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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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" 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 #6 to #7: <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> </div> </div> <p>By a <strong>generalized cohomology theory</strong> is usually meant a contravariant functor on a <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a> satisfying all abstract properties of <a class='existingWikiWord' href='/nlab/show/diff/ordinary+cohomology'>ordinary cohomology</a>, except possibly for the dimension axiom. For more on this see at</p> <ul> <li> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/generalized+%28Eilenberg-Steenrod%29+cohomology'><span><del class='diffmod'> generalized</del><ins class='diffmod'> Whitehead-generalized</ins><del class='diffdel'> (Eilenberg-Steenrod)</del> cohomology</span></a></strong>.</p> <p>(and for the dual concept see at <em><a class='existingWikiWord' href='/nlab/show/diff/generalized+homology'>generalized homology</a></em>).</p> </li> </ul> <p>But there are more general generalizations of the concept of ordinary cohomology, too. For instance there is also</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nonabelian+cohomology'>nonabelian cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sheaf+cohomology'>sheaf cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology'>differential cohomology</a></p> </li> </ul> <p>etc.</p> <p>For a fully general concept of generalized cohomology, see at</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a>.</li> </ul> <table><thead><tr><th /><th><a class='existingWikiWord' href='/nlab/show/diff/homotopy+%28as+an+operation%29'>homotopy</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/homology'>homology</a></th></tr></thead><tbody><tr><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_a598d95055d01a56fe12d38042a77321cd66ae59_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[S^n,-]</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_a598d95055d01a56fe12d38042a77321cd66ae59_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[-,A]</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_a598d95055d01a56fe12d38042a77321cd66ae59_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>⊗</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>(-) \otimes A</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/functor'>covariant</a> <a class='existingWikiWord' href='/nlab/show/diff/hom-functor'>hom</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/contravariant+functor'>contravariant</a> <a class='existingWikiWord' href='/nlab/show/diff/hom-functor'>hom</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/homological+algebra'>homological algebra</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Tor'>Tor</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/end'>end</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/end'>end</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/end'>coend</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-categorical+hom-space'>derived hom space</a> <math class='maruku-mathml' display='inline' id='mathml_a598d95055d01a56fe12d38042a77321cd66ae59_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy='false'>(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbb{R}Hom(S^n,-)</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycles</a> <math class='maruku-mathml' display='inline' id='mathml_a598d95055d01a56fe12d38042a77321cd66ae59_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi><mi>Hom</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbb{R}Hom(-,A)</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/derived+tensor+product'>derived tensor product</a> <math class='maruku-mathml' display='inline' id='mathml_a598d95055d01a56fe12d38042a77321cd66ae59_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><msup><mo>⊗</mo> <mi>𝕃</mi></msup><mi>A</mi></mrow><annotation encoding='application/x-tex'>(-) \otimes^{\mathbb{L}} A</annotation></semantics></math></td></tr> </tbody></table> <p> </p> </div> <div class="revisedby"> <p> Last revised on November 29, 2021 at 07:04:48. 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