As <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> varies this constitutes an <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf'>abelian sheaf</a> of complexes.</p> <h2 id='definition'>Definition</h2> <h3 id='for_smooth_manifolds'>For smooth manifolds</h3> <p>The <strong>de Rham complex</strong> of a <a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a> is the <a class='existingWikiWord' href='/nlab/show/diff/cochain+complex'>cochain complex</a> which in degree <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math> has the <a class='existingWikiWord' href='/nlab/show/diff/vector+space'>vector space</a> <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Omega^n(X)</annotation></semantics></math> of degree-<math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/differential+form'>differential forms</a> on <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. The coboundary map is the deRham <em><a class='existingWikiWord' href='/nlab/show/diff/de+Rham+differential'>exterior derivative</a></em>.</p> <p>Explicitly, given a differential <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-form <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>, its de Rham differential <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mi>ω</mi></mrow><annotation encoding='application/x-tex'>d\omega</annotation></semantics></math> can be computed as</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mi>ω</mi><mo stretchy='false'>(</mo><msub><mi>v</mi> <mn>0</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo>=</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>i</mi></munder><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><msup><mo stretchy='false'>)</mo> <mi>i</mi></msup><msub><mi>ℒ</mi> <mrow><msub><mi>v</mi> <mi>i</mi></msub></mrow></msub><mi>ω</mi><mo stretchy='false'>(</mo><msub><mi>v</mi> <mn>0</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>v</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo>+</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mrow><mi>i</mi><mo><</mo><mi>j</mi></mrow></munder><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><msup><mo stretchy='false'>)</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><mi>ω</mi><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><msub><mi>v</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>v</mi> <mi>j</mi></msub><mo stretchy='false'>]</mo><mo>,</mo><msub><mi>v</mi> <mn>0</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>v</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>v</mi> <mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>v</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo>,</mo></mrow><annotation encoding='application/x-tex'>d\omega(v_0,\ldots,v_k)=\sum_i (-1)^i \mathcal{L}_{v_i} \omega(v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_k)+\sum_{i\lt j}(-1)^{i+j}\omega([v_i,v_j],v_0,\ldots,v_{i-1},v_{i+1},\ldots,v_{j-1},v_{j+1},\ldots,v_k),</annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>v_i</annotation></semantics></math> are <a class='existingWikiWord' href='/nlab/show/diff/vector+field'>vector fields</a> on <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[-,-]</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/Lie+bracket+of+vector+fields'>Lie bracket of vector fields</a>, and <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℒ</mi> <mi>v</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{L}_{v}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/Lie+derivative'>Lie derivative</a> of a smooth function with respect to a vector field <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi></mrow><annotation encoding='application/x-tex'>v</annotation></semantics></math>.</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/chain+homology+and+cohomology'>cohomology</a> of the de Rham complex (hence the <a class='existingWikiWord' href='/nlab/show/diff/quotient+object'>quotient</a> of <a class='existingWikiWord' href='/nlab/show/diff/closed+differential+form'>closed differential forms</a> by <a class='existingWikiWord' href='/nlab/show/diff/closed+differential+form'>exact differential forms</a>) is <strong>de Rham cohomology</strong>. Under the <a class='existingWikiWord' href='/nlab/show/diff/exterior+algebra'>wedge product</a>, the deRham complex becomes a <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>differential graded algebra</a>. This may be regarded as the <a class='existingWikiWord' href='/nlab/show/diff/Chevalley-Eilenberg+algebra'>Chevalley-Eilenberg algebra</a> of the <a class='existingWikiWord' href='/nlab/show/diff/tangent+Lie+algebroid'>tangent Lie algebroid</a> <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>T X</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <p>The corresponding <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf'>abelian sheaf</a> in this case defines a <a class='existingWikiWord' href='/nlab/show/diff/smooth+spectrum'>smooth spectrum</a> via the <a class='existingWikiWord' href='/nlab/show/diff/stable+Dold-Kan+correspondence'>stable Dold-Kan correspondence</a>, see at <em><a href='smooth+spectrum#ExamplesDeRhamSpectra'>smooth spectrum – Examples – De Rham spectra</a></em>.</p> <h3 id='for_algebraic_objects'>For algebraic objects</h3> <p>For <a class='existingWikiWord' href='/nlab/show/diff/smooth+scheme'>smooth varieties</a> <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, algebraic de Rham cohomology is defined to be the <a class='existingWikiWord' href='/nlab/show/diff/hypercohomology'>hypercohomology</a> of the de Rham complex <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Ω</mi> <mi>X</mi> <mo>•</mo></msubsup></mrow><annotation encoding='application/x-tex'>\Omega_X^\bullet</annotation></semantics></math>.</p> <p>De Rham cohomology has a rather subtle generalization for possibly singular algebraic varieties due to (<a href='#Grothendieck'>Grothendieck</a>).</p> <p>For <a class='existingWikiWord' href='/nlab/show/diff/analytic+space'>analytic spaces</a></p> <ul> <li>T. Bloom, M. Herrera, <em>De Rham cohomology of an analytic space</em>, Inv. Math. <strong>7</strong> (1969), 275-296, <a href='http://dx.doi.org/10.1007/BF01425536'>doi</a></li> </ul> <h3 id='ForCohesiveHomotopyTypes'>For cohesive homotopy types</h3> <p>In the general context of <a class='existingWikiWord' href='/nlab/show/diff/cohesive+homotopy+theory'>cohesive homotopy theory</a> in a <a class='existingWikiWord' href='/nlab/show/diff/cohesive+%28infinity%2C1%29-topos'>cohesive (∞,1)-topos</a> <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math>, for <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∈</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>A \in \mathbf{H}</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/cohesive+homotopy+type+theory'>cohesive homotopy type</a>, then the <a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>homotopy fiber</a> of the <a class='existingWikiWord' href='/nlab/show/diff/unit+of+a+monad'>counit</a> of the <a class='existingWikiWord' href='/nlab/show/diff/flat+modality'>flat modality</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mi>A</mi><mo>≔</mo><mi>fib</mi><mo stretchy='false'>(</mo><mo>♭</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \flat_{dR} A \coloneqq fib(\flat A \to A) </annotation></semantics></math></div> <p>may be interpreted as the de Rham complex with <a class='existingWikiWord' href='/nlab/show/diff/coefficient'>coefficients</a> in <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>.</p> <p>This is the <a class='existingWikiWord' href='/nlab/show/diff/target'>codomain</a> for the <a class='existingWikiWord' href='/nlab/show/diff/Maurer-Cartan+form'>Maurer-Cartan form</a> <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>θ</mi> <mrow><mi>Ω</mi><mi>A</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\theta_{\Omega A}</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ω</mi><mi>A</mi></mrow><annotation encoding='application/x-tex'>\Omega A</annotation></semantics></math> in this generality. The <a class='existingWikiWord' href='/nlab/show/diff/shape+modality'>shape</a> of <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>θ</mi> <mrow><mi>Ω</mi><mi>A</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\theta_{\Omega A}</annotation></semantics></math> is the general <a class='existingWikiWord' href='/nlab/show/diff/Chern+character'>Chern character</a> on <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Π</mi><mo stretchy='false'>(</mo><mi>Ω</mi><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Pi(\Omega A)</annotation></semantics></math>.</p> <p>For more on this see at</p> <ul> <li><em><a href='cohesive+%28infinity%2C1%29-topos+--+structures#deRhamCohomology'>structures in a cohesive infinity-topos – de Rham cohomology</a></em></li> </ul> <p>More precisely, <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>♭</mo> <mi>dR</mi></msub><mi>Σ</mi><mi>A</mi></mrow><annotation encoding='application/x-tex'>\flat_{dR} \Sigma A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Π</mi> <mi>dR</mi></msub><mi>Ω</mi><mi>A</mi></mrow><annotation encoding='application/x-tex'>\Pi_{dR} \Omega A</annotation></semantics></math> play the role of the non-negative degree and negative degree part, respectively of the de Rham complex with coefficients in <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Π</mi><msub><mo>♭</mo> <mi>dR</mi></msub><mi>Σ</mi><mi>A</mi></mrow><annotation encoding='application/x-tex'>\Pi \flat_{dR} \Sigma A</annotation></semantics></math>. For more on this see at</p> <ul> <li><em><a href='differential%20cohomology%20diagram#DeRhamCoefficients'>differential cohomology diagram – de Rham coefficients</a></em>.</li> </ul> <h2 id='examples'>Examples</h2> <h3 id='de_rham_cohomology_of_spheres'>de Rham cohomology of spheres</h3> <div class='num_prop'> <h6 id='proposition'>Proposition</h6> <p>For positive <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>, the de Rham cohomology of the <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/sphere'>sphere</a> <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>S^n</annotation></semantics></math> is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>ℝ</mi></mtd> <mtd><mi>if</mi><mspace width='thickmathspace' /><mi>p</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>n</mi></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> H^p(S^n) = \left\{ \array{ \mathbb{R} & if\; p = 0,n \\ 0 & otherwise } \right. \,. </annotation></semantics></math></div> <p>For <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>n=0</annotation></semantics></math>, we have</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy='false'>(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy='false'>)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>ℝ</mi><mo>⊕</mo><mi>ℝ</mi></mtd> <mtd><mi>if</mi><mspace width='thickmathspace' /><mi>p</mi><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> H^p(S^0) = \left\{ \array{ \mathbb{R} \oplus \mathbb{R} & if\; p = 0 \\ 0 & otherwise } \right. \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>This follows from the <a class='existingWikiWord' href='/nlab/show/diff/Mayer-Vietoris+sequence'>Mayer-Vietoris sequence</a> associated to the open cover of <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>S^n</annotation></semantics></math> by the subset excluding just the north pole and the subset excluding just the south pole, together with the fact that the dimension of the <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mn>0</mn> <mi>th</mi></msup></mrow><annotation encoding='application/x-tex'>0^{th}</annotation></semantics></math> de Rham cohomology of a smooth manifold is its number of connected components.</p> </div> <h2 id='properties'>Properties</h2> <h3 id='basic_theorems'>Basic theorems</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Poincar%C3%A9+lemma'>Poincare lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/de+Rham+theorem'>de Rham theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/non-abelian+de+Rham+cohomology'>nonabelian de Rham cohomology</a></p> </li> </ul> <h3 id='RelaionToPLDeRhamComplex'>Relation to PL de Rham complex</h3> <div class='num_prop' id='PLdeRhamComplex'> <h6 id='proposition_2'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/PL+de+Rham+complex+of+smooth+manifold+is+equivalent+to+de+Rham+complex'>PL de Rham complex of smooth manifold is equivalent to de Rham complex</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a>.</p> <p>We have the following <a class='existingWikiWord' href='/nlab/show/diff/zigzag'>zig-zag</a> of <a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>dgc-algebra</a> <a class='existingWikiWord' href='/nlab/show/diff/quasi-isomorphism'>quasi-isomorphisms</a> between the <a class='existingWikiWord' href='/nlab/show/diff/PL+de+Rham+complex'>PL de Rham complex</a> of (the <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> underlying) <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and the smooth <a class='existingWikiWord' href='/nlab/show/diff/de+Rham+complex'>de Rham complex</a> of <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><msubsup><mi>Ω</mi> <mi>PLdR</mi> <mo>•</mo></msubsup><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>S</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd> <mtd /> <mtd /> <mtd /> <mtd><msubsup><mi>Ω</mi> <mi>dR</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd /> <mtd><msup><mo>↘</mo> <mpadded width='0'><mrow><msub><mi>i</mi> <mi>poly</mi></msub></mrow></mpadded></msup></mtd> <mtd /> <mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msup><mi>p</mi> <mo>*</mo></msup></mrow></mpadded></msup><mo>↙</mo></mtd></mtr> <mtr><mtd><mpadded lspace='-100%width' width='0'><mrow><msubsup><mi>Ω</mi> <mi>PLdR</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>=</mo><mspace width='thickmathspace' /></mrow></mpadded><msubsup><mi>Ω</mi> <mi>PLdR</mi> <mo>•</mo></msubsup><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>Sing</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd> <mtd /> <mtd /> <mtd /> <mtd><msubsup><mi>Ω</mi> <mi>PSdR</mi> <mo>•</mo></msubsup><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>S</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && \Omega^\bullet_{PLdR} \big( S(X) \big) && && \Omega^\bullet_{dR}(X) \\ & {}^{ \mathllap{ i^\ast } } \nearrow & & \searrow^{ \mathrlap{ i_{poly} } } & & {}^{ \mathllap{ p^\ast } } \swarrow \\ \mathllap{ \Omega^\bullet_{PLdR}(X) \;=\; } \Omega^\bullet_{PLdR} \big( Sing(X) \big) && && \Omega^\bullet_{PSdR} \big( S(X) \big) } </annotation></semantics></math></div> <p>Here <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>S(X)</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/simplicial+complex'>simplicial complex</a> corresponding to any smooth <a class='existingWikiWord' href='/nlab/show/diff/triangulation'>triangulation</a> of <math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> </div> <h3 id='relation_to_deligne_complex'>Relation to Deligne complex</h3> <p>See at <em><a class='existingWikiWord' href='/nlab/show/diff/Deligne+cohomology'>Deligne complex</a></em></p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pullback+of+a+differential+form'>pullback of differential forms</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivariant+de+Rham+cohomology'>equivariant de Rham cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/de+Rham-Witt+complex'>de Rham-Witt complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/twisted+de+Rham+cohomology'>twisted de Rham cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Deligne+cohomology'>Deligne cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/de+Rham+space'>de Rham space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/groupoid+of+Lie-algebra+valued+forms'>Lie algebra valued differential forms</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-Lie+algebroid-valued+differential+form'>L-infinity algebra valued differential forms</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/absolute+de+Rham+cohomology'>absolute de Rham cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Dolbeault+complex'>Dolbeault complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dolbeault+cohomology'>Dolbeault cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/holomorphic+de+Rham+complex'>holomorphic de Rham complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hodge%E2%80%93de+Rham+spectral+sequence'>Hodge-de Rham spectral sequence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/chiral+de+Rham+complex'>chiral de Rham complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/crystalline+cohomology'>crystalline cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/comparison+theorem+%28crystalline+cohomology%29'>comparison theorem (crystalline cohomology)</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hodge+cohomology'>Hodge cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/PL+de+Rham+complex'>PL de Rham complex</a></p> </li> </ul> <h2 id='references'>References</h2> <h3 id='in_differential_geometry'>In differential geometry</h3> <p>Discussion in <a class='existingWikiWord' href='/nlab/show/diff/differential+geometry'>differential geometry</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Raoul+Bott'>Raoul Bott</a>, <a class='existingWikiWord' href='/nlab/show/diff/Loring+Tu'>Loring Tu</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Differential+Forms+in+Algebraic+Topology'>Differential Forms in Algebraic Topology</a></em>, Graduate Texts in Mathematics 82, Springer 1982 (<a href='https://link.springer.com/book/10.1007/978-1-4757-3951-0'>doi:10.1007/978-1-4757-3951-0</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Georges+de+Rham'>Georges de Rham</a>, Chapter II of: <em>Differentiable Manifolds – Forms, Currents, Harmonic Forms</em>, Grundlehren <strong>266</strong>, Springer (1984) [[doi:10.1007/978-3-642-61752-2](https://doi.org/10.1007/978-3-642-61752-2)]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Dominic+G.+B.+Edelen'>Dominic G. B. Edelen</a>, <em>Applied exterior calculus</em>, Wiley (1985) [<a href='https://books.google.de/books?id=GUkViODKZ2oC=PP1=ucNgef4HKM=Edelen%20%22Applied%20exterior%20calculus%22%20Wiley=PP1#v=onepage=Edelen%20%22Applied%20exterior%20calculus%22%20Wiley=false'>GoogleBooks</a>]</p> </li> </ul> <p>With an eye towards application in <a class='existingWikiWord' href='/nlab/show/diff/mathematical+physics'>mathematical physics</a>:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Mikio+Nakahara'>Mikio Nakahara</a>, Chapter 6 of: <em><a class='existingWikiWord' href='/nlab/show/diff/Geometry%2C+Topology+and+Physics'>Geometry, Topology and Physics</a></em>, IOP 2003 (<a href='https://doi.org/10.1201/9781315275826'>doi:10.1201/9781315275826</a>, <a href='http://alpha.sinp.msu.ru/~panov/LibBooks/GRAV/(Graduate_Student_Series_in_Physics)Mikio_Nakahara-Geometry,_Topology_and_Physics,_Second_Edition_(Graduate_Student_Series_in_Physics)-Institute_of_Physics_Publishing(2003).pdf'>pdf</a>)</li> </ul> <h3 id='in_algebraic_geometry'>In algebraic geometry</h3> <p>Discussion in <a class='existingWikiWord' href='/nlab/show/diff/algebraic+geometry'>algebraic geometry</a></p> <p>A useful introduction is</p> <ul> <li>Kiran Kedlaya, <em><math class='maruku-mathml' display='inline' id='mathml_a443142cc7e598991dc603068608f5bb3e3a5f55_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>-adic cohomology, from theory to practice</em> (<a href='http://swc.math.arizona.edu/aws/2007/KedlayaNotes11Mar.pdf'>pdf</a>)</li> </ul> <p>A classical reference on the algebraic version is</p> <ul id='Grothendieck'> <li><a class='existingWikiWord' href='/nlab/show/diff/Alexander+Grothendieck'>Alexander Grothendieck</a>, <em>On the De Rham cohomology of algebraic varieties</em>, Publications Mathématiques de l’IHÉS <strong>29</strong>, 351-359 (1966), <a href='http://www.numdam.org/item?id=PMIHES_1966__29__95_0'>numdam</a>.</li> </ul> <ul> <li>A. Grothendieck, <em>Crystals and the de Rham cohomology of schemes</em>, in Giraud, Jean; Grothendieck, Alexander; Kleiman, Steven L. et al., Dix Exposés sur la Cohomologie des Schémas, Advanced studies in pure mathematics <strong>3</strong>, Amsterdam: North-Holland, pp. 306–358, <a href='http://www.ams.org/mathscinet-getitem?mr=0269663'>MR0269663</a>, <a href='http://www.math.jussieu.fr/~leila/grothendieckcircle/DixExp.pdf'>pdf</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Robin+Hartshorne'>Robin Hartshorne</a>, <em>On the de Rham cohomology of algebraic varieties</em>, Publ. Mathématiques de l’IHÉS <strong>45</strong> (1975), p. 5-99 <a href='http://www.ams.org/mathscinet-getitem?mr=55:5633'>MR55#5633</a></li> <li>P. Monsky, <em>Finiteness of de Rham cohomology</em>, Amer. J. Math. <strong>94</strong> (1972), 237–245, <a href='http://www.ams.org/mathscinet-getitem?mr=301017'>MR301017</a>, <a href='http://dx.doi.org/10.2307/2373603'>doi</a></li> </ul> <p>See also</p> <ul> <li> <p>Yves André, <em>Comparison theorems between algebraic and analytic De Rham cohomology</em> (<a href='http://www.emis.de/journals/JTNB/2004-2/pages335-355.pdf'>pdf</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Mikhail+Kapranov'>Mikhail Kapranov</a>, <em>DG-Modules over the de Rham complex and the vanishing cycles functor</em>, Lecture Notes in Mathematics <strong>1479</strong>, Springer (1991) [[doi:10.1007/BFb0086264](https://doi.org/10.1007/BFb0086264)]</p> </li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on August 26, 2024 at 07:37:43. 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