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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 #73 to #74: <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'>Čech 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> <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='#by_sections_of_associated_bundles'>By sections of associated <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mi>U</mi></mrow><annotation encoding='application/x-tex'>K U</annotation></semantics></math>-bundles</a></li><li><a href='#ByBundlesOfFredholmOperators'>By bundles of Fredholm operators</a></li><li><a href='#by_twisted_vector_bundles_gerbe_modules'>By twisted vector bundles (gerbe modules)</a></li><li><a href='#by_kktheory_of_twisted_convolution_algebras'>By KK-theory of twisted convolution algebras</a></li></ul></li><li><a href='#other_constructions'>Other constructions</a></li><li><a href='#twists'>Twists</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><ul><li><a href='#general'>General</a></li><li><a href='#ReferencesHigherTwists'>Higher twists</a></li></ul></li></ul></div> <h2 id='idea'>Idea</h2> <p><em>Twisted K-theory</em> is a <a class='existingWikiWord' href='/nlab/show/diff/twisted+cohomology'>twisted cohomology</a> version of (<a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological</a>) <a class='existingWikiWord' href='/nlab/show/diff/K-theory'>K-theory</a>.</p> <p>The most famous twist is by a class in degree 3 <a class='existingWikiWord' href='/nlab/show/diff/ordinary+cohomology'>ordinary cohomology</a> (geometrically a <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>U(1)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/bundle+gerbe'>bundle gerbe</a> or <a class='existingWikiWord' href='/nlab/show/diff/circle+n-group'>circle 2-group</a>-<a class='existingWikiWord' href='/nlab/show/diff/principal+2-bundle'>principal 2-bundle</a>), but there are various other twists.</p> <h2 id='definition'>Definition</h2> <h3 id='by_sections_of_associated_bundles'>By sections of associated <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mi>U</mi></mrow><annotation encoding='application/x-tex'>K U</annotation></semantics></math>-bundles</h3> <p>Write <a class='existingWikiWord' href='/nlab/show/diff/K-theory+spectrum'>KU</a> for the <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a> of complex <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a>. Its degree-0 space is, up to <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a>, the space</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mi>U</mi><mo>×</mo><mi>ℤ</mi><mo>=</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>n</mi></msub><mi>B</mi><mi>U</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'> B U \times \mathbb{Z} = {\lim_\to}_n B U(n) \times \mathbb{Z} </annotation></semantics></math></div> <p>or the space <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Fred</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Fred(\mathcal{H})</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/Fredholm+operator'>Fredholm operator</a>s on some separable <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+space'>Hilbert space</a> <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding='application/x-tex'>\mathcal{H}</annotation></semantics></math>.</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>K</mi><mi>U</mi><msub><mo stretchy='false'>)</mo> <mn>0</mn></msub><mo>≃</mo><mi>B</mi><mi>U</mi><mo>×</mo><mi>ℤ</mi><mo>≃</mo><mi>Fred</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (K U)_0 \simeq B U \times \mathbb{Z} \simeq Fred(\mathcal{H}) \,. </annotation></semantics></math></div> <p>The ordinary <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a> of a suitable <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is given by the set of <a class='existingWikiWord' href='/nlab/show/diff/homotopy+class'>homotopy classes</a> of maps from (the <a class='existingWikiWord' href='/nlab/show/diff/suspension+spectrum'>suspension spectrum</a> of) <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>KU</mi></mrow><annotation encoding='application/x-tex'>KU</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mo>•</mo></msub><mo>≃</mo><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mo stretchy='false'>(</mo><mi>K</mi><mi>U</mi><msub><mo stretchy='false'>)</mo> <mo>•</mo></msub><mo stretchy='false'>]</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> K(X)_\bullet \simeq [X, (K U)_\bullet] \,. </annotation></semantics></math></div> <p>The <a class='existingWikiWord' href='/nlab/show/diff/projective+unitary+group'>projective unitary group</a> <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mi>U</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>P U(\mathcal{H})</annotation></semantics></math> (a <a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a>) acts canonically by <a class='existingWikiWord' href='/nlab/show/diff/automorphism'>automorphism</a>s on <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>K</mi><mi>U</mi><msub><mo stretchy='false'>)</mo> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>(K U)_0</annotation></semantics></math>. (This follows by the identification of <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><msub><mi>U</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>K U_0</annotation></semantics></math> with the space of <a class='existingWikiWord' href='/nlab/show/diff/Fredholm+operator'>Fredholm operators</a>, see <a href='#ByBundlesOfFredholmOperators'>below</a>) Therefore for <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>P \to X</annotation></semantics></math> any <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>PU</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>PU(\mathcal{H})</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/principal+bundle'>principal bundle</a>, we can form the <a class='existingWikiWord' href='/nlab/show/diff/associated+bundle'>associated bundle</a> <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mrow><mi>P</mi><mi>U</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>K</mi><mi>U</mi><msub><mo stretchy='false'>)</mo> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>P \times_{P U(\mathcal{H})} (K U)_0</annotation></semantics></math>.</p> <p>Since the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a> of <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mi>U</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>P U(\mathcal{H})</annotation></semantics></math> is that of an <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Mac+Lane+space'>Eilenberg-MacLane space</a> <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>ℤ</mi><mo>,</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(\mathbb{Z},2)</annotation></semantics></math>, there is precisely one isomorphism class of such bundles representing a class <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha \in H^3(X, \mathbb{Z})</annotation></semantics></math>.</p> <div class='num_defn' id='SpectrumBundDefinition'> <h6 id='definition_2'>Definition</h6> <p>The <strong>twisted K-theory</strong> with twist <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha \in H^3(X, \mathbb{Z})</annotation></semantics></math> is the set of <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a>-classes of <a class='existingWikiWord' href='/nlab/show/diff/section'>section</a>s of such a bundle</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>K</mi> <mi>α</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><msup><mo stretchy='false'>)</mo> <mn>0</mn></msup><mo>:</mo><mo>=</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy='false'>(</mo><msup><mi>P</mi> <mi>α</mi></msup><msub><mo>×</mo> <mrow><mi>P</mi><mi>U</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>K</mi><mi>U</mi><msub><mo stretchy='false'>)</mo> <mn>0</mn></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> K_\alpha(X)^0 := \Gamma_X(P^\alpha \times_{P U(\mathcal{H})} (K U)_0) \,. </annotation></semantics></math></div> <p>Similarily the reduced <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math>-twisted K-theory is the subset</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>K</mi><mo stretchy='false'>˜</mo></mover> <mi>α</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><msup><mo stretchy='false'>)</mo> <mn>0</mn></msup><mo>:</mo><mo>=</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy='false'>(</mo><msup><mi>P</mi> <mi>α</mi></msup><msub><mo>×</mo> <mrow><mi>P</mi><mi>U</mi><mo stretchy='false'>(</mo><mi>ℋ</mi><mo stretchy='false'>)</mo></mrow></msub><mi>B</mi><mi>U</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \tilde K_\alpha(X)^0 := \Gamma_X(P^\alpha \times_{P U(\mathcal{H})} B U) \,. </annotation></semantics></math></div></div> <h3 id='ByBundlesOfFredholmOperators'>By bundles of Fredholm operators</h3> <p>The following is due to (<a href='#AtiyahSinger69'>Atiyah-Singer 69</a>, <a href='#AtiyahSegal04'>Atiyah-Segal 04</a>).</p> <p>Write</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Cl</mi> <mi>n</mi></msub><mo>≔</mo><msup><mi>Cl</mi> <mi>ℂ</mi></msup><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mo stretchy='false'>⟨</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>⟩</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Cl_n \coloneqq Cl^{\mathbb{C}}(\mathbb{R}^n,\langle -,-\rangle)</annotation></semantics></math></p> <p>for the <a class='existingWikiWord' href='/nlab/show/diff/complexification'>complexification</a> of the <a class='existingWikiWord' href='/nlab/show/diff/Clifford+algebra'>Clifford algebra</a> of the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+space'>Cartesian space</a> <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^n</annotation></semantics></math> with its standard <a class='existingWikiWord' href='/nlab/show/diff/inner+product+space'>inner product</a>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>S_n</annotation></semantics></math> for its <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/2\mathbb{Z}</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/graded+vector+space'>graded</a> <a class='existingWikiWord' href='/nlab/show/diff/irreducible+representation'>irreducible module</a> (see at <em><a class='existingWikiWord' href='/nlab/show/diff/spin+representation'>spin representation</a></em>);</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>H_0</annotation></semantics></math> for <a class='existingWikiWord' href='/nlab/show/diff/generalized+the'>the</a> <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/2\mathbb{Z}</annotation></semantics></math>-graded <a class='existingWikiWord' href='/nlab/show/diff/separable+Hilbert+space'>separable Hilbert space</a> whose even and odd part are both infinite-dimensional.</p> </li> </ul> <div class='num_defn' id='Fredn'> <h6 id='definition_3'>Definition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_31' 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>, the <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Fred</mi> <mrow><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>Fred^{(n)}</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/Fredholm+operator'>Fredholm operators</a> on <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub><mo>⊗</mo><msub><mi>H</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>S_n \otimes H_0</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Fred</mi> <mrow><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></msup><mo>≔</mo><mrow><mo>{</mo><mi>F</mi><mo>∈</mo><mi>ℬ</mi><mo stretchy='false'>(</mo><msub><mi>S</mi> <mi>n</mi></msub><mo>⊗</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo stretchy='false'>|</mo><mspace width='thickmathspace' /><mi>F</mi><mspace width='thinmathspace' /><mi>odd</mi><mspace width='thinmathspace' /><mo>,</mo><msup><mi>F</mi> <mo>*</mo></msup><mo>=</mo><mi>F</mi><mspace width='thinmathspace' /><mo>,</mo><msup><mi>F</mi> <mn>2</mn></msup><mo>−</mo><mn>1</mn><mo>∈</mo><mi>𝒦</mi><mo stretchy='false'>(</mo><msub><mi>S</mi> <mi>n</mi></msub><mo>⊗</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo><mo stretchy='false'>[</mo><mi>F</mi><mo>,</mo><mi>γ</mi><mo stretchy='false'>]</mo><mo>=</mo><mn>0</mn><mspace width='thinmathspace' /><mi>for</mi><mspace width='thinmathspace' /><mi>γ</mi><mo>∈</mo><msub><mi>Cl</mi> <mi>n</mi></msub><mo>}</mo></mrow></mrow><annotation encoding='application/x-tex'> Fred^{(n)} \coloneqq \left\{ F \in \mathcal{B}(S_n \otimes H_0) \;|\; F \, odd\,, F^\ast = F \,, F^2 - 1 \in \mathcal{K}(S_n \otimes H_0)\,, [F,\gamma] = 0 \, for\, \gamma \in Cl_n \right\} </annotation></semantics></math></div> <p>(where <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}</annotation></semantics></math> denotes <a class='existingWikiWord' href='/nlab/show/diff/bounded+operator'>bounded operators</a> and <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒦</mi></mrow><annotation encoding='application/x-tex'>\mathcal{K}</annotation></semantics></math> denotes <a class='existingWikiWord' href='/nlab/show/diff/compact+operator'>compact operators</a> and where <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_37' 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> denotes the <a class='existingWikiWord' href='/nlab/show/diff/graded+commutator'>graded commutator</a>) and the <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topology</a> on this set is the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a> induced by the embedding</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Fred</mi> <mrow><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></msup><mo>↪</mo><mi>ℬ</mi><mo stretchy='false'>(</mo><msub><mi>S</mi> <mi>n</mi></msub><mo>⊗</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo><mo>×</mo><mi>𝒦</mi><mo stretchy='false'>(</mo><msub><mi>S</mi> <mi>n</mi></msub><mo>⊗</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> Fred^{(n)} \hookrightarrow \mathcal{B}(S_n \otimes H_0) \times \mathcal{K}(S_n \otimes H_0) </annotation></semantics></math></div> <p>given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>↦</mo><mo stretchy='false'>(</mo><mi>F</mi><mo>,</mo><msup><mi>F</mi> <mn>2</mn></msup><mo>−</mo><mn>1</mn><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> F\mapsto (F, F^2 - 1) \,, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}</annotation></semantics></math> is equipped with the <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a> and <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒦</mi></mrow><annotation encoding='application/x-tex'>\mathcal{K}</annotation></semantics></math> with the <a class='existingWikiWord' href='/nlab/show/diff/norm+topology'>norm topology</a>.</p> </div> <p>(<a href='#AtiyahSinger69'>Atiyah-Singer 69, p. 7</a>, <a href='#AtiyahSegal04'>Atiyah-Segal 04, p. 21</a>, <a href='#FreedHopkinsTeleman11'>Freed-Hopkins-Teleman 11, def. A.40</a>)</p> <p>These spaces indeed form a model for the <a class='existingWikiWord' href='/nlab/show/diff/K-theory+spectrum'>KU</a> <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a>:</p> <div class='num_prop'> <h6 id='proposition'>Proposition</h6> <p>For all <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_42' 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> there are <a class='existingWikiWord' href='/nlab/show/diff/natural+equivalence'>natural</a> <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalences</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Fred</mi> <mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msup><mover><mo>⟶</mo><mo>≃</mo></mover><mi>Ω</mi><msup><mi>Fred</mi> <mrow><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></msup></mrow><annotation encoding='application/x-tex'> Fred^{(n+1)} \stackrel{\simeq}{\longrightarrow} \Omega Fred^{(n)} </annotation></semantics></math></div> <p>and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Fred</mi> <mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow></msup><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>Fred</mi> <mrow><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></msup></mrow><annotation encoding='application/x-tex'> Fred^{(n+2)} \stackrel{\simeq}{\longrightarrow} Fred^{(n)} </annotation></semantics></math></div> <p>between the spaces of graded <a class='existingWikiWord' href='/nlab/show/diff/Fredholm+operator'>Fredholm operators</a> of def. <a class='maruku-ref' href='#Fredn'>2</a> and their <a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop spaces</a>.</p> </div> <p>(<a href='#AtiyahSinger69'>Atiyah-Singer 69, theorem B(k)</a>, <a href='#AtiyahSegal04'>Atiyah-Segal 04 (4.2)</a>, <a href='#FreedHopkinsTeleman11'>Freed-Hopkins-Teleman 11, below def. A.40</a>)</p> <p>Regard the <a class='existingWikiWord' href='/nlab/show/diff/stable+unitary+group'>stable unitary group</a> <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo stretchy='false'>(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>U(H_0)</annotation></semantics></math> as equipped with the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a> induced by the inclusion</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo stretchy='false'>(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo><mover><mo>↪</mo><mrow><mo stretchy='false'>(</mo><mi>id</mi><mo>,</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>)</mo></mrow></mover><mi>ℬ</mi><mo stretchy='false'>(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo><mo>×</mo><mi>ℬ</mi><mo stretchy='false'>(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> U(H_0) \stackrel{(id,(-)^{-1})}{\hookrightarrow} \mathcal{B}(H_0)\times\mathcal{B}(H_0) </annotation></semantics></math></div> <p>from the <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a> on the <a class='existingWikiWord' href='/nlab/show/diff/bounded+operator'>bounded linear operators</a>.</p> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/conjugation+action'>conjugation action</a> of the <a class='existingWikiWord' href='/nlab/show/diff/stable+unitary+group'>stable unitary group</a> <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo stretchy='false'>(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>U(H_0)</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Fred</mi> <mrow><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>Fred^{(n)}</annotation></semantics></math>, def. <a class='maruku-ref' href='#Fredn'>2</a>, is <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous</a>.</p> </div> <p>This follows with (<a href='#AtiyahSegal04'>Atiyah-Segal 04, prop. A1.1</a>).</p> <div class='num_defn'> <h6 id='definition_4'>Definition</h6> <p>Given a class <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>χ</mi><mo>∈</mo><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\chi \in H^3(X,\mathbb{Z})</annotation></semantics></math> represented by a <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>PU</mi><mo stretchy='false'>(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>PU(H_0)</annotation></semantics></math>-bundle <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>P \to X</annotation></semantics></math> with <a class='existingWikiWord' href='/nlab/show/diff/associated+bundle'>associated</a> Fredholm bundle</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Fred</mi> <mrow><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mo>+</mo><mi>χ</mi></mrow></msup><mo>≔</mo><mi>P</mi><munder><mo>×</mo><mrow><mi>PU</mi><mo stretchy='false'>(</mo><msub><mi>H</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo></mrow></munder><msup><mi>Fred</mi> <mrow><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></msup><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> Fred^{(n)+ \chi} \coloneqq P \underset{PU(H_0)}{\times} Fred^{(n)} \,, </annotation></semantics></math></div> <p>then the corresponding <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>χ</mi></mrow><annotation encoding='application/x-tex'>\chi</annotation></semantics></math>-twisted <a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a> <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a> is that consisting of the <a class='existingWikiWord' href='/nlab/show/diff/space+of+sections'>spaces of sections</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Γ</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><msup><mi>Fred</mi> <mrow><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mo>+</mo><mi>χ</mi></mrow></msup><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \Gamma(X, Fred^{(n)+ \chi}) \,. </annotation></semantics></math></div></div> <p>(<a href='#FreedHopkinsTeleman11'>Freed-Hopkins-Teleman 11, def. 3.14</a>)</p> <h3 id='by_twisted_vector_bundles_gerbe_modules'>By twisted vector bundles (gerbe modules)</h3> <div class='num_defn' id='TwBundDefinition'> <h6 id='definition_5'>Definition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha \in H^3(X, \mathbb{Z})</annotation></semantics></math> be a class in degree-3 <a class='existingWikiWord' href='/nlab/show/diff/integral+cohomology'>integral cohomology</a> and let <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo>∈</mo><msup><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mn>3</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant='bold'><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>P \in \mathbf{H}^3(X, \mathbf{B}^2 U(1))</annotation></semantics></math> be any <a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycle</a> representative, which we may think of either as giving a <a class='existingWikiWord' href='/nlab/show/diff/circle+n-bundle+with+connection'>circle 2-bundle</a> or a <a class='existingWikiWord' href='/nlab/show/diff/bundle+gerbe'>bundle gerbe</a>.</p> <p>Write <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>TwBund</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>P</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>TwBund(X, P)</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/diff/groupoid'>groupoid</a> of <a class='existingWikiWord' href='/nlab/show/diff/twisted+bundle'>twisted bundle</a>s on <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with twist given by <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math>. Then let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>K</mi><mo stretchy='false'>˜</mo></mover> <mi>α</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><mi>TwBund</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>P</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \tilde K_\alpha(X) := TwBund(X,P) </annotation></semantics></math></div> <p>be the set of <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> classes of twisted bundles. Call this the <strong>twisted K-theory</strong> of <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with twist <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math>.</p> </div> <blockquote> <p>(Some technical details need to be added for the non-torsion case.)</p> </blockquote> <div class='num_prop'> <h6 id='proposition_3'>Proposition</h6> <p>This definition of twisted <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>K</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>K_0</annotation></semantics></math> is equivalent to that of prop. <a class='maruku-ref' href='#SpectrumBundDefinition'>1</a>.</p> </div> <p>This is (<a href='#CBMMS'>CBMMS, prop. 6.4, prop. 7.3</a>).</p> <h3 id='by_kktheory_of_twisted_convolution_algebras'>By KK-theory of twisted convolution algebras</h3> <p>A <a class='existingWikiWord' href='/nlab/show/diff/circle+n-group'>circle 2-group</a> <a class='existingWikiWord' href='/nlab/show/diff/principal+2-bundle'>principal 2-bundle</a> is also incarnated as a <a class='existingWikiWord' href='/nlab/show/diff/centrally+extended+groupoid'>centrally extended Lie groupoid</a>. The corresponding <a class='existingWikiWord' href='/nlab/show/diff/category+algebra'>twisted groupoid convolution algebra</a> has as its <a class='existingWikiWord' href='/nlab/show/diff/operator+K-theory'>operator K-theory</a> the twisted K-theory of the base space (or base-<a class='existingWikiWord' href='/nlab/show/diff/stack'>stack</a>). See at <em><a class='existingWikiWord' href='/nlab/show/diff/KK-theory'>KK-theory</a></em> for more on this.</p> <h2 id='other_constructions'>Other constructions</h2> <p>Let <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Vectr</mi></mrow><annotation encoding='application/x-tex'>Vectr</annotation></semantics></math> be the <a class='existingWikiWord' href='/nlab/show/diff/stack'>stack</a> of <a class='existingWikiWord' href='/nlab/show/diff/vectorial+bundle'>vectorial bundles</a>. (If we just take <a class='existingWikiWord' href='/nlab/show/diff/vector+bundle'>vector bundle</a>s we get a notion of twisted K-theory that only allows twists that are finite order elements in their <a class='existingWikiWord' href='/nlab/show/diff/cohomology+group'>cohomology group</a>).</p> <p>There is a canonical morphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ρ</mi><mo>:</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>U</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo>→</mo><mi>Vect</mi><mo>↪</mo><mi>Vectr</mi></mrow><annotation encoding='application/x-tex'> \rho : \mathbf{B} U(1) \to Vect \hookrightarrow Vectr </annotation></semantics></math></div> <p>coming from the standard <a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a> of the <a class='existingWikiWord' href='/nlab/show/diff/group'>group</a> <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>U(1)</annotation></semantics></math>.</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mstyle mathvariant='bold'><mi>B</mi></mstyle> <mo>⊗</mo></msub><mi>Vectr</mi></mrow><annotation encoding='application/x-tex'>\mathbf{B}_{\otimes} Vectr</annotation></semantics></math> be the <a class='existingWikiWord' href='/nlab/show/diff/delooping'>delooping</a> of <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Vectr</mi></mrow><annotation encoding='application/x-tex'>Vectr</annotation></semantics></math> with respect to the <a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a> <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal structure</a> (not the additive structure).</p> <p>Then we have a <a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>fibration sequence</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Vectr</mi><mo>→</mo><mo>*</mo><mo>→</mo><msub><mstyle mathvariant='bold'><mi>B</mi></mstyle> <mo>⊗</mo></msub><mi>Vectr</mi></mrow><annotation encoding='application/x-tex'> Vectr \to {*} \to \mathbf{B}_\otimes Vectr </annotation></semantics></math></div> <p>of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(infinity,1)-categories</a> (instead of <a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>infinity-groupoid</a>s).</p> <p>The entire morphism above deloops</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>ρ</mi><mo>:</mo><msup><mstyle mathvariant='bold'><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo>→</mo><msub><mstyle mathvariant='bold'><mi>B</mi></mstyle> <mo>⊗</mo></msub><mi>Vect</mi><mo>↪</mo><msub><mstyle mathvariant='bold'><mi>B</mi></mstyle> <mo>⊗</mo></msub><mi>Vectr</mi></mrow><annotation encoding='application/x-tex'> \mathbf{B}\rho : \mathbf{B}^2 U(1) \to \mathbf{B}_\otimes Vect \hookrightarrow \mathbf{B}_{\otimes} Vectr </annotation></semantics></math></div> <p>being the standard representation of the <a class='existingWikiWord' href='/nlab/show/diff/2-group'>2-group</a> <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>U</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{B}U(1)</annotation></semantics></math>.</p> <p>From the general nonsense of <a class='existingWikiWord' href='/nlab/show/diff/twisted+cohomology'>twisted cohomology</a> this induces canonically now for every <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mstyle mathvariant='bold'><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{B}^2 U(1)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cocycle</a> <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> (for instance given by a <a class='existingWikiWord' href='/nlab/show/diff/bundle+gerbe'>bundle gerbe</a>) a notion of <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math>-twisted <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Vectr</mi></mrow><annotation encoding='application/x-tex'>Vectr</annotation></semantics></math>-cohomology:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mi>c</mi></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>Vectr</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mrow><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>ρ</mi><mo>∘</mo><mi>c</mi></mrow></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant='bold'><mi>H</mi></mstyle><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><msub><mstyle mathvariant='bold'><mi>B</mi></mstyle> <mo>⊗</mo></msub><mi>Vectr</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ \mathbf{H}^c(X, Vectr) &\to& {*} \\ \downarrow && \downarrow^{\mathbf{B}\rho \circ c} \\ {*} &\to& \mathbf{H}(X,\mathbf{B}_\otimes Vectr) } \,. </annotation></semantics></math></div> <p>After unwrapping what this means, the result of (<a href='#Gomi'>Gomi</a>) shows that <a class='existingWikiWord' href='/nlab/show/diff/concordance'>concordance</a> classes in <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mstyle mathvariant='bold'><mi>H</mi></mstyle> <mi>c</mi></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>Vectr</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbf{H}^c(X,Vectr)</annotation></semantics></math> yield twisted K-theory.</p> <h2 id='twists'>Twists</h2> <p>By the general discussion of <a class='existingWikiWord' href='/nlab/show/diff/twisted+cohomology'>twisted cohomology</a> the <a class='existingWikiWord' href='/nlab/show/diff/moduli+space'>moduli space</a> for the twists of <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>periodic complex K-theory</a> <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>KU</mi></mrow><annotation encoding='application/x-tex'>KU</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/Picard+infinity-group'>Picard ∞-group</a> in <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>KU</mi><mi>Mod</mi></mrow><annotation encoding='application/x-tex'>KU Mod</annotation></semantics></math>. The “geometric” twists among these have as moduli space the non-connected delooping <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>bgl</mi> <mn>1</mn> <mo>*</mo></msubsup><mo stretchy='false'>(</mo><mi>KU</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>bgl_1^\ast(KU)</annotation></semantics></math> of the <a class='existingWikiWord' href='/nlab/show/diff/infinity-group+of+units'>∞-group of units</a> of <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>KU</mi></mrow><annotation encoding='application/x-tex'>KU</annotation></semantics></math>.</p> <p>A model for this in 4-truncation is given by <a class='existingWikiWord' href='/nlab/show/diff/super+line+2-bundle'>super line 2-bundles</a>. For the moment see there for further discussion and further references.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <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/topological+K-theory'>topological K-theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/KR+cohomology+theory'>KR-theory</a></p> </li> <li> <p><strong>twisted K-theory</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/twisted+equivariant+K-theory'>twisted equivariant K-theory</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/twisted+ad-equivariant+K-theory'>twisted ad-equivariant K-theory</a></li> </ul> </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/twisted+differential+K-theory'>twisted differential K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+integration+in+K-theory'>fiber integration in K-theory</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/twisted+ordinary+cohomology'>twisted ordinary cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/twisted+Bredon+cohomology'>twisted Bredon cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/twisted+cohomotopy'>twisted Cohomotopy</a></p> </li> </ul> <h2 id='references'>References</h2> <h3 id='general'>General</h3> <p>The concept of twisted K-theory originates in</p> <ul> <li id='Karoubi68'> <p><a class='existingWikiWord' href='/nlab/show/diff/Max+Karoubi'>Max Karoubi</a>, <em>Algèbres de Clifford et K-théorie.</em> Ann. Sci. Ecole Norm. Sup. (4), pp. 161-270 (1968) (<a href='http://www.numdam.org/item/ASENS_1968_4_1_2_161_0'>numdam:ASENS_1968_4_1_2_161_0</a>)</p> </li> <li id='DonovanKaroubi70'> <p><a class='existingWikiWord' href='/nlab/show/diff/Peter+Donovan'>Peter Donovan</a>, <a class='existingWikiWord' href='/nlab/show/diff/Max+Karoubi'>Max Karoubi</a>, <em>Graded Brauer groups and <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>-theory with local coefficients</em>, Publications Mathématiques de l’IHÉS, 38 (1970), p. 5-25 (<a href='http://www.numdam.org/item?id=PMIHES_1970__38__5_0'>numdam:PMIHES_1970__38__5_0</a>)</p> </li> </ul> <p>which discusses twists of <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>KO</mi></mrow><annotation encoding='application/x-tex'>KO</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>KU</mi></mrow><annotation encoding='application/x-tex'>KU</annotation></semantics></math> over some <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> by elements in <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H^0(X,\mathbb{Z}_2) \times H^1(X,\mathbb{Z}_2) \times H^3(X, \mathbb{Z})</annotation></semantics></math>.</p> <p>Including the twist in degree 1 (see also at <em><a class='existingWikiWord' href='/nlab/show/diff/PU%28%E2%84%8B%29'>PU(ℋ)</a></em>):</p> <ul> <li id='Parker88'><a class='existingWikiWord' href='/nlab/show/diff/Ellen+Maycock+Parker'>Ellen Maycock Parker</a>, around Prop. 2.2 of: <em>The Brauer Group of Graded Continuous Trace <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-Algebras</em>, Transactions of the American Mathematical Society <strong>308</strong> 1 (1988) [[jstor:2000953](https://www.jstor.org/stable/2000953)]</li> </ul> <p>reviewed in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Alan+Carey'>Alan Carey</a>, <a class='existingWikiWord' href='/nlab/show/diff/Bai-Ling+Wang'>Bai-Ling Wang</a>, p. 5 of: <em>Thom isomorphism and Push-forward map in twisted K-theory</em>, Journal of K-Theory <strong>1</strong> 2 (2008) 357-393 (<a href='https://arxiv.org/abs/math/0507414'>arXiv:math/0507414</a>, <a href='https://doi.org/10.1017/is007011015jkt011'>doi:10.1017/is007011015jkt011</a>)</li> </ul> <p>The formulation in terms of sections of Fredholm bundles seems to go back to</p> <ul> <li id='Rosenberg89'><a class='existingWikiWord' href='/nlab/show/diff/Jonathan+Rosenberg'>Jonathan Rosenberg</a>, <em>Continuous-trace algebras from the bundle theoretic point of view</em>, J. Austral. Math. Soc. Ser. A 47 (1989), no. 3, 368-381 (<a href='https://doi.org/10.1017/S1446788700033097'>doi:10.1017/S1446788700033097</a>)</li> </ul> <p>and is expanded on in:</p> <ul> <li id='FreedHopkinsTeleman02'> <p><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Freed'>Daniel Freed</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michael+Hopkins'>Michael Hopkins</a>, <a class='existingWikiWord' href='/nlab/show/diff/Constantin+Teleman'>Constantin Teleman</a>, Diagram (2.6) in: <em>Twisted equivariant K-theory with complex coefficients</em>, Journal of Topology, Volume 1, Issue 1, 2007 (<a href='https://arxiv.org/abs/math/0206257'>arXiv:math/0206257</a>, <a href='https://doi.org/10.1112/jtopol/jtm001'>doi:10.1112/jtopol/jtm001</a>)</p> </li> <li id='AtiyahSegal04'> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael+Atiyah'>Michael Atiyah</a>, <a class='existingWikiWord' href='/nlab/show/diff/Graeme+Segal'>Graeme Segal</a>, <em>Twisted K-theory</em>, Ukrainian Math. Bull. <strong>1</strong><span><del class='diffmod'> ,</del><ins class='diffmod'> </ins> 3 (2004)<del class='diffmod'> (</del><ins class='diffmod'> [[arXiv:math/0407054](http://arxiv.org/abs/math/0407054),</ins></span><del class='diffdel'><a href='http://arxiv.org/abs/math/0407054'>arXiv:math/0407054</a></del><del class='diffdel'>, </del><a href='http://iamm.su/en/journals/j879/?VID=10'>journal page</a>, <a href='http://iamm.su/upload/iblock/45e/t1-n3-287-330.pdf'>published pdf</a><span><del class='diffmod'> )</del><ins class='diffmod'> ]</ins></span></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>Peter May</a>, <a class='existingWikiWord' href='/nlab/show/diff/Johann+Sigurdsson'>Johann Sigurdsson</a>, Section 22.3 of: <em><a class='existingWikiWord' href='/nlab/show/diff/Parametrized+Homotopy+Theory'>Parametrized Homotopy Theory</a></em>, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (<a href='https://bookstore.ams.org/surv-132'>ISBN:978-0-8218-3922-5</a>, <a href='https://arxiv.org/abs/math/0411656'>arXiv:math/0411656</a>, <a href='http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf'>pdf</a>)</p> </li> <li id='AtiyahSegal05'> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael+Atiyah'>Michael Atiyah</a>, <a class='existingWikiWord' href='/nlab/show/diff/Graeme+Segal'>Graeme Segal</a>, <em>Twisted K-theory and cohomology</em> (<a href='http://arxiv.org/abs/math/0510674'>arXiv:math/0510674</a>)</p> </li> <li id='AndoBlumbergGepner10'> <p><a class='existingWikiWord' href='/nlab/show/diff/Matthew+Ando'>Matthew Ando</a>, <a class='existingWikiWord' href='/nlab/show/diff/Andrew+Blumberg'>Andrew Blumberg</a>, <a class='existingWikiWord' href='/nlab/show/diff/David+Gepner'>David Gepner</a>, sections 2.1 and 7 of: <em>Twists of K-theory and TMF</em>, in <a class='existingWikiWord' href='/nlab/show/diff/Jonathan+Rosenberg'>Jonathan Rosenberg</a> et al. (eds.), <em>Superstrings, Geometry, Topology, and <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebras</em>, volume 81 of <em>Proceedings of Symposia in Pure Mathematics</em>, 2009 (<a href='http://arxiv.org/abs/1002.3004'>arXiv:1002.3004</a>, <a href='https://doi.org/10.1090/pspum/081'>doi:10.1090/pspum/081</a>)</p> </li> </ul> <p>A comprehensive account of twisted K-theory with twists in <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H^3(X, \mathbb{Z})</annotation></semantics></math> is in:</p> <ul> <li id='AtiyahSinger69'><a class='existingWikiWord' href='/nlab/show/diff/Michael+Atiyah'>Michael Atiyah</a>, <a class='existingWikiWord' href='/nlab/show/diff/Isadore+Singer'>Isadore Singer</a>, <em>Index theory for skew-adjoint Fredholm operators</em>, Publications Mathématiques de l’Institut des Hautes Études Scientifiques January 1969, Volume 37, Issue 1, pp 5-26 (<a href='http://www.maths.ed.ac.uk/~aar/papers/askew.pdf'>pdf</a>)</li> </ul> <p>and for more general twists in</p> <ul> <li id='FreedHopkinsTeleman11'><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Freed'>Daniel Freed</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michael+Hopkins'>Michael Hopkins</a>, <a class='existingWikiWord' href='/nlab/show/diff/Constantin+Teleman'>Constantin Teleman</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Loop+Groups+and+Twisted+K-Theory'>Loop Groups and Twisted K-Theory</a> I</em> , J. Topology, 4 (2011), 737-789 (<a href='http://arxiv.org/abs/0711.1906'>arXiv:0711.1906</a>)</li> </ul> <p>See also</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Max+Karoubi'>Max Karoubi</a>, <em>Twisted bundles and twisted K-theory</em>, in: Guillermo Cortiñas (ed.) <em>Topics in Noncommutative Geometry</em> (<a href='https://arxiv.org/abs/1012.2512'>arXiv:1012.2512</a>, <a href='https://bookstore.ams.org/cmip-16'>ISBN:978-0-8218-6864-5</a>)</li> </ul> <p>Textbook accounts:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Dale+Husem%C3%B6ller'>Dale Husemöller</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michael+Joachim'>Michael Joachim</a>, <a class='existingWikiWord' href='/nlab/show/diff/Branislav+Jurco'>Branislav Jurčo</a>, <a class='existingWikiWord' href='/nlab/show/diff/Martin+Schottenloher'>Martin Schottenloher</a>, Part IV of: <em><a class='existingWikiWord' href='/nlab/show/diff/Basic+Bundle+Theory+and+K-Cohomology+Invariants'>Basic Bundle Theory and K-Cohomology Invariants</a></em>, Springer Lecture Notes in Physics <strong>726</strong>, 2008, (<a href='http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726.pdf'>pdf</a>, <a href='https://link.springer.com/book/10.1007/978-3-540-74956-1'>doi:10.1007/978-3-540-74956-1</a>)</li> </ul> <p>Discussion of <a class='existingWikiWord' href='/nlab/show/diff/twisted+ad-equivariant+K-theory'>twisted ad-equivariant K-theory</a> and relation to <a class='existingWikiWord' href='/nlab/show/diff/loop+group'>loop group</a> <a class='existingWikiWord' href='/nlab/show/diff/representation'>representations</a> and the <a class='existingWikiWord' href='/nlab/show/diff/Verlinde+ring'>Verlinde ring</a>, now again with twists in <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H^0(X,\mathbb{Z}_2) \times H^1(X,\mathbb{Z}_2) \times H^3(X, \mathbb{Z})</annotation></semantics></math>, is in the series of articles</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Freed'>Daniel Freed</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michael+Hopkins'>Michael Hopkins</a>, <a class='existingWikiWord' href='/nlab/show/diff/Constantin+Teleman'>Constantin Teleman</a>, <em>Twisted K-theory and loop group representations</em> <a href='http://arxiv.org/abs/math/0312155'>arXiv:math/0312155</a>; <em><a class='existingWikiWord' href='/nlab/show/diff/Loop+Groups+and+Twisted+K-Theory'>Loop Groups and Twisted K-Theory</a> I</em> (<a href='http://arxiv.org/abs/0711.1906'>arXiv:0711.1906</a>) ; <em><a class='existingWikiWord' href='/nlab/show/diff/Loop+Groups+and+Twisted+K-Theory'>Loop Groups and Twisted K-Theory</a> II</em> (<a href='http://arxiv.org/abs/math/0511232'>arXiv:math/0511232</a>)</li> </ul> <p>The result on twisted K-groups has been lifted to an equivalence of categories in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Freed'>Daniel Freed</a>, <a class='existingWikiWord' href='/nlab/show/diff/Constantin+Teleman'>Constantin Teleman</a>, <em>Dirac families for loop groups as matrix factorizations</em>, <a href='http://arxiv.org/abs/1409.6051'>arxiv/1409.6051</a></li> </ul> <p>Discussion in terms of <a class='existingWikiWord' href='/nlab/show/diff/Karoubi+K-theory'>Karoubi K-theory</a>/<a class='existingWikiWord' href='/nlab/show/diff/Clifford+module+bundle'>Clifford module bundles</a> is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Max+Karoubi'>Max Karoubi</a>, <em>Clifford modules and twisted K-theory</em>, Proceedings of the International Conference on Clifford algebras (ICCA7) (<a href='http://arxiv.org/abs/0801.2794'>arXiv:0801.2794</a>)</li> </ul> <p>See the references at <em><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module+bundle'>(infinity,1)-vector bundle</a></em> for more on this.</p> <p>Discussion in terms of <a class='existingWikiWord' href='/nlab/show/diff/twisted+bundle'>twisted bundles</a>/<a class='existingWikiWord' href='/nlab/show/diff/bundle+gerbe+module'>bundle gerbe modules</a> is in</p> <ul> <li id='CBMMS'><a class='existingWikiWord' href='/nlab/show/diff/Peter+Bouwknegt'>Peter Bouwknegt</a>, <a class='existingWikiWord' href='/nlab/show/diff/Alan+Carey'>Alan Carey</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mathai+Varghese'>Varghese Mathai</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michael+Murray'>Michael Murray</a>, <a class='existingWikiWord' href='/nlab/show/diff/Danny+Stevenson'>Danny Stevenson</a>, <em>Twisted K-theory and K-theory of bundle gerbes</em>, Commun Math Phys, 228 (2002) 17-49 (<a href='http://arxiv.org/abs/hep-th/0106194'>arXiv:hep-th/0106194</a>, <a href='https://doi.org/10.1007/s002200200646'>doi:10.1007/s002200200646</a>)</li> </ul> <p>but apparently contains a mistake, as pointed out in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Alan+Carey'>Alan Carey</a>, <a class='existingWikiWord' href='/nlab/show/diff/Bai-Ling+Wang'>Bai-Ling Wang</a>, top of p. 10 in <em>Thom isomorphism and Push-forward map in twisted K-theory</em>, Journal of K-Theory <strong>1</strong> 2 (2008) 357-393 (<a href='https://arxiv.org/abs/math/0507414'>arXiv:math/0507414</a>, <a href='https://doi.org/10.1017/is007011015jkt011'>doi:10.1017/is007011015jkt011</a>)</li> </ul> <p>The generalization of this to <a class='existingWikiWord' href='/nlab/show/diff/groupoid+K-theory'>groupoid K-theory</a> is in (<a href='#FHT07'>FHT 07, around p. 26</a>) and</p> <ul> <li id='TuXuLG03'><a class='existingWikiWord' href='/nlab/show/diff/Jean-Louis+Tu'>Jean-Louis Tu</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ping+Xu'>Ping Xu</a>, <a class='existingWikiWord' href='/nlab/show/diff/Camille+Laurent-Gengoux'>Camille Laurent-Gengoux</a>, <em>Twisted K-theory of differentiable stacks</em> (<a href='http://arxiv.org/abs/math/0306138'>arXiv:math/0306138</a>)</li> </ul> <p>(which establishes the relation to <a class='existingWikiWord' href='/nlab/show/diff/KK-theory'>KK-theory</a>).</p> <ul> <li id='Karoubi'> <p><a class='existingWikiWord' href='/nlab/show/diff/Max+Karoubi'>Max Karoubi</a>, <em>Twisted bundles and twisted K-theory</em>, <a href='http://arxiv.org/abs/1012.2512'>arxiv/1012.2512</a></p> </li> <li id='Pennig'> <p><a class='existingWikiWord' href='/nlab/show/diff/Ulrich+Pennig'>Ulrich Pennig</a>, <em>Twisted K-theory with coefficients in <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebras</em>, (<a href='http://arxiv.org/abs/1103.4096'>arXiv:1103.4096</a>)</p> </li> </ul> <p>Comparison between <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theoretic</a> and <a class='existingWikiWord' href='/nlab/show/diff/operator+algebra'>operator algebraic</a> constructions:</p> <ul> <li id='HebestreitSagave19'><a class='existingWikiWord' href='/nlab/show/diff/Fabian+Hebestreit'>Fabian Hebestreit</a>, <a class='existingWikiWord' href='/nlab/show/diff/Steffen+Sagave'>Steffen Sagave</a>, <em>Homotopical and operator algebraic twisted K-theory</em>, Mathematische Annalen <strong>378</strong> (2020) 1021-1059 (<a href='https://arxiv.org/abs/1904.01872'>arXiv:1904.01872</a>, <a href='https://doi.org/10.1007/s00208-020-02066-6'>doi:10.1007/s00208-020-02066-6</a>)</li> </ul> <p>Discussion in terms of <a class='existingWikiWord' href='/nlab/show/diff/vectorial+bundle'>vectorial bundles</a> is in</p> <ul> <li id='Gomi'> <p><a class='existingWikiWord' href='/nlab/show/diff/Kiyonori+Gomi'>Kiyonori Gomi</a>, <em>Twisted K-theory and finite-dimensional approximation</em> (<a href='http://arxiv.org/abs/0803.2327'>arXiv:0803.2327</a>)</p> </li> <li id='GomiTerashima'> <p><a class='existingWikiWord' href='/nlab/show/diff/Kiyonori+Gomi'>Kiyonori Gomi</a>, <a class='existingWikiWord' href='/nlab/show/diff/Yuji+Terashima'>Yuji Terashima</a>, <em>Chern-Weil Construction for Twisted K-Theory</em>, Communication ins Mathematical Physics, Volume 299, Number 1, 225-254 (<a href='https://doi.org/10.1007/s00220-010-1080-1'>doi:10.1007/s00220-010-1080-1</a>)</p> </li> </ul> <p>Discussion of combined <a class='existingWikiWord' href='/nlab/show/diff/twisted+K-theory'>twisted</a> <a class='existingWikiWord' href='/nlab/show/diff/equivariant+K-theory'>equivariant</a> <a class='existingWikiWord' href='/nlab/show/diff/KR+cohomology+theory'>KR-theory</a> on <a class='existingWikiWord' href='/nlab/show/diff/orbifold'>orbi-</a> <a class='existingWikiWord' href='/nlab/show/diff/orientifold'>orientifolds</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/El-ka%C3%AFoum+M.+Moutuou'>El-kaïoum M. Moutuou</a>, <em>Twistings of KR for Real groupoids</em> (<a href='http://arxiv.org/abs/1110.6836'>arXiv:1110.6836</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/El-ka%C3%AFoum+M.+Moutuou'>El-kaïoum M. Moutuou</a>, <em>Graded Brauer groups of a groupoid with involution</em>, J. Funct. Anal. 266 (2014), no.5 (<a href='https://arxiv.org/abs/1202.2057'>arXiv:1202.2057</a>)</p> </li> <li id='Freed12'> <p><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Freed'>Daniel Freed</a>, <em>Lectures on twisted K-theory and orientifolds</em>, lectures at ESI Vienna, 2012 (<a class='existingWikiWord' href='/nlab/files/FreedESI2012.pdf' title='pdf'>pdf</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Freed'>Daniel Freed</a>, <a class='existingWikiWord' href='/nlab/show/diff/Gregory+Moore'>Gregory Moore</a>, Section 7 of: <em>Twisted equivariant matter</em>, Ann. Henri Poincaré (2013) 14: 1927 (<a href='https://arxiv.org/abs/1208.5055'>arXiv:1208.5055</a>)</p> </li> <li id='Gomi17'> <p><a class='existingWikiWord' href='/nlab/show/diff/Kiyonori+Gomi'>Kiyonori Gomi</a>, <em>Freed-Moore K-theory</em> (<a href='https://arxiv.org/abs/1705.09134'>arXiv:1705.09134</a>, <a href='http://inspirehep.net/record/1601772'>spire:1601772</a>)</p> </li> </ul> <p>Review:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jonathan+Rosenberg'>Jonathan Rosenberg</a>, <em>Twisted cohomology</em>, in <em><a class='existingWikiWord' href='/nlab/show/diff/Encyclopedia+of+Mathematical+Physics+2nd+ed'>Encyclopedia of Mathematical Physics 2nd ed</a></em>, Elsevier (2024) [[arXiv:2401.03966](https://arxiv.org/abs/2401.03966)]</li> </ul> <p>Discussion of twisted <a class='existingWikiWord' href='/nlab/show/diff/K-homology'>K-homology</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Bai-Ling+Wang'>Bai-Ling Wang</a>, <em>Gemometric cycles, index theory and twisted K-homology</em> (<a href='https://arxiv.org/abs/0710.1625'>arXiv:0710.1625</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Eckhard+Meinrenken'>Eckhard Meinrenken</a>, <em>Twisted K-homology and group-valued moment maps</em>, International Mathematics Research Notices 2012 (20) (2012), 4563–4618 (<a href='https://arxiv.org/abs/1008.1261'>arXiv:1008.1261</a>)</p> </li> <li> <p>Bei Liu, <em>Twisted K-homology,Geometric cycles and T-duality</em> (<a href='https://arxiv.org/abs/1411.1575'>arXiv:1411.1575</a>)</p> </li> </ul> <p>Discussion of combined <a class='existingWikiWord' href='/nlab/show/diff/twisted+K-theory'>twisted</a> and <a class='existingWikiWord' href='/nlab/show/diff/equivariant+K-theory'>equivariant</a> and <a class='existingWikiWord' href='/nlab/show/diff/KR+cohomology+theory'>real</a> K-theory</p> <ul> <li id='Gomi17'><a class='existingWikiWord' href='/nlab/show/diff/Kiyonori+Gomi'>Kiyonori Gomi</a>, <em>Freed-Moore K-theory</em> (<a href='https://arxiv.org/abs/1705.09134'>arXiv:1705.09134</a>)</li> </ul> <p>Discussion of <a class='existingWikiWord' href='/nlab/show/diff/twisted+K-theory'>twisted</a> <a class='existingWikiWord' href='/nlab/show/diff/differential+K-theory'>differential</a> <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>K-theory</a> and its relation to <a class='existingWikiWord' href='/nlab/show/diff/D-brane'>D-brane charge</a> in <a class='existingWikiWord' href='/nlab/show/diff/type+II+string+theory'>type II string theory</a> (see at <em><a class='existingWikiWord' href='/nlab/show/diff/D-brane+charge+quantization+in+K-theory'>K-theory classification of D-brane charge</a></em>):</p> <ul> <li id='GradySati19a'><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Grady'>Daniel Grady</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hisham+Sati'>Hisham Sati</a>, <em>Ramond-Ramond fields and twisted differential K-theory</em> (<a href='https://arxiv.org/abs/1903.08843'>arXiv:1903.08843</a>)</li> </ul> <p>Discussion of <a class='existingWikiWord' href='/nlab/show/diff/twisted+K-theory'>twisted</a> <a class='existingWikiWord' href='/nlab/show/diff/differential+K-theory'>differential</a> <a class='existingWikiWord' href='/nlab/show/diff/KO-theory'>orthogonal</a> <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>K-theory</a> and its relation to <a class='existingWikiWord' href='/nlab/show/diff/D-brane'>D-brane charge</a> in <a class='existingWikiWord' href='/nlab/show/diff/type+I+string+theory'>type I string theory</a> (on <a class='existingWikiWord' href='/nlab/show/diff/orientifold'>orientifolds</a>):</p> <ul> <li id='GradySati19b'><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Grady'>Daniel Grady</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hisham+Sati'>Hisham Sati</a>, <em>Twisted differential KO-theory</em> (<a href='https://arxiv.org/abs/1905.09085'>arXiv:1905.09085</a>)</li> </ul> <h3 id='ReferencesHigherTwists'>Higher twists</h3> <p>Discussion of higher twists of K-theory (above degree 3):</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Ib+Madsen'>Ib Madsen</a>, <a class='existingWikiWord' href='/nlab/show/diff/Victor+Snaith'>Victor Snaith</a>, <a class='existingWikiWord' href='/nlab/show/diff/J%C3%B8rgen+Tornehave'>Jørgen Tornehave</a>, <em>Infinite loop maps in geometric topology</em>, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 81, Issue 3, (1977)(<a href='https://doi.org/10.1017/S0305004100053482'>doi:10.1017/S0305004100053482</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Constantin+Teleman'>Constantin Teleman</a>, Section 3 of: <em>K-theory of the moduli of bundles over a Riemann surface and deformations of the Verlinde algebra</em> (<a href='https://arxiv.org/abs/math/0306347'>arXiv:math/0306347</a>), in <a class='existingWikiWord' href='/nlab/show/diff/Ulrike+Tillmann'>Ulrike Tillmann</a> (ed.) <em>Topology, Geometry and Quantum Field Theory</em>, Cambridge University Press 2004 (<a href='https://doi.org/10.1017/CBO9780511526398'>doi:10.1017/CBO9780511526398</a>)</p> </li> </ul> <p>Via <a class='existingWikiWord' href='/nlab/show/diff/C-star-algebra'>$C^\ast$-algebras</a>:</p> <ul> <li id='Pennig16'> <p><a class='existingWikiWord' href='/nlab/show/diff/Ulrich+Pennig'>Ulrich Pennig</a>, <em>A noncommutative model for higher twisted K-Theory</em>, J Topology (2016) 9 (1): 27-50 (<a href='https://arxiv.org/abs/1502.02807'>arXiv:1502.02807</a>)</p> </li> <li id='DardalatPennig16'> <p><a class='existingWikiWord' href='/nlab/show/diff/Marius+Dadarlat'>Marius Dadarlat</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ulrich+Pennig'>Ulrich Pennig</a>, <em>A Dixmier-Douady theory for strongly self-absorbing <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebras</em>, J. Reine Angew. Math. 718 (2016) 153-181 (<a href='https://arxiv.org/abs/1302.4468'>arXiv:1302.4468</a>, <a href='https://doi.org/10.1515/crelle-2014-0044'>doi:10.1515/crelle-2014-0044</a>)</p> </li> <li id='DardalatPennig15'> <p><a class='existingWikiWord' href='/nlab/show/diff/Marius+Dadarlat'>Marius Dadarlat</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ulrich+Pennig'>Ulrich Pennig</a>, <em>Unit spectra of K-theory from strongly self-absorbing <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebras</em>, Algebr. Geom. Topol. 15 (2015) 137-168 [[arXiv:1306.2583](https://arxiv.org/abs/1306.2583), <a href='http://dx.doi.org/10.2140/agt.2015.15.137'>doi:10.2140/agt.2015.15.137</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/David+Brook'>David Brook</a>, <em>Computations in higher twisted K-theory</em> (<a href='https://arxiv.org/abs/2007.08964A'>arXiv:2007.08964A</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/David+Brook'>David Brook</a>, <em>Higher twisted K-theory</em> (<a href='https://digital.library.adelaide.edu.au/dspace/handle/2440/125740'>dspace:2440/125740</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/David+E.+Evans'>David E. Evans</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ulrich+Pennig'>Ulrich Pennig</a>, <em>Spectral Sequence Computation of Higher Twisted K-Groups of <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>SU</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>SU(n)</annotation></semantics></math></em> [[arXiv:2307.00423](https://arxiv.org/abs/2307.00423)]</p> </li> </ul> <p>see also:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Ulrich+Pennig'>Ulrich Pennig</a>, <em>Equivariant Higher Twisted K-Theory of <math class='maruku-mathml' display='inline' id='mathml_3a6ee9e8ef2673282136e7ac12fffffd7bad2b84_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>SU</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>SU(n)</annotation></semantics></math> via Exponential Functors</em>, <a href='CQTS#PennigSep2023'>talk at</a> <a class='existingWikiWord' href='/nlab/show/diff/Center+for+Quantum+and+Topological+Systems'>CQTS</a> (20 Sep 2023) [video:YT]</li> </ul> <p>Discussion of the <a class='existingWikiWord' href='/nlab/show/diff/twisted+Chern+character'>twisted Chern character</a> for higher twisted K-theory:</p> <ul> <li>Lachlan Macdonald, <a class='existingWikiWord' href='/nlab/show/diff/Mathai+Varghese'>Varghese Mathai</a>, Hemanth Saratchandran, <em>On the Chern character in Higher Twisted K-theory and spherical T-duality</em>, Commun. Math. Phys. 385, 331-368 (2021) (<a href='https://arxiv.org/abs/2007.02507'>arXiv:2007.02507</a>)</li> </ul> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on February 9, 2024 at 16:30:31. 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