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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 #71 to #72: <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='elliptic_cohomology'>Elliptic cohomology</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/elliptic+cohomology'>elliptic cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/tmf'>tmf</a>, <a class='existingWikiWord' href='/nlab/show/diff/string+theory'>string theory</a></strong></p> <p><a class='existingWikiWord' href='/nlab/show/diff/complex+oriented+cohomology+theory'>complex oriented</a> <a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a> of <a class='existingWikiWord' href='/nlab/show/diff/chromatic+homotopy+theory'>chromatic level</a> 2</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/elliptic+curve'>elliptic curve</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/supersingular+elliptic+curve'>supersingular elliptic curve</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derived+elliptic+curve'>derived elliptic curve</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/moduli+stack+of+elliptic+curves'>moduli stack of elliptic curves</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/modular+form'>modular form</a>, <a class='existingWikiWord' href='/nlab/show/diff/Jacobi+form'>Jacobi form</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Eisenstein+series'>Eisenstein series</a>, <a class='existingWikiWord' href='/nlab/show/diff/j-invariant'>j-invariant</a>, <a class='existingWikiWord' href='/nlab/show/diff/Weierstrass+sigma-function'>Weierstrass sigma-function</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dedekind+eta+function'>Dedekind eta function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/elliptic+genus'>elliptic genus</a>, <a class='existingWikiWord' href='/nlab/show/diff/Witten+genus'>Witten genus</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+modular+form'>topological modular form</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/string+orientation+of+tmf'>string orientation of tmf</a></li> </ul> </li> </ul> </div> <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></li><li><a href='#constructions'>Constructions</a><ul><li><a href='#DecomopositionViaArithmeticSquares'>Decomposition via Arithmetic fracture squares</a></li><li><a href='#stacks_from_spectra'>Stacks from spectra</a><ul><li><a href='#the_context__derived_geometry_over_formal_duals_of_rings'>The context – derived geometry over formal duals of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>E_\infty</annotation></semantics></math>-rings</a></li><li><a href='#coverings_by_the_thom_spectrum'>Coverings by the Thom spectrum</a></li><li><a href='#decategorification_the_ordinary_moduli_stack_of_elliptic_curves'>Decategorification: the ordinary moduli stack of elliptic curves</a></li><li><a href='#explicit_computation_of_homotopy_groups_by_a_spectral_sequence'>Explicit computation of homotopy groups by a spectral sequence</a></li></ul></li><li><a href='#WithLevelStructure'>With Level structure</a></li></ul></li><li><a href='#properties'>Properties</a><ul><li><a href='#HomotopyGroups'>Homotopy groups</a></li><li><a href='#BoardmanHomomorphism'>Boardman homomorphism</a></li><li><a href='#InclusionOfCircle2Bundles'>Inclusion of circle 2-bundles</a></li><li><a href='#MapToTateKTheory'>Maps to K-theory and to Tate K-theory</a></li><li><a href='#witten_genus_and_string_orientation'>Witten genus and string orientation</a></li><li><a href='#chromatic_filtration'>Chromatic filtration</a></li><li><a href='#anderson_selfduality'>Anderson self-duality</a></li><li><a href='#modular_equivariant_versions'>Modular equivariant versions</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>The <a class='existingWikiWord' href='/nlab/show/diff/generalized+%28Eilenberg-Steenrod%29+cohomology'>generalized (Eilenberg-Steenrod) cohomology</a> theory/<a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a> called <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> – for <em><a class='existingWikiWord' href='/nlab/show/diff/topological+modular+form'>topological modular forms</a></em> – is in a precise sense the union of all <a class='existingWikiWord' href='/nlab/show/diff/elliptic+cohomology'>elliptic cohomology theories</a>/<a class='existingWikiWord' href='/nlab/show/diff/elliptic+spectrum'>elliptic spectra</a> (<a href='#Hopkins94'>Hopkins 94</a>).</p> <p>More precisely, <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a> in <a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ rings</a> of the <a class='existingWikiWord' href='/nlab/show/diff/elliptic+spectrum'>elliptic spectra</a> of all <a class='existingWikiWord' href='/nlab/show/diff/elliptic+cohomology'>elliptic cohomology</a> theories, parameterized over the <a class='existingWikiWord' href='/nlab/show/diff/moduli+stack+of+elliptic+curves'>moduli stack of elliptic curves</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mi>ell</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{ell}</annotation></semantics></math>. That such a parameterization exists, coherently, in the first place is due to the <a class='existingWikiWord' href='/nlab/show/diff/Goerss-Hopkins-Miller+theorem'>Goerss-Hopkins-Miller theorem</a>. In the language of <a class='existingWikiWord' href='/nlab/show/diff/derived+algebraic+geometry'>derived algebraic geometry</a> this refines the <a class='existingWikiWord' href='/nlab/show/diff/ring'>commutative ring</a>-valued <a class='existingWikiWord' href='/nlab/show/diff/structure+sheaf'>structure sheaf</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding='application/x-tex'>\mathcal{O}</annotation></semantics></math> of the <a class='existingWikiWord' href='/nlab/show/diff/moduli+stack+of+elliptic+curves'>moduli stack of elliptic curves</a> to an <a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ ring</a>-valued sheaf <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝒪</mi> <mi>top</mi></msup></mrow><annotation encoding='application/x-tex'>\mathcal{O}^{top}</annotation></semantics></math>, making <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>ℳ</mi> <mi>ell</mi></msub><mo>,</mo><msup><mi>𝒪</mi> <mi>top</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{M}_{ell}, \mathcal{O}^{top})</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/derived+Deligne-Mumford+stack'>spectral Deligne-Mumford stack</a>, and <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ ring</a> of <a class='existingWikiWord' href='/nlab/show/diff/global+section'>global sections</a> of that structure sheaf (<a class='existingWikiWord' href='/nlab/show/diff/A+Survey+of+Elliptic+Cohomology'>Lurie</a>).</p> <p>The construction of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> has motivation from <a class='existingWikiWord' href='/nlab/show/diff/physics'>physics</a> (<a class='existingWikiWord' href='/nlab/show/diff/string+theory'>string theory</a>) and from <a class='existingWikiWord' href='/nlab/show/diff/chromatic+homotopy+theory'>chromatic homotopy theory</a>:</p> <ol> <li> <p><strong>from <a class='existingWikiWord' href='/nlab/show/diff/string+theory'>string theory</a>.</strong> Associating to a <a class='existingWikiWord' href='/nlab/show/diff/space'>space</a>, roughly, the <a class='existingWikiWord' href='/nlab/show/diff/partition+function'>partition function</a> of the <a class='existingWikiWord' href='/nlab/show/diff/spinning+string'>spinning string</a>/<a class='existingWikiWord' href='/nlab/show/diff/superstring'>superstring</a> <a class='existingWikiWord' href='/nlab/show/diff/sigma-model'>sigma-model</a> with that space as <a class='existingWikiWord' href='/nlab/show/diff/target+space'>target</a> <a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetime</a> defines a <a class='existingWikiWord' href='/nlab/show/diff/genus'>genus</a> known as the <em><a class='existingWikiWord' href='/nlab/show/diff/Witten+genus'>Witten genus</a></em>, with <a class='existingWikiWord' href='/nlab/show/diff/coefficient'>coefficients</a> in ordinary <a class='existingWikiWord' href='/nlab/show/diff/modular+form'>modular forms</a>. Now, the interesting genera typically appear as the values on <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy groups</a> (the <a class='existingWikiWord' href='/nlab/show/diff/decategorification'>decategorification</a>) of <a class='existingWikiWord' href='/nlab/show/diff/orientation+in+generalized+cohomology'>orientations</a> of <a class='existingWikiWord' href='/nlab/show/diff/multiplicative+cohomology+theory'>multiplicative cohomology theories</a>; for instance the <a class='existingWikiWord' href='/nlab/show/diff/A-hat+genus'>A-hat genus</a>, which is the <a class='existingWikiWord' href='/nlab/show/diff/partition+function'>partition function</a> of the <a class='existingWikiWord' href='/nlab/show/diff/spinning+particle'>spinning particle</a>/<a class='existingWikiWord' href='/nlab/show/diff/superparticle'>superparticle</a> is a shadow of the <a class='existingWikiWord' href='/nlab/show/diff/K-orientation'>Atiyah-Bott-Shapiro</a> <a class='existingWikiWord' href='/nlab/show/diff/spin+structure'>Spin structure</a>-orientation of the <a class='existingWikiWord' href='/nlab/show/diff/KO-theory'>KO</a> spectrum. Therefore an obvious question is which spectrum lifts this classical statement from point <a class='existingWikiWord' href='/nlab/show/diff/particle'>particles</a> to <a class='existingWikiWord' href='/nlab/show/diff/string'>strings</a>. The spectrum <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> solves this: there is a <a class='existingWikiWord' href='/nlab/show/diff/string+structure'>String structure</a> <a class='existingWikiWord' href='/nlab/show/diff/string+orientation+of+tmf'>orientation of tmf</a> such that on homotopy groups it reduces to the <a class='existingWikiWord' href='/nlab/show/diff/Witten+genus'>Witten genus</a> of the <a class='existingWikiWord' href='/nlab/show/diff/superstring'>superstring</a> (<a href='#AndoHopkinsRezk10'>Ando-Hopkins-Rezk 10</a>).</p> <p>Mathematically this means for instance that <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math>-cohomology classes help to detect elements in the <a class='existingWikiWord' href='/nlab/show/diff/string+structure'>string</a> <a class='existingWikiWord' href='/nlab/show/diff/bordism+ring'>cobordism ring</a>. Physically it means that the small aspect of <a class='existingWikiWord' href='/nlab/show/diff/string+theory'>string theory</a> which is captured by the <a class='existingWikiWord' href='/nlab/show/diff/Witten+genus'>Witten genus</a> is realized more deeply as part of fundamental mathematics (<a class='existingWikiWord' href='/nlab/show/diff/chromatic+homotopy+theory'>chromatic</a> <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable homotopy theory</a>, see the next point) and specifically of <a class='existingWikiWord' href='/nlab/show/diff/elliptic+cohomology'>elliptic cohomology</a>. Since the full mathematical structure of <a class='existingWikiWord' href='/nlab/show/diff/string+theory'>string theory</a> is still under investigation, this might point the way:</p> <blockquote> <p>A properly developed theory of elliptic cohomology is likely to shed some light on what <a class='existingWikiWord' href='/nlab/show/diff/string+theory'>string theory</a> really means. (<a href='#Witten87'>Witten 87, very last sentence</a>)</p> </blockquote> </li> <li> <p><strong>from <a class='existingWikiWord' href='/nlab/show/diff/chromatic+homotopy+theory'>chromatic homotopy theory</a>.</strong> The <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+%28infinity%2C1%29-category'>symmetric monoidal</a> <a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable</a> <a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category+of+spectra'>(∞,1)-category of spectra</a> (<a class='existingWikiWord' href='/nlab/show/diff/finite+spectrum'>finite spectra</a>) has its <a class='existingWikiWord' href='/nlab/show/diff/prime+spectrum+of+a+symmetric+monoidal+stable+%28%E2%88%9E%2C1%29-category'>prime spectrum</a> parameterized by <a class='existingWikiWord' href='/nlab/show/diff/prime+number'>prime numbers</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> and <a class='existingWikiWord' href='/nlab/show/diff/Morava+K-theory'>Morava K-theory</a> spectra <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(n)</annotation></semantics></math> at these primes, for natural numbers <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>. The level <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> here is called the <em><a class='existingWikiWord' href='/nlab/show/diff/chromatic+homotopy+theory'>chromatic level</a></em>. In some sense the part of this prime spectrum at chromatic level 0 is <a class='existingWikiWord' href='/nlab/show/diff/ordinary+cohomology'>ordinary cohomology</a> and that at level 1 is <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a>. Therefore an obvious question is what the part at level 2 would be, and in some sense the answer is <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math>. (This point of view has been particularly amplified in the review (<a href='#MazelGee13'>Mazel-Gee 13</a>) of the writeup of the construction in (<a href='#Behrens13'>Behrens 13</a>), which in turn is based on unpublished results based on (<a href='#Hopkins02'>Hopkins 02</a>)). For purposes of <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable homotopy theory</a> this means for instance that <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> provides new tools for computing more <a class='existingWikiWord' href='/nlab/show/diff/homotopy+groups+of+spheres'>homotopy groups of spheres</a> via an <a class='existingWikiWord' href='/nlab/show/diff/Adams%E2%80%93Novikov+spectral+sequence'>Adams-Novikov spectral sequence</a>.</p> </li> </ol> <h2 id='definition'>Definition</h2> <p>Write</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mi>cub</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{cub}</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/diff/moduli+space+of+curves'>moduli stack of curves</a> for <a class='existingWikiWord' href='/nlab/show/diff/cubic+curve'>cubic curves</a>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mi>ell</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{ell}</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/diff/moduli+stack+of+elliptic+curves'>moduli stack of elliptic curves</a>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mover><mi>ell</mi><mo>¯</mo></mover></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{\overline{ell}}</annotation></semantics></math> for its <a class='existingWikiWord' href='/nlab/show/diff/Deligne-Mumford+compactification'>Deligne-Mumford compactification</a> obtained by adding the <a class='existingWikiWord' href='/nlab/show/diff/nodal+curve'>nodal cubic curve</a>.</p> </li> </ul> <p>(Here <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mi>cub</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{cub}</annotation></semantics></math> is obatined by furthermor adding also the <a class='existingWikiWord' href='/nlab/show/diff/cusp'>cuspidal cubic curve</a>, hence we have canonical maps <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mi>ell</mi></msub><mo>→</mo><msub><mi>ℳ</mi> <mover><mi>ell</mi><mo>¯</mo></mover></msub><mo>→</mo><msub><mi>ℳ</mi> <mi>cusp</mi></msub><mo>→</mo><msub><mi>ℳ</mi> <mi>FG</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{ell}\to \mathcal{M}_{\overline{ell}}\to \mathcal{M}_{cusp} \to \mathcal{M}_{FG}</annotation></semantics></math>).</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/Goerss-Hopkins-Miller+theorem'>Goerss-Hopkins-Miller theorem</a> equips these three moduli stacks with <a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ ring</a>-valued <a class='existingWikiWord' href='/nlab/show/diff/structure+sheaf'>structure sheaves</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝒪</mi> <mi>top</mi></msup></mrow><annotation encoding='application/x-tex'>\mathcal{O}^{top}</annotation></semantics></math> (and by <a class='existingWikiWord' href='/nlab/show/diff/A+Survey+of+Elliptic+Cohomology'>Lurie (Survey)</a> that makes them into <a class='existingWikiWord' href='/nlab/show/diff/derived+Deligne-Mumford+stack'>spectral Deligne-Mumford stacks</a> which are moduli spaces for <a class='existingWikiWord' href='/nlab/show/diff/derived+elliptic+curve'>derived elliptic curves</a> etc.)</p> <p>The <em><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math>-spectrum</em> is defined to be the <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>E_\infty</annotation></semantics></math>-ring of <a class='existingWikiWord' href='/nlab/show/diff/global+section'>global sections</a> of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝒪</mi> <mi>top</mi></msup></mrow><annotation encoding='application/x-tex'>\mathcal{O}^{top}</annotation></semantics></math> (in the sense of <a class='existingWikiWord' href='/nlab/show/diff/derived+algebraic+geometry'>derived algebraic geometry</a>, hence the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a> of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝒪</mi> <mi>top</mi></msup></mrow><annotation encoding='application/x-tex'>\mathcal{O}^{top}</annotation></semantics></math> over the <a class='existingWikiWord' href='/nlab/show/diff/%C3%A9tale+site'>etale site</a> of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℳ</mi></mrow><annotation encoding='application/x-tex'>\mathcal{M}</annotation></semantics></math>). More precisely one sets</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>TMF</mi><mo>≔</mo><mi>Γ</mi><mo stretchy='false'>(</mo><msub><mi>ℳ</mi> <mi>ell</mi></msub><mo>,</mo><msup><mi>𝒪</mi> <mi>top</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>TMF \coloneqq \Gamma(\mathcal{M}_{ell}, \mathcal{O}^{top})</annotation></semantics></math>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Tmf</mi><mo>≔</mo><mi>Γ</mi><mo stretchy='false'>(</mo><msub><mi>ℳ</mi> <mover><mi>ell</mi><mo>¯</mo></mover></msub><mo>,</mo><msup><mi>𝒪</mi> <mi>top</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Tmf \coloneqq \Gamma(\mathcal{M}_{\overline{ell}}, \mathcal{O}^{top})</annotation></semantics></math>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi><mo>≔</mo></mrow><annotation encoding='application/x-tex'>tmf \coloneqq</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/connective+cover'>connective cover</a> of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Tmf</mi></mrow><annotation encoding='application/x-tex'>Tmf</annotation></semantics></math> (also <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>≃</mo><mi>Γ</mi><mo stretchy='false'>(</mo><msub><mi>ℳ</mi> <mover><mi>cub</mi><mo>¯</mo></mover></msub><mo>,</mo><msup><mi>𝒪</mi> <mi>top</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\simeq \Gamma(\mathcal{M}_{\overline{cub}}, \mathcal{O}^{top})</annotation></semantics></math> (<a href='#HillLawson13}'>Hill-Lawson 13, p. 2</a> (?)).</p> </li> </ul> <h2 id='constructions'>Constructions</h2> <h3 id='DecomopositionViaArithmeticSquares'>Decomposition via Arithmetic fracture squares</h3> <p>We survey here some aspects of the explicit construction in (<a href='#Behrens13'>Behrens 13</a>), a review is also in (<a href='#MazelGee13'>Mazel-Gee 13</a>),</p> <p>The basic strategy here is to use <a class='existingWikiWord' href='/nlab/show/diff/fracture+theorem'>arithmetic squares</a> in order to decompose the problem into smaller more manageable pieces.</p> <p>Write <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mrow><msub><mi>ℳ</mi> <mi>ell</mi></msub></mrow><mo>¯</mo></mover></mrow><annotation encoding='application/x-tex'>\overline{\mathcal{M}_{ell}}</annotation></semantics></math> for the compactified <a class='existingWikiWord' href='/nlab/show/diff/moduli+stack+of+elliptic+curves'>moduli stack of elliptic curves</a>. In there one finds the pieces</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mover><mrow><msub><mi>ℳ</mi> <mi>ell</mi></msub></mrow><mo>¯</mo></mover></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>ι</mi> <mi>p</mi></msub></mrow></mover></mtd> <mtd><mo stretchy='false'>(</mo><mover><mrow><msub><mi>ℳ</mi> <mi>ell</mi></msub></mrow><mo>¯</mo></mover><msub><mo stretchy='false'>)</mo> <mi>p</mi></msub></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>ι</mi> <mi>ℚ</mi></msub></mrow></mpadded></msup><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>(</mo><mover><mrow><msub><mi>ℳ</mi> <mi>ell</mi></msub></mrow><mo>¯</mo></mover><msub><mo stretchy='false'>)</mo> <mi>ℚ</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ \overline{\mathcal{M}_{ell}} &\stackrel{\iota_{p}}{\leftarrow}& (\overline{\mathcal{M}_{ell}})_p \\ {}^{\mathllap{\iota_{\mathbb{Q}}}}\uparrow \\ (\overline{\mathcal{M}_{ell}})_{\mathbb{Q}} } </annotation></semantics></math></div> <p>given by <a class='existingWikiWord' href='/nlab/show/diff/rationalization'>rationalization</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mover><mrow><msub><mi>ℳ</mi> <mi>ell</mi></msub></mrow><mo>¯</mo></mover><msub><mo stretchy='false'>)</mo> <mi>ℚ</mi></msub><mo>=</mo><mover><mrow><msub><mi>ℳ</mi> <mi>ell</mi></msub></mrow><mo>¯</mo></mover><munder><mo>×</mo><mrow><mi>Spec</mi><mo stretchy='false'>(</mo><mi>ℤ</mi><mo stretchy='false'>)</mo></mrow></munder><mi>Spec</mi><mo stretchy='false'>(</mo><mi>ℚ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (\overline{\mathcal{M}_{ell}})_{\mathbb{Q}} = \overline{\mathcal{M}_{ell}} \underset{Spec(\mathbb{Z})}{\times} Spec(\mathbb{Q}) </annotation></semantics></math></div> <p>(hence this is the moduli of <a href='elliptic+curve#OverTheRationalNumbers'>elliptic curves over the rational numbers</a>) and by <a class='existingWikiWord' href='/nlab/show/diff/p-adic+completion'>p-completion</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mover><mrow><msub><mi>ℳ</mi> <mi>ell</mi></msub></mrow><mo>¯</mo></mover><msub><mo stretchy='false'>)</mo> <mi>p</mi></msub><mo>=</mo><mo stretchy='false'>(</mo><mover><mrow><msub><mi>ℳ</mi> <mi>ell</mi></msub></mrow><mo>¯</mo></mover><mo stretchy='false'>)</mo><munder><mo>×</mo><mrow><mi>Spec</mi><mo stretchy='false'>(</mo><mi>ℤ</mi><mo stretchy='false'>)</mo></mrow></munder><mi>Spf</mi><mo stretchy='false'>(</mo><msub><mi>ℤ</mi> <mi>p</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (\overline{\mathcal{M}_{ell}})_p = (\overline{\mathcal{M}_{ell}}) \underset{Spec(\mathbb{Z})}{\times} Spf(\mathbb{Z}_p) </annotation></semantics></math></div> <p>for any <a class='existingWikiWord' href='/nlab/show/diff/prime+number'>prime number</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding='application/x-tex'>\mathbb{Z}_p</annotation></semantics></math> denotes the <a class='existingWikiWord' href='/nlab/show/diff/p-adic+integer'>p-adic integers</a> and <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Spf</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Spf(-)</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/formal+spectrum'>formal spectrum</a>. (Hence this is the moduli of <a href='elliptic+curve#OverpAdics'>elliptic curves over p-adic integers</a>).</p> <p>This induces the <a class='existingWikiWord' href='/nlab/show/diff/fracture+theorem'>arithmetic square</a> decomposition which realizes <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝒪</mi> <mi>top</mi></msup></mrow><annotation encoding='application/x-tex'>\mathcal{O}^{top}</annotation></semantics></math> as the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+pullback'>homotopy fiber product</a> in</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>𝒪</mi> <mi>top</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mi>p</mi></munder><mo stretchy='false'>(</mo><msub><mi>ι</mi> <mi>p</mi></msub><msub><mo stretchy='false'>)</mo> <mo>*</mo></msub><msubsup><mi>𝒪</mi> <mi>p</mi> <mi>top</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>L</mi> <mi>ℚ</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy='false'>(</mo><msub><mi>ι</mi> <mi>ℚ</mi></msub><msub><mo stretchy='false'>)</mo> <mo>*</mo></msub><msubsup><mi>𝒪</mi> <mi>ℚ</mi> <mi>top</mi></msubsup></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>α</mi> <mi>arith</mi></msub></mrow></mover></mtd> <mtd><msub><mrow><mo>(</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mi>p</mi></munder><mo stretchy='false'>(</mo><msub><mi>ι</mi> <mi>p</mi></msub><msub><mo stretchy='false'>)</mo> <mo>*</mo></msub><msubsup><mi>𝒪</mi> <mi>p</mi> <mi>top</mi></msubsup><mo>)</mo></mrow> <mi>ℚ</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ \mathcal{O}^{top} &\to& \prod_p (\iota_p)_\ast \mathcal{O}^{top}_p \\ \downarrow && \downarrow^{\mathrlap{L_{\mathbb{Q}}}} \\ (\iota_{\mathbb{Q}})_\ast \mathcal{O}^{top}_{\mathbb{Q}} &\stackrel{\alpha_{arith}}{\to}& \left( \prod_p (\iota_p)_\ast \mathcal{O}^{top}_p \right)_{\mathbb{Q}} } </annotation></semantics></math></div> <p>Here <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>𝒪</mi> <mi>ℚ</mi> <mi>top</mi></msubsup></mrow><annotation encoding='application/x-tex'>\mathcal{O}^{top}_{\mathbb{Q}}</annotation></semantics></math> can be obtained directly, and to obtain <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>𝒪</mi> <mi>p</mi> <mi>top</mi></msubsup></mrow><annotation encoding='application/x-tex'>\mathcal{O}^{top}_p</annotation></semantics></math> one uses in turn another <a class='existingWikiWord' href='/nlab/show/diff/fracture+theorem'>fracture square</a>, now decomposing via <a class='existingWikiWord' href='/nlab/show/diff/K%28n%29-local+stable+homotopy+theory'>K(n)-localization</a> into <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(1)</annotation></semantics></math>-local and <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(2)</annotation></semantics></math>-local pieces.</p> <p>(…)</p> <h3 id='stacks_from_spectra'>Stacks from spectra</h3> <p>There is a way to “construct” the tmf-spectrum as the <a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ ring</a> of <a class='existingWikiWord' href='/nlab/show/diff/global+section'>global section</a>s of a <a class='existingWikiWord' href='/nlab/show/diff/structured+%28infinity%2C1%29-topos'>structured (∞,1)-topos</a> whose underlying space is essentially the <a class='existingWikiWord' href='/nlab/show/diff/moduli+stack'>moduli stack</a> of <a class='existingWikiWord' href='/nlab/show/diff/elliptic+curve'>elliptic curve</a>s. We sketch some main ideas of this construction.</p> <h4 id='the_context__derived_geometry_over_formal_duals_of_rings'>The context – derived geometry over formal duals of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>E_\infty</annotation></semantics></math>-rings</h4> <p>The discussion happens in the context of <a class='existingWikiWord' href='/nlab/show/diff/derived+geometry'>derived geometry</a> in the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> over a <a class='existingWikiWord' href='/nlab/show/diff/essentially+small+%28infinity%2C1%29-category'>small</a> version of the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-site'>(∞,1)-site</a> of formal duals of <a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ ring</a>s (<a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>ring spectra</a>). This is equipped with some <a class='existingWikiWord' href='/nlab/show/diff/subcanonical+coverage'>subcanonical coverage</a>. For <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo>∈</mo><msub><mi>E</mi> <mn>∞</mn></msub><mi>Ring</mi></mrow><annotation encoding='application/x-tex'>R \in E_\infty Ring</annotation></semantics></math> we write <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Spec</mi><mi>R</mi></mrow><annotation encoding='application/x-tex'>Spec R</annotation></semantics></math> for its image under the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories'>(∞,1)-Yoneda embedding</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>∞</mn></msub><mi>Ring</mi><msup><mo stretchy='false'>)</mo> <mi>op</mi></msup><mo>↪</mo><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>(E_\infty Ring)^{op} \hookrightarrow \mathbf{H}</annotation></semantics></math>.</p> <p>\begin{remark} The <a class='existingWikiWord' href='/nlab/show/diff/terminal+object+in+a+quasi-category'>terminal object</a> in <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math> is the formal dual of the <a class='existingWikiWord' href='/nlab/show/diff/sphere+spectrum'>sphere spectrum</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo><mo>≃</mo><mi>Spec</mi><mo stretchy='false'>(</mo><mi>𝕊</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> * \simeq Spec(\mathbb{S}) \,. </annotation></semantics></math></div> <p>Because the sphere spectrum is the <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a> in <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub><mi>Ring</mi></mrow><annotation encoding='application/x-tex'>E_\infty Ring</annotation></semantics></math>. \end{remark}</p> <h4 id='coverings_by_the_thom_spectrum'>Coverings by the Thom spectrum</h4> <p>The crucial input for the entire construction is the following statement.</p> <p>The idea is that the formal dual of the <a class='existingWikiWord' href='/nlab/show/diff/MU'>complex cobordism</a> <a class='existingWikiWord' href='/nlab/show/diff/Thom+spectrum'>Thom spectrum</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi><mi>U</mi></mrow><annotation encoding='application/x-tex'>M U</annotation></semantics></math> is in a suitable sense a covering</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Spec</mi><mi>M</mi><mi>U</mi><mo>→</mo><mi>Spec</mi><mi>𝕊</mi></mrow><annotation encoding='application/x-tex'> Spec M U \to Spec \mathbb{S} </annotation></semantics></math></div> <p>of the <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> in <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mi>H</mi></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{H}</annotation></semantics></math>. (See at <em><a href='Adams%20spectral%20sequence#DefinitionInHigherAlgebra'>Adams spectral sequence – As derived descent</a></em>)</p> <p>This means that <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Spec</mi><mi>M</mi><mi>U</mi></mrow><annotation encoding='application/x-tex'>Spec M U</annotation></semantics></math> plays the role of a <a class='existingWikiWord' href='/nlab/show/diff/cover'>cover</a> of the point. This allows to do some computations with ring spectra <em>locally on the cover <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Spec</mi><mi>M</mi><mi>U</mi></mrow><annotation encoding='application/x-tex'>Spec M U</annotation></semantics></math></em> . Since <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi><msup><mi>U</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>M U^*</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/Lazard+ring'>Lazard ring</a>, this explains why <a class='existingWikiWord' href='/nlab/show/diff/formal+group'>formal group law</a>s show up all over the place.</p> <p>To see this, first notice that the problem of realizing <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo>=</mo><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>R = tmf</annotation></semantics></math> or any other ring spectrum as the ring of global sections on something has a <em>tautological solution</em> : almost by definition (see <a class='existingWikiWord' href='/nlab/show/diff/generalized+scheme'>generalized scheme</a>) there is an <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>E_\infty</annotation></semantics></math>-ring valued <a class='existingWikiWord' href='/nlab/show/diff/structure+sheaf'>structure sheaf</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi><mi>Spec</mi><mo stretchy='false'>(</mo><mi>R</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{O}Spec(R)</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Spec</mi><mi>R</mi></mrow><annotation encoding='application/x-tex'>Spec R</annotation></semantics></math> and its global sections is <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>. So we have in particular</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi><mo>≃</mo><mi>𝒪</mi><mo stretchy='false'>(</mo><mi>Spec</mi><mo stretchy='false'>(</mo><mi>tmf</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> tmf \simeq \mathcal{O}(Spec(tmf)) \,. </annotation></semantics></math></div> <p>In order to get a less tautological and more insightful characterization, the strategy is now to pass on the right to the <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Spec</mi><mi>M</mi><mi>U</mi></mrow><annotation encoding='application/x-tex'>Spec M U</annotation></semantics></math>-cover by forming the <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-pullback'>(∞,1)-pullback</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>Spec</mi><mo stretchy='false'>(</mo><mi>tmf</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>Spec</mi><mo stretchy='false'>(</mo><mi>M</mi><mi>U</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Spec</mi><mo stretchy='false'>(</mo><mi>tmf</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>Spec</mi><mo stretchy='false'>(</mo><mi>M</mi><mi>U</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo><mo>≃</mo><mi>Spec</mi><mo stretchy='false'>(</mo><mi>𝕊</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ Spec(tmf) \times Spec(M U) &\to& Spec(tmf) \\ \downarrow && \downarrow \\ Spec(M U) &\to& * \simeq Spec(\mathbb{S}) } \,. </annotation></semantics></math></div> <p>The resulting <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+nerve'>Cech nerve</a> is a <a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>groupoid object in an (∞,1)-category</a> given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><mi>Spec</mi><mo stretchy='false'>(</mo><mi>tmf</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>Spec</mi><mo stretchy='false'>(</mo><mi>MU</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>Spec</mi><mo stretchy='false'>(</mo><mi>MU</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>→</mo></mover><mi>Spec</mi><mo stretchy='false'>(</mo><mi>tmf</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>Spec</mi><mo stretchy='false'>(</mo><mi>MU</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \cdots \stackrel{\to}{\stackrel{\to}{\to}} Spec(tmf) \times Spec(MU) \times Spec(MU) \stackrel{\to}{\to} Spec(tmf) \times Spec(MU) </annotation></semantics></math></div> <p>which by formal duality is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><mi>Spec</mi><mo stretchy='false'>(</mo><mi>tmf</mi><mo>∧</mo><mi>MU</mi><mo>∧</mo><mi>MU</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>→</mo></mover><mi>Spec</mi><mo stretchy='false'>(</mo><mi>tmf</mi><mo>∧</mo><mi>MU</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \cdots \stackrel{\to}{\stackrel{\to}{\to}} Spec (tmf \wedge MU \wedge MU) \stackrel{\to}{\to} Spec ( tmf \wedge MU) </annotation></semantics></math></div> <p>where the <a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash product</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∧</mo></mrow><annotation encoding='application/x-tex'>\wedge</annotation></semantics></math> of ring spectra over the <a class='existingWikiWord' href='/nlab/show/diff/sphere+spectrum'>sphere spectrum</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝕊</mi></mrow><annotation encoding='application/x-tex'>\mathbb{S}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a> operation on function algebras formally dual to forming products of spaces.</p> <p>As a <a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>groupoid object</a> this is still equivalent to just <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Spec</mi><mo stretchy='false'>(</mo><mi>tmf</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Spec(tmf)</annotation></semantics></math>.</p> <h4 id='decategorification_the_ordinary_moduli_stack_of_elliptic_curves'>Decategorification: the ordinary moduli stack of elliptic curves</h4> <p>To simplify this we take a drastic step and apply a lot of <a class='existingWikiWord' href='/nlab/show/diff/decategorification'>decategorification</a>: by applying the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a> to all the <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>E_\infty</annotation></semantics></math>-rings involved these are sent to graded ordinary <a class='existingWikiWord' href='/nlab/show/diff/ring'>ring</a>s <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mo>*</mo></msub><mo stretchy='false'>(</mo><mi>tmf</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_*(tmf)</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mo>*</mo></msub><mo stretchy='false'>(</mo><mi>M</mi><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_*(M U)</annotation></semantics></math> etc. The result is an ordinary <a class='existingWikiWord' href='/nlab/show/diff/simplicial+object'>simplicial</a> <a class='existingWikiWord' href='/nlab/show/diff/scheme'>scheme</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><mi>Spec</mi><mo stretchy='false'>(</mo><msub><mi>π</mi> <mo>*</mo></msub><mo stretchy='false'>(</mo><mi>tmf</mi><mo>∧</mo><mi>M</mi><mi>U</mi><mo>∧</mo><mi>M</mi><mi>U</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>→</mo></mover><mi>Spec</mi><mo stretchy='false'>(</mo><msub><mi>π</mi> <mo>*</mo></msub><mo stretchy='false'>(</mo><mi>tmf</mi><mo>∧</mo><mi>M</mi><mi>U</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \cdots \stackrel{\to}{\stackrel{\to}{\to}} Spec (\pi_*(tmf \wedge M U \wedge M U)) \stackrel{\to}{\to} Spec ( \pi_*(tmf \wedge M U)) \,, </annotation></semantics></math></div> <p>which remembers the fact that its structure rings are graded by being equipped with an <a class='existingWikiWord' href='/nlab/show/diff/action'>action</a> of the multiplicative group <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔾</mi><mo>=</mo><msup><mi>𝔸</mi> <mo>×</mo></msup></mrow><annotation encoding='application/x-tex'>\mathbb{G} = \mathbb{A}^\times</annotation></semantics></math> (see <a class='existingWikiWord' href='/nlab/show/diff/line+object'>line object</a>).</p> <p>This general Ansatz is discussed in (<a href='#Hopkins'>Hopkins</a>).</p> <p>This simplicial scheme, which is degreewise the formal dual of a graded ring of <a class='existingWikiWord' href='/nlab/show/diff/generalized+%28Eilenberg-Steenrod%29+cohomology'>generalized homology</a>-groups one can show is in fact a <a class='existingWikiWord' href='/nlab/show/diff/groupoid'>groupoid</a>, hence a <a class='existingWikiWord' href='/nlab/show/diff/stack'>stack</a>: effectively the <a class='existingWikiWord' href='/nlab/show/diff/moduli+stack+of+elliptic+curves'>moduli stack of elliptic curves</a>. <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mi>ell</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{ell}</annotation></semantics></math>. See (<a href='#HenriquesModuli'>Henriques</a>).</p> <p>In fact if in this construction one replaced <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Spec</mi><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>Spec tmf</annotation></semantics></math> by the point, one obtains the simplicial scheme</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><mi>Spec</mi><mo stretchy='false'>(</mo><msub><mi>π</mi> <mo>*</mo></msub><mo stretchy='false'>(</mo><mi>M</mi><mi>U</mi><mo>∧</mo><mi>M</mi><mi>U</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mover><mo>→</mo><mo>→</mo></mover><mi>Spec</mi><mo stretchy='false'>(</mo><msub><mi>π</mi> <mo>*</mo></msub><mo stretchy='false'>(</mo><mi>M</mi><mi>U</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \cdots \stackrel{\to}{\stackrel{\to}{\to}} Spec (\pi_*(M U \wedge M U)) \stackrel{\to}{\to} Spec ( \pi_*(M U)) </annotation></semantics></math></div> <p>which one finds is the <a class='existingWikiWord' href='/nlab/show/diff/moduli+stack+of+formal+groups'>moduli stack of formal group laws</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mi>fg</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{fg}</annotation></semantics></math>.</p> <h4 id='explicit_computation_of_homotopy_groups_by_a_spectral_sequence'>Explicit computation of homotopy groups by a spectral sequence</h4> <p>Now, a priori these underived stacks remember little about the original <a class='existingWikiWord' href='/nlab/show/diff/derived+scheme'>derived scheme</a>s <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Spec</mi><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>Spec tmf</annotation></semantics></math> etc. They may not even carry any <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>E_\infty</annotation></semantics></math>-ring valued structure sheaf anymore (though some of them do).</p> <p>If they do carry an <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>E_\infty</annotation></semantics></math>-ring valued structure sheaf <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding='application/x-tex'>\mathcal{O}</annotation></semantics></math>, one can compute the homotopy groups of its global sections by a <a class='existingWikiWord' href='/nlab/show/diff/spectral+sequence'>spectral sequence</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy='false'>(</mo><msub><mi>ℳ</mi> <mi>ell</mi></msub><mo>,</mo><msub><mi>π</mi> <mi>q</mi></msub><mo stretchy='false'>(</mo><mi>𝒪</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>⇒</mo><msub><mi>π</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msub><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>ℳ</mi> <mi>ell</mi></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> H^p(\mathcal{M}_{ell}, \pi_q(\mathcal{O})) \Rightarrow \pi_{p+q} \mathcal{O}(\mathcal{M}_{ell}) \,. </annotation></semantics></math></div> <p>But it turns out that even if the derived structure sheaf does not exist, this spectral sequence may still converge and may still compute the homotopy groups of the ring spectrum that one started with. This gives one way to compute the homotopy groups of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math>.</p> <p>For the case of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> one finds that the homotopy sheaves <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>q</mi></msub><mo stretchy='false'>(</mo><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>ℳ</mi> <mi>ell</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_q(\mathcal{O}(\mathcal{M}_{ell}))</annotation></semantics></math> are simple: they vanish in odd degree and are <a class='existingWikiWord' href='/nlab/show/diff/tensor+power'>tensor power</a>s <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ω</mi> <mrow><mo>⊗</mo><mi>k</mi></mrow></msup></mrow><annotation encoding='application/x-tex'>\omega^{\otimes k}</annotation></semantics></math> of the canonical line bundle <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math> in even degree <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding='application/x-tex'>2 k</annotation></semantics></math>, where the fiber of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math> over an <a class='existingWikiWord' href='/nlab/show/diff/elliptic+curve'>elliptic curve</a> is the <a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent space</a> of that curve at its identity element. A <a class='existingWikiWord' href='/nlab/show/diff/section'>section</a> of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ω</mi> <mrow><mo>⊗</mo><mi>k</mi></mrow></msup></mrow><annotation encoding='application/x-tex'>\omega^{\otimes k}</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/modular+form'>modular form</a> of weight <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>. So the whole problem of computing the homotopy groups of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> boils down to computing the <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>abelian sheaf cohomology</a> of the moduli stack of elliptic curves with coefficients in these abelian groups of modular forms — and then examining the resulting spectral sequence.</p> <p>This can be done quite explicitly in terms of a long but fairly elementary computation in ordinary algebra. A detailed discussion of this computation is in (<a href='#Henriques'>Henriques</a>)</p> <h3 id='WithLevelStructure'>With Level structure</h3> <p>The <a class='existingWikiWord' href='/nlab/show/diff/moduli+stack+of+elliptic+curves'>moduli stack of elliptic curves</a> has covers by that of <a class='existingWikiWord' href='/nlab/show/diff/level+structure+on+an+elliptic+curve'>elliptic curves with level structure</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Γ</mi></mrow><annotation encoding='application/x-tex'>\Gamma</annotation></semantics></math>. Under some conditions these covers inherit derived structure sheaves <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝒪</mi> <mi>top</mi></msup></mrow><annotation encoding='application/x-tex'>\mathcal{O}^{top}</annotation></semantics></math> and hence induce spectra of “topological forms with level structure”, <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi><mo stretchy='false'>(</mo><mi>Γ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>tmf(\Gamma)</annotation></semantics></math> (<a href='#MahowaldRezk09'>Mahowald-Rezk 09</a>). For more on this see at <em><a class='existingWikiWord' href='/nlab/show/diff/modular+equivariant+elliptic+cohomology'>modular equivariant elliptic cohomology</a></em>.</p> <p>For instance <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>tmf</mi> <mn>0</mn></msub><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>tmf_0(2)</annotation></semantics></math> (for the <a class='existingWikiWord' href='/nlab/show/diff/congruence+subgroup'>congruence subgroup</a> which preserves an NS-R <a class='existingWikiWord' href='/nlab/show/diff/spin+structure'>spin structure</a> on elliptic curves over the complex numbers) is the <a class='existingWikiWord' href='/nlab/show/diff/elliptic+spectrum'>elliptic spectrum</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ell</mi></mrow><annotation encoding='application/x-tex'>Ell</annotation></semantics></math> of (<a href='#spin+orientation+of+Ochanine+elliptic+cohomology#LandweberRavenelStong93'>Landweber-Ravenel-Stong 93</a>), see at <em><a class='existingWikiWord' href='/nlab/show/diff/tmf0%282%29'>tmf0(2)</a></em>.</p> <p>Discussion of level structure also governs the relation of tmf to K-theory, see at <em><a href='#MapToTateKTheory'>Maps to K-theory and Tate K-theory</a></em>.</p> <h2 id='properties'>Properties</h2> <h3 id='HomotopyGroups'>Homotopy groups</h3> <p>The first few <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy groups</a> of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> are (<a href='#Hopkins02'>Hopkins 02, section 4.3</a>)</p> <table><thead><tr><th><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math></th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th><th>13</th><th>14</th></tr></thead><tbody><tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mi>tmf</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_k(tmf)</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_107' 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></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_108' 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></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>24</mn><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/24\mathbb{Z}</annotation></semantics></math></td><td style='text-align: left;'>0</td><td style='text-align: left;'>0</td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_110' 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></td><td style='text-align: left;'>0</td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo>⊕</mo><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn><mi>ℤ</mi><msup><mo stretchy='false'>)</mo> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>(\mathbb{Z}/2\mathbb{Z})^2</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>6</mn><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/6\mathbb{Z}</annotation></semantics></math></td><td style='text-align: left;'>0</td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>3</mn><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/3\mathbb{Z}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_116' 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></td></tr> </tbody></table> <h3 id='BoardmanHomomorphism'>Boardman homomorphism</h3> <p>Write <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝕊</mi></mrow><annotation encoding='application/x-tex'>\mathbb{S}</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/diff/sphere+spectrum'>sphere spectrum</a> and <a class='existingWikiWord' href='/nlab/show/diff/tmf'>tmf</a> for the <a class='existingWikiWord' href='/nlab/show/diff/connective+spectrum'>connective spectrum</a> of <a class='existingWikiWord' href='/nlab/show/diff/topological+modular+form'>topological modular forms</a>. Since <a class='existingWikiWord' href='/nlab/show/diff/tmf'>tmf</a> is an <a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞</a><a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>ring spectrum</a>, there is an essentially unique homomorphism of <a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞</a><a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>ring spectra</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝕊</mi><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>tmf</mi></msub></mrow></mover><mi>tmf</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \mathbb{S} \overset{e_{tmf}}{\longrightarrow} tmf \,. </annotation></semantics></math></div> <p>Regarded as a morphism of <a class='existingWikiWord' href='/nlab/show/diff/generalized+homology'>generalized homology</a>-theories, this is also called the <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+theorem'>Hurewicz homomorphism</a>, or rather the <a class='existingWikiWord' href='/nlab/show/diff/Boardman+homomorphism'>Boardman homomorphism</a> for <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math></p> <div class='num_prop' id='BoardmanHomomorphismInTmfIs6Connected'> <h6 id='proposition'>Proposition</h6> <p><strong>(Boardman homomorphism in <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> is 6-connected)</strong></p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/Boardman+homomorphism+in+tmf'>Boardman homomorphism in tmf</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝕊</mi><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>tmf</mi></msub></mrow></mover><mi>tmf</mi></mrow><annotation encoding='application/x-tex'> \mathbb{S} \overset{e_{tmf}}{\longrightarrow} tmf </annotation></semantics></math></div> <p>induces an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> on <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group+of+a+spectrum'>stable homotopy groups</a> (hence from the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+groups+of+spheres'>stable homotopy groups of spheres</a>), up to degree 6:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mrow><mo>•</mo><mo>≤</mo><mn>6</mn></mrow></msub><mo stretchy='false'>(</mo><mi>𝕊</mi><mo stretchy='false'>)</mo><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>π</mi> <mrow><mo>•</mo><mo>≤</mo><mn>6</mn></mrow></msub><mo stretchy='false'>(</mo><msub><mi>e</mi> <mi>tmf</mi></msub><mo stretchy='false'>)</mo></mrow></munderover><msub><mi>π</mi> <mrow><mo>•</mo><mo>≤</mo><mn>6</mn></mrow></msub><mo stretchy='false'>(</mo><mi>tmf</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \pi_{\bullet \leq 6}(\mathbb{S}) \underoverset{\simeq}{\pi_{\bullet \leq 6}(e_{tmf})}{\longrightarrow} \pi_{\bullet\leq 6}(tmf) \,. </annotation></semantics></math></div></div> <p>(<a href='#Hopkins02'>Hopkins 02, Prop. 4.6</a>, <a href='#DFHH14'>DFHH 14, Ch. 13</a>)</p> <h3 id='InclusionOfCircle2Bundles'>Inclusion of circle 2-bundles</h3> <p>Write <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>B</mi> <mn>2</mn></msup><mi>U</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo>≃</mo><mi>K</mi><mo stretchy='false'>(</mo><mi>ℤ</mi><mo>,</mo><mn>3</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>B^2 U(1) \simeq K(\mathbb{Z},3)</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/diff/abelian+infinity-group'>abelian ∞-group</a> whose underlying <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a> is the <a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a> for <a class='existingWikiWord' href='/nlab/show/diff/circle+2-bundle'>circle 2-bundle</a>. Write <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝕊</mi><mo stretchy='false'>[</mo><msup><mi>B</mi> <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'>\mathbb{S}[B^2 U(1)]</annotation></semantics></math> for its <a class='existingWikiWord' href='/nlab/show/diff/%E2%88%9E-group+%E2%88%9E-ring'>∞-group ∞-ring</a>.</p> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>There is a canonical homomorphism of <a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ rings</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝕊</mi><mo stretchy='false'>[</mo><msup><mi>B</mi> <mn>2</mn></msup><mi>U</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo stretchy='false'>]</mo><mo>→</mo><mi>tmf</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \mathbb{S}[B^2 U(1)] \to tmf \,. </annotation></semantics></math></div></div> <p>See (<a href='#ABG10'>Ando-Blumberg-Gepner 10, section 8</a>).</p> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>This means that every <a class='existingWikiWord' href='/nlab/show/diff/circle+2-bundle'>circle 2-bundle</a> (<math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_126' 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>) given by a modulating map <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>χ</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><msup><mi>B</mi> <mn>2</mn></msup><mi>U</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\chi \colon X \to B^2 U(1)</annotation></semantics></math> determines a class represented by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>χ</mi></mover><msup><mi>B</mi> <mn>2</mn></msup><mi>U</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo>→</mo><mi>𝕊</mi><mo stretchy='false'>[</mo><msup><mi>B</mi> <mn>2</mn></msup><mi>U</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo stretchy='false'>]</mo><mo>→</mo><mi>tmf</mi></mrow><annotation encoding='application/x-tex'> X \stackrel{\chi}{\to} B^2 U(1) \to \mathbb{S}[B^2 U(1)] \to tmf </annotation></semantics></math></div> <p>in the <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/generalized+cohomology'>generalized cohomology</a> of its base space <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> </div> <h3 id='MapToTateKTheory'>Maps to K-theory and to Tate K-theory</h3> <p>The inclusion of the compactification point (representing the <a class='existingWikiWord' href='/nlab/show/diff/nodal+curve'>nodal curve</a> but being itself the <a class='existingWikiWord' href='/nlab/show/diff/cusp'>cusp</a> of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mover><mi>ell</mi><mo>¯</mo></mover></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{\overline{ell}}</annotation></semantics></math>) into the <a class='existingWikiWord' href='/nlab/show/diff/Deligne-Mumford+compactification'>compactified</a> <a class='existingWikiWord' href='/nlab/show/diff/moduli+stack+of+elliptic+curves'>moduli stack of elliptic curves</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mover><mi>ell</mi><mo>¯</mo></mover></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{\overline{ell}}</annotation></semantics></math> is equivalently the inclusion of the <a class='existingWikiWord' href='/nlab/show/diff/moduli+stack+of+tori'>moduli stack of 1-dimensional tori</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mrow><mn>1</mn><mi>dtori</mi></mrow></msub><mo>=</mo><msub><mi>ℳ</mi> <mrow><msub><mi>𝔾</mi> <mi>m</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{1dtori} = \mathcal{M}_{\mathbb{G}_m}</annotation></semantics></math> (<a href='#LawsonNaumann12'>Lawson-Naumann 12, Appendix A</a>)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mrow><msub><mi>𝔾</mi> <mi>m</mi></msub></mrow></msub><mo>≃</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>⟶</mo><msub><mi>ℳ</mi> <mover><mi>ell</mi><mo>¯</mo></mover></msub><mo>→</mo><msub><mi>ℳ</mi> <mi>FG</mi></msub></mrow><annotation encoding='application/x-tex'> \mathcal{M}_{\mathbb{G}_m} \simeq \mathbf{B}\mathbb{Z}_2 \longrightarrow \mathcal{M}_{\overline{ell}} \to \mathcal{M}_{FG} </annotation></semantics></math></div> <p>and pullback of <a class='existingWikiWord' href='/nlab/show/diff/global+section'>global sections</a> of <a class='existingWikiWord' href='/nlab/show/diff/Goerss-Hopkins-Miller+theorem'>Goerss-Hopkins-Miller-Lurie theorem</a>-wise <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>E_\infty</annotation></semantics></math>-ring valued structure sheaves yields maps</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>KO</mi><mo>⟵</mo><mo>⟵</mo><mi>𝕊</mi></mrow><annotation encoding='application/x-tex'> KO \longleftarrow \longleftarrow \mathbb{S} </annotation></semantics></math></div> <p>exhibiting <a class='existingWikiWord' href='/nlab/show/diff/KO-theory'>KO</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>=</mo><mi>Γ</mi><mo stretchy='false'>(</mo><msub><mi>ℳ</mi> <mrow><msub><mi>𝔾</mi> <mi>m</mi></msub></mrow></msub><mo>,</mo><msup><mi>𝒪</mi> <mi>top</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>= \Gamma(\mathcal{M}_{\mathbb{G}_m}, \mathcal{O}^{top})</annotation></semantics></math>.</p> <p>At least after <a class='existingWikiWord' href='/nlab/show/diff/localization+of+a+ring'>2-localization</a> the canonical <a class='existingWikiWord' href='/nlab/show/diff/double+cover'>double cover</a> of the compactification of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mrow><msub><mi>𝔾</mi> <mi>m</mi></msub></mrow></msub><mo>≃</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{\mathbb{G}_m} \simeq \mathbf{B}\mathbb{Z}_2</annotation></semantics></math> similarly yields under <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Γ</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><msup><mi>𝒪</mi> <mi>top</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Gamma(-,\mathcal{O}^{top})</annotation></semantics></math> the inclusion of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ko</mi></mrow><annotation encoding='application/x-tex'>ko</annotation></semantics></math> as the <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>\mathbb{Z}_2</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/homotopy+fixed+point'>homotopy fixed points</a> of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ku</mi></mrow><annotation encoding='application/x-tex'>ku</annotation></semantics></math> (see at <em><a class='existingWikiWord' href='/nlab/show/diff/KR+cohomology+theory'>KR-theory</a></em> for more on this)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>ku</mi> <mrow><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><msub><mi>ko</mi> <mrow><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ ku_{(2)} \\ \uparrow \\ ko_{(2)} } </annotation></semantics></math></div> <p>and combined with the above this comes with maps from <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> by restriction along the inclusion of the <a class='existingWikiWord' href='/nlab/show/diff/nodal+curve'>nodal curve</a> <a class='existingWikiWord' href='/nlab/show/diff/cusp'>cusp</a> as</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>ku</mi> <mrow><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow></msub></mtd> <mtd><mo>⟵</mo></mtd> <mtd><msub><mi>tmf</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mn>3</mn><msub><mo stretchy='false'>)</mo> <mrow><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><msub><mi>ko</mi> <mrow><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow></msub></mtd> <mtd><mo>⟵</mo></mtd> <mtd><msub><mi>tmf</mi> <mrow><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow></msub></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \array{ ku_{(2)} & \longleftarrow & tmf_1(3)_{(2)} \\ \uparrow && \uparrow \\ ko_{(2)} & \longleftarrow & tmf_{(2)} } \,, </annotation></semantics></math></div> <p>(<a href='#LawsonNaumann12'>Lawson-Naumann 12, theorem 1.2</a>), where <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>tmf</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mn>3</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>tmf_1(3)</annotation></semantics></math> denotes <a class='existingWikiWord' href='/nlab/show/diff/topological+modular+form'>topological modular forms</a> with <a class='existingWikiWord' href='/nlab/show/diff/level+structure+on+an+elliptic+curve'>level-3 structure</a> (<a href='#MahowaldRezk09'>Mahowald-Rezk 09</a>).</p> <p>Moreover, including not just the nodal curve cusp but its <a class='existingWikiWord' href='/nlab/show/diff/formal+neighbourhood'>formal neighbourhood</a>, which is the <a class='existingWikiWord' href='/nlab/show/diff/Tate+curve'>Tate curve</a>, there is analogously a canonical map of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>E_\infty</annotation></semantics></math>-rings</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi><mo>⟶</mo><mi>KO</mi><mo stretchy='false'>[</mo><mo stretchy='false'>[</mo><mi>q</mi><mo stretchy='false'>]</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'> tmf \longrightarrow KO[ [ q ] ] </annotation></semantics></math></div> <p>to <a class='existingWikiWord' href='/nlab/show/diff/Tate+K-theory'>Tate K-theory</a> (this is originally asserted in <a href='#AndoHopkinsStrickland01'>Ando-Hopkins-Strickland 01</a>, details are in <a href='#HillLawson13'>Hill-Lawson 13, appendix A</a>).</p> <h3 id='witten_genus_and_string_orientation'>Witten genus and string orientation</h3> <p>The <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a> is the codomain of the <a class='existingWikiWord' href='/nlab/show/diff/Witten+genus'>Witten genus</a>, or rather of its refinements to the <a class='existingWikiWord' href='/nlab/show/diff/string+orientation+of+tmf'>string orientation of tmf</a> with value in <a class='existingWikiWord' href='/nlab/show/diff/topological+modular+form'>topological modular forms</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi><mo>:</mo><mi>M</mi><mi>String</mi><mo>→</mo><mi>tmf</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \sigma : M String \to tmf \,. </annotation></semantics></math></div> <p>The original Witten genus is the value of the composite of this with the <a href='#MapToTateKTheory'>map to Tate K-theory</a> on <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy groups</a>. (<a href='#AndoHopkinsRezk10'>Ando-Hopkins-Rezk 10</a>)</p> <h3 id='chromatic_filtration'>Chromatic filtration</h3> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/chromatic+homotopy+theory'>chromatic homotopy theory</a></strong></p> <table><thead><tr><th><a class='existingWikiWord' href='/nlab/show/diff/chromatic+homotopy+theory'>chromatic level</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/complex+oriented+cohomology+theory'>complex oriented cohomology theory</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ ring</a>/<a class='existingWikiWord' href='/nlab/show/diff/A-infinity-ring'>A-∞ ring</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/real+oriented+cohomology+theory'>real oriented cohomology theory</a></th></tr></thead><tbody><tr><td style='text-align: left;'>0</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/ordinary+cohomology'>ordinary cohomology</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Mac+Lane+spectrum'>Eilenberg-MacLane spectrum</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>H \mathbb{Z}</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/HZR-theory'>HZR-theory</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'>0th <a class='existingWikiWord' href='/nlab/show/diff/Morava+K-theory'>Morava K-theory</a></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mn>0</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(0)</annotation></semantics></math></td><td style='text-align: left;' /></tr> <tr><td style='text-align: left;'>1</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>complex K-theory</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/K-theory+spectrum'>complex K-theory spectrum</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>KU</mi></mrow><annotation encoding='application/x-tex'>KU</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/KR+cohomology+theory'>KR-theory</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'>first <a class='existingWikiWord' href='/nlab/show/diff/Morava+K-theory'>Morava K-theory</a></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(1)</annotation></semantics></math></td><td style='text-align: left;' /></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'>first <a class='existingWikiWord' href='/nlab/show/diff/Morava+E-theory'>Morava E-theory</a></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E(1)</annotation></semantics></math></td><td style='text-align: left;' /></tr> <tr><td style='text-align: left;'>2</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/elliptic+cohomology'>elliptic cohomology</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/elliptic+spectrum'>elliptic spectrum</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ell</mi> <mi>E</mi></msub></mrow><annotation encoding='application/x-tex'>Ell_E</annotation></semantics></math></td><td style='text-align: left;' /></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'>second <a class='existingWikiWord' href='/nlab/show/diff/Morava+K-theory'>Morava K-theory</a></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(2)</annotation></semantics></math></td><td style='text-align: left;' /></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'>second <a class='existingWikiWord' href='/nlab/show/diff/Morava+E-theory'>Morava E-theory</a></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E(2)</annotation></semantics></math></td><td style='text-align: left;' /></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/algebraic+K-theory'>algebraic K-theory</a> of <a class='existingWikiWord' href='/nlab/show/diff/K-theory+spectrum'>KU</a></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>KU</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(KU)</annotation></semantics></math></td><td style='text-align: left;' /></tr> <tr><td style='text-align: left;'>3 …10</td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/K3-spectrum'>K3 cohomology</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/K3-spectrum'>K3 spectrum</a></td><td style='text-align: left;' /></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>th <a class='existingWikiWord' href='/nlab/show/diff/Morava+K-theory'>Morava K-theory</a></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(n)</annotation></semantics></math></td><td style='text-align: left;' /></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>th <a class='existingWikiWord' href='/nlab/show/diff/Morava+E-theory'>Morava E-theory</a></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E(n)</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/BPR-theory'>BPR-theory</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n+1</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/algebraic+K-theory'>algebraic K-theory</a> applied to chrom. level <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(E_n)</annotation></semantics></math> (<a class='existingWikiWord' href='/nlab/show/diff/red-shift+conjecture'>red-shift conjecture</a>)</td><td style='text-align: left;' /></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/MU'>complex cobordism cohomology</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/MU'>MU</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/MR+cohomology+theory'>MR-theory</a></td></tr> </tbody></table> <h3 id='anderson_selfduality'>Anderson self-duality</h3> <p>The <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Tmf</mi></mrow><annotation encoding='application/x-tex'>Tmf</annotation></semantics></math> is self-dual under <a class='existingWikiWord' href='/nlab/show/diff/Anderson+duality'>Anderson duality</a>, more precisley <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Tmf</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>Tmf[1/2]</annotation></semantics></math> is Anderson-dual to <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Σ</mi> <mn>21</mn></msup><mi>Tmf</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Sigma^{21} Tmf[1/2]</annotation></semantics></math> (<a href='#Stojanoska11'>Stojanoska 11, theorem 13.1</a>)</p> <h3 id='modular_equivariant_versions'>Modular equivariant versions</h3> <p>See at <em><a class='existingWikiWord' href='/nlab/show/diff/modular+equivariant+elliptic+cohomology'>modular equivariant elliptic cohomology</a></em> and at <em><a class='existingWikiWord' href='/nlab/show/diff/Tmf%28n%29'>Tmf(n)</a></em>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/topological+automorphic+form'>taf</a></li> </ul> <p><strong>Substructure of the <a class='existingWikiWord' href='/nlab/show/diff/moduli+space+of+curves'>moduli stack of curves</a> and the (<a class='existingWikiWord' href='/nlab/show/diff/equivariant+cohomology'>equivariant</a>) <a class='existingWikiWord' href='/nlab/show/diff/generalized+cohomology'>cohomology theory</a> associated with it via the <a class='existingWikiWord' href='/nlab/show/diff/Goerss-Hopkins-Miller+theorem'>Goerss-Hopkins-Miller-Lurie theorem</a>:</strong></p> <table><thead><tr><th /><th /><th /><th /><th /><th /><th /><th /><th /><th /></tr></thead><tbody><tr><td style='text-align: left;'>covering</td><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;'>by <a class='existingWikiWord' href='/nlab/show/diff/level+structure+on+an+elliptic+curve'>of level-n structures</a> (<a class='existingWikiWord' href='/nlab/show/diff/modular+curve'>modular curve</a>)</td><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;' /></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo><mo>=</mo><mi>Spec</mi><mo stretchy='false'>(</mo><mi>ℤ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\ast = Spec(\mathbb{Z})</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>→</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_174' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Spec</mi><mo stretchy='false'>(</mo><mi>ℤ</mi><mo stretchy='false'>[</mo><mo stretchy='false'>[</mo><mi>q</mi><mo stretchy='false'>]</mo><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Spec(\mathbb{Z}[ [q] ])</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_175' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>→</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_176' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mover><mi>ell</mi><mo>¯</mo></mover></msub><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{\overline{ell}}[n]</annotation></semantics></math></td><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;' /></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/structure+group'>structure group</a> of covering</td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_177' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo stretchy='false'>↓</mo> <mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn><mi>ℤ</mi></mrow></msup></mrow><annotation encoding='application/x-tex'>\downarrow^{\mathbb{Z}/2\mathbb{Z}}</annotation></semantics></math></td><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_178' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo stretchy='false'>↓</mo> <mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn><mi>ℤ</mi></mrow></msup></mrow><annotation encoding='application/x-tex'>\downarrow^{\mathbb{Z}/2\mathbb{Z}}</annotation></semantics></math></td><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_179' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo stretchy='false'>↓</mo> <mrow><msub><mi>SL</mi> <mn>2</mn></msub><mo stretchy='false'>(</mo><mi>ℤ</mi><mo stretchy='false'>/</mo><mi>n</mi><mi>ℤ</mi><mo stretchy='false'>)</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})}</annotation></semantics></math> (<a class='existingWikiWord' href='/nlab/show/diff/modular+group'>modular group</a>)</td><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;' /></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/moduli+stack'>moduli stack</a></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_180' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mrow><mn>1</mn><mi>dTori</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{1dTori}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_181' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>↪</mo></mrow><annotation encoding='application/x-tex'>\hookrightarrow</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_182' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mi>Tate</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{Tate}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_183' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>↪</mo></mrow><annotation encoding='application/x-tex'>\hookrightarrow</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_184' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mover><mi>ell</mi><mo>¯</mo></mover></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{\overline{ell}}</annotation></semantics></math> (<a class='existingWikiWord' href='/nlab/show/diff/moduli+stack+of+elliptic+curves'>M_ell</a>)</td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_185' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>↪</mo></mrow><annotation encoding='application/x-tex'>\hookrightarrow</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_186' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mi>cub</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{cub}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_187' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>→</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_188' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℳ</mi> <mi>fg</mi></msub></mrow><annotation encoding='application/x-tex'>\mathcal{M}_{fg}</annotation></semantics></math> (<a class='existingWikiWord' href='/nlab/show/diff/moduli+stack+of+formal+groups'>M_fg</a>)</td></tr> <tr><td style='text-align: left;'>of</td><td style='text-align: left;'><a href='torus#InAlgebraicGeometry'>1d tori</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Tate+curve'>Tate curves</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/elliptic+curve'>elliptic curves</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/cubic+curve'>cubic curves</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/moduli+stack+of+formal+groups'>1d commutative formal groups</a></td></tr> <tr><td style='text-align: left;'>value <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_189' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>𝒪</mi> <mi>Σ</mi> <mi>top</mi></msubsup></mrow><annotation encoding='application/x-tex'>\mathcal{O}^{top}_{\Sigma}</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/structure+sheaf'>structure sheaf</a> over curve <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_190' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi></mrow><annotation encoding='application/x-tex'>\Sigma</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/K-theory+spectrum'>KU</a></td><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_191' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>KU</mi><mo stretchy='false'>[</mo><mo stretchy='false'>[</mo><mi>q</mi><mo stretchy='false'>]</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>KU[ [q] ]</annotation></semantics></math></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/elliptic+spectrum'>elliptic spectrum</a></td><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/complex+oriented+cohomology+theory'>complex oriented cohomology theory</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_192' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Γ</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><msup><mi>𝒪</mi> <mi>top</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Gamma(-, \mathcal{O}^{top})</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/global+section'>global sections</a> of <a class='existingWikiWord' href='/nlab/show/diff/structure+sheaf'>structure sheaf</a></td><td style='text-align: left;'>(<a class='existingWikiWord' href='/nlab/show/diff/KO-theory'>KO</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_193' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>↪</mo></mrow><annotation encoding='application/x-tex'>\hookrightarrow</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/K-theory+spectrum'>KU</a>) = <a class='existingWikiWord' href='/nlab/show/diff/KR+cohomology+theory'>KR-theory</a></td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Tate+K-theory'>Tate K-theory</a> (<math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_194' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>KO</mi><mo stretchy='false'>[</mo><mo stretchy='false'>[</mo><mi>q</mi><mo stretchy='false'>]</mo><mo stretchy='false'>]</mo><mo>↪</mo><mi>KU</mi><mo stretchy='false'>[</mo><mo stretchy='false'>[</mo><mi>q</mi><mo stretchy='false'>]</mo><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>KO[ [q] ] \hookrightarrow KU[ [q] ]</annotation></semantics></math>)</td><td style='text-align: left;' /><td style='text-align: left;'>(<a class='existingWikiWord' href='/nlab/show/diff/tmf'>Tmf</a> <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_195' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>→</mo></mrow><annotation encoding='application/x-tex'>\to</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/Tmf%28n%29'>Tmf(n)</a>) (<a class='existingWikiWord' href='/nlab/show/diff/modular+equivariant+elliptic+cohomology'>modular equivariant elliptic cohomology</a>)</td><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/tmf'>tmf</a></td><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_196' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝕊</mi></mrow><annotation encoding='application/x-tex'>\mathbb{S}</annotation></semantics></math></td></tr> </tbody></table> <h2 id='references'>References</h2> <p>The idea of a <a class='existingWikiWord' href='/nlab/show/diff/generalized+cohomology'>generalized cohomology theory</a> with <a class='existingWikiWord' href='/nlab/show/diff/coefficient'>coefficients</a> the ring of <a class='existingWikiWord' href='/nlab/show/diff/topological+modular+form'>topological modular forms</a> providing a home for the refined <a class='existingWikiWord' href='/nlab/show/diff/Witten+genus'>Witten genus</a> of</p> <ul> <li id='Witten87a'><a class='existingWikiWord' href='/nlab/show/diff/Edward+Witten'>Edward Witten</a>, <em>Elliptic Genera And Quantum Field Theory</em>, Commun. Math. Phys. 109 525 (1987) (<a href='http://projecteuclid.org/euclid.cmp/1104117076'>euclid.cmp/1104117076</a>, <a href='https://people.maths.ox.ac.uk/beem/papers/elliptic_genus_witten.pdf'>pdf</a>)</li> </ul> <p>and produced as a <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a> of <a class='existingWikiWord' href='/nlab/show/diff/elliptic+cohomology'>elliptic cohomology</a> theories over the <a class='existingWikiWord' href='/nlab/show/diff/moduli+stack+of+elliptic+curves'>moduli stack of elliptic curves</a> was originally announced, as joint work with <a class='existingWikiWord' href='/nlab/show/diff/Mark+Mahowald'>Mark Mahowald</a> and <a class='existingWikiWord' href='/nlab/show/diff/Haynes+Miller'>Haynes Miller</a>, in</p> <ul> <li id='Hopkins94'><a class='existingWikiWord' href='/nlab/show/diff/Michael+Hopkins'>Michael Hopkins</a>, section 9 of <em>Topological modular forms, the Witten Genus, and the theorem of the cube</em>, Proceedings of the International Congress of Mathematics, Zürich 1994 (<a class='existingWikiWord' href='/nlab/files/Hopkins_TopModFormsAtICM.pdf' title='pdf'>pdf</a>, <a href='https://doi.org/10.1007/978-3-0348-9078-6_49'>doi:10.1007/978-3-0348-9078-6_49</a>)</li> </ul> <p>(There the spectrum was still called “<math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_197' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>eo</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>eo_2</annotation></semantics></math>” instead of “<math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_198' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math>”.) The details of the definition then appeared in</p> <ul> <li id='Hopkins02'><a class='existingWikiWord' href='/nlab/show/diff/Michael+Hopkins'>Michael Hopkins</a>, section 4 of <em>Algebraic topology and modular forms</em>, Proceedings of the ICM, Beijing 2002, vol. 1, 283–309 (<a href='http://arxiv.org/abs/math/0212397'>arXiv:math/0212397</a>)</li> </ul> <p>A central tool that goes into the construction is the <a class='existingWikiWord' href='/nlab/show/diff/Goerss-Hopkins-Miller+theorem'>Goerss-Hopkins-Miller theorem</a>, see there for references on that.</p> <p>Textbook account:</p> <ul> <li id='DFHH14'><a class='existingWikiWord' href='/nlab/show/diff/Christopher+Douglas'>Christopher Douglas</a>, <a class='existingWikiWord' href='/nlab/show/diff/John+Francis'>John Francis</a>, <a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Henriques'>André Henriques</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michael+Hill'>Michael Hill</a> (eds.), <em>Topological Modular Forms</em>, Mathematical Surveys and Monographs Volume 201, AMS 2014 (<a href='https://bookstore.ams.org/surv-201'>ISBN:978-1-4704-1884-7</a>)</li> </ul> <p>Expositions include</p> <ul> <li id='MazelGee13'> <p><a class='existingWikiWord' href='/nlab/show/diff/Aaron+Mazel-Gee'>Aaron Mazel-Gee</a>, <em>You could’ve invented <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_199' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math></em>, April 2013 (<a href='http://math.berkeley.edu/~aaron/writing/ustars-tmf-beamer.pdf'>pdf slides</a>, <a href='http://math.berkeley.edu/~aaron/writing/tmf-seminar-talk.pdf'>notes pdf</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/A+Survey+of+Elliptic+Cohomology'>A Survey of Elliptic Cohomology</a></em></p> </li> </ul> <p>See also</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Christopher+Douglas'>Chris Douglas</a>, <a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Henriques'>André Henriques</a>, <em>Topological modular forms and conformal nets</em>, in <a class='existingWikiWord' href='/nlab/show/diff/Hisham+Sati'>Hisham Sati</a>, <a class='existingWikiWord' href='/nlab/show/diff/Urs+Schreiber'>Urs Schreiber</a> (eds.), <em><a class='existingWikiWord' href='/schreiber/show/diff/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory' title='schreiber'>Mathematical Foundations of Quantum Field and Perturbative String Theory</a></em>, Proceedings of Symposia in Pure Mathematics, AMS (2011) (<a href='https://arxiv.org/abs/1103.4187'>arXiv:1103.4187</a>, <a href='https://doi.org/10.1090/pspum/083'>doi:10.1090/pspum/083</a>)</li> </ul> <p>An actual detailed account of the construction of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_200' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> (via decomposition by <a class='existingWikiWord' href='/nlab/show/diff/fracture+theorem'>arithmetic squares</a>) is spelled out in</p> <ul> <li id='Behrens13'><a class='existingWikiWord' href='/nlab/show/diff/Mark+Behrens'>Mark Behrens</a>, <em>Notes on the construction of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_201' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math></em>, 2013 (<a href='http://math.mit.edu/~mbehrens/papers/buildTMF.pdf'>pdf</a>)</li> </ul> <p>A complete account of the computation of the homotopy groups of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_202' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> (following previous unpublished computations) is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Tilman+Bauer'>Tilman Bauer</a>, <em>Computation of the homotopy groups of the spectrum <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_203' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math></em> (<a href='http://www.math.rochester.edu/people/faculty/doug/otherpapers/eo2ss.pdf'>pdf</a>)</li> </ul> <p>A survey of how this works is in</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Akhil+Mathew'>Akhil Mathew</a>, <em>The homotopy groups of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_204' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>TMF</mi></mrow><annotation encoding='application/x-tex'>TMF</annotation></semantics></math></em> (<a href='http://math.uchicago.edu/~amathew/tmfhomotopy.pdf'>pdf</a>)</p> <p>(This presents as an instructive much simpler but analogous case the construction of <a class='existingWikiWord' href='/nlab/show/diff/KO-theory'>KO</a> in analogy to the construction of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_205' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math>, more details on this are in <a href='#Mathew13'>Mathew 13, section 3</a>.)</p> </li> </ul> <p>and course notes that go through the construction of tmf and the computation of its homotopy groups are here:</p> <ul> <li> <p><em>Talbot workshop on TMF</em> (<a href='http://math.mit.edu/conferences/talbot/2007/tmfproc/'>web</a>)</p> <ul> <li id='Hopkins'> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael+Hopkins'>Mike Hopkins</a> (talk notes by <a class='existingWikiWord' href='/nlab/show/diff/Michael+Hill'>Michael Hill</a>), <em>Stacks and complex oriented cohomology theories</em> (<a href='http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter08/MikesTalk1.pdf'>pdf</a>)</p> </li> <li id='Henriques'> <p><a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Henriques'>André Henriques</a>, <em>The homotopy groups of tmf</em> (<a href='http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter16/TmfHomotopy.pdf'>pdf</a>)</p> </li> <li id='HenriquesModuli'> <p><a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Henriques'>André Henriques</a>, <em>The moduli stack of elliptic curves</em> (<a href='http://math.mit.edu/conferences/talbot/2007/tmfproc/Chapter04/henriques.pdf'>pdf</a>)</p> </li> </ul> </li> </ul> <p>The non-connective version of this is discussed in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Johan+Konter'>Johan Konter</a>, <em>The homotopy groups of the spectrum Tmf</em> (<a href='http://arxiv.org/abs/1212.3656'>arXiv:1212.3656</a>)</li> </ul> <p>Supplementary material graphically displaying parts of these intricate computations is in</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Henriques'>André Henriques</a>,</p> <p>the graded ring <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_206' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>tmf</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>pt</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>tmf^\ast(pt)</annotation></semantics></math> (<a href='http://www.staff.science.uu.nl/~henri105/PDF/TmfRing.pdf'>pdf</a>);</p> <p>the <a class='existingWikiWord' href='/nlab/show/diff/spectral+sequence'>spectral sequence</a> used to compute it (<a href='http://math.mit.edu/conferences/talbot/2007/tmfproc/henriques-tmfSS.pdf'>pdf</a>);</p> <p>whose <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_207' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>E_2</annotation></semantics></math>-page is <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_208' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Ext</mi> <mrow><mi>A</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ext_{A(2)}(\mathbb{F}_2, \mathbb{F}_2)</annotation></semantics></math> where <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_209' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>A(2)</annotation></semantics></math> is displayed here: <a href='http://www.staff.science.uu.nl/~henri105/PDF/A2.pdf'>pdf</a>;</p> <p>the spectral sequence that computes <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_210' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Tmf</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>pt</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Tmf^\ast(pt)</annotation></semantics></math> (<a href='http://math.mit.edu/conferences/talbot/2007/tmfproc/EllipticSpectralSequence.pdf'>pdf</a>)</p> </li> </ul> <p>The <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_211' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>\mathbb{Z}_2</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/homology'>homology</a> of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_212' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> is discussed in</p> <ul> <li id='Mathew13'><a class='existingWikiWord' href='/nlab/show/diff/Akhil+Mathew'>Akhil Mathew</a>, <em>The homology of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_213' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math></em> (<a href='http://arxiv.org/abs/1305.6100'>arXiv:1305.6100</a>)</li> </ul> <p>The refinement of the <a class='existingWikiWord' href='/nlab/show/diff/Witten+genus'>Witten genus</a> to a morphism of <a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ rings</a> to <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_214' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math>, hence the <a class='existingWikiWord' href='/nlab/show/diff/string+orientation+of+tmf'>string orientation of tmf</a> is due to</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael+Hopkins'>Michael Hopkins</a>, <em>Topological modular forms, the Witten Genus, and the theorem of the cube</em>, Proceedings of the International Congress of Mathematics, Zürich 1994 (<a href='http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf'>pdf</a>)</p> </li> <li id='AndoHopkinsStrickland01'> <p><a class='existingWikiWord' href='/nlab/show/diff/Matthew+Ando'>Matthew Ando</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michael+Hopkins'>Michael Hopkins</a>, <a class='existingWikiWord' href='/nlab/show/diff/Neil+Strickland'>Neil Strickland</a>, <em>Elliptic spectra, the Witten genus and the theorem of the cube</em>, Invent. Math. 146 (2001) 595–687 MR1869850</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael+Hopkins'>Michael Hopkins</a>, <em>Algebraic topology and modular forms</em>, Proceedings of the ICM, Beijing 2002, vol. 1, 283–309 (<a href='http://arxiv.org/abs/math/0212397'>arXiv:math/0212397</a>)</p> </li> <li id='AndoHopkinsRezk10'> <p><a class='existingWikiWord' href='/nlab/show/diff/Matthew+Ando'>Matthew Ando</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michael+Hopkins'>Michael Hopkins</a>, <a class='existingWikiWord' href='/nlab/show/diff/Charles+Rezk'>Charles Rezk</a>, <em>Multiplicative orientations of KO-theory and the spectrum of topological modular forms</em>, 2010 (<a href='http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf'>pdf</a>)</p> </li> </ul> <p>see also remark 1.4 of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Paul+Goerss'>Paul Goerss</a>, <em>Topological modular forms (after Hopkins, Miller and Lurie)</em> (<a href='http://arxiv.org/PS_cache/arxiv/pdf/0910/0910.5130v1.pdf'>pdf</a>).</li> </ul> <p>and for more on the <a class='existingWikiWord' href='/nlab/show/diff/sigma-orientation'>sigma-orientation</a> see</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Matthew+Ando'>Matthew Ando</a>, <em>The sigma orientation for analytic circle-equivariant elliptic cohomology</em>, Geom. Topol. 7 (2003) 91-153 (<a href='http://arxiv.org/abs/math/0201092'>arXiv:math/0201092</a>)</li> </ul> <p>Discussion of <a class='existingWikiWord' href='/nlab/show/diff/twisted+cohomology'>twisted cohomology</a> with <a class='existingWikiWord' href='/nlab/show/diff/coefficient'>coefficients</a> in <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_215' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> is in</p> <ul> <li id='ABG10'><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>, section 8 of <em>Twists of K-theory and TMF</em>, in Robert S. Doran, Greg Friedman, <a class='existingWikiWord' href='/nlab/show/diff/Jonathan+Rosenberg'>Jonathan Rosenberg</a>, <em>Superstrings, Geometry, Topology, and <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_216' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^*</annotation></semantics></math>-algebras</em>, Proceedings of Symposia in Pure Mathematics <a href='http://www.ams.org/bookstore-getitem/item=PSPUM-81'>vol 81</a>, American Mathematical Society (<a href='http://arxiv.org/abs/1002.3004'>arXiv:1002.3004</a>)</li> </ul> <p>Topological modular forms with <em><a class='existingWikiWord' href='/nlab/show/diff/level+structure+on+an+elliptic+curve'>level N-structure</a></em> – <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_217' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi><mo stretchy='false'>(</mo><mi>N</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>tmf(N)</annotation></semantics></math> – is discussed in</p> <ul> <li id='MahowaldRezk09'> <p><a class='existingWikiWord' href='/nlab/show/diff/Mark+Mahowald'>Mark Mahowald</a>, <a class='existingWikiWord' href='/nlab/show/diff/Charles+Rezk'>Charles Rezk</a>, <em>Topological modular forms of level 3</em>, Pure Appl. Math. Quar. 5 (2009) 853-872 (<a href='http://www.math.uiuc.edu/~rezk/tmf3-paper-final.pdf'>pdf</a>)</p> </li> <li id='DavisMahowald10'> <p><a class='existingWikiWord' href='/nlab/show/diff/Donald+Davis'>Donald Davis</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mark+Mahowald'>Mark Mahowald</a>, <em>Connective versions of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_218' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>TMF</mi><mo stretchy='false'>(</mo><mn>3</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>TMF(3)</annotation></semantics></math></em> (<a href='http://arxiv.org/abs/1005.3752'>arXiv:1005.3752</a>)</p> </li> <li id='Stojanoska11'> <p><a class='existingWikiWord' href='/nlab/show/diff/Vesna+Stojanoska'>Vesna Stojanoska</a>, Duality for Topological Modular Forms (<a href='http://arxiv.org/abs/1105.3968'>arXiv:1105.3968</a>)</p> </li> <li id='LawsonNaumann12'> <p><a class='existingWikiWord' href='/nlab/show/diff/Tyler+Lawson'>Tyler Lawson</a>, <a class='existingWikiWord' href='/nlab/show/diff/Niko+Naumann'>Niko Naumann</a>, <em>Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2</em>, Int. Math. Res. Not. (2013) (<a href='http://arxiv.org/abs/1203.1696'>arXiv:1203.1696</a>)</p> </li> <li id='HillLawson13'> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael+Hill'>Michael Hill</a>, <a class='existingWikiWord' href='/nlab/show/diff/Tyler+Lawson'>Tyler Lawson</a>, <em>Topological modular forms with level structure</em>, Inventiones mathematicae volume 203, pages 359–416 (2016) (<a href='http://arxiv.org/abs/1312.7394'>arXiv:1312.7394</a>, <a href='https://doi.org/10.1007/s00222-015-0589-5'>doi:10.1007/s00222-015-0589-5</a>)</p> <blockquote> <p>(with <a class='existingWikiWord' href='/nlab/show/diff/level+structure+on+an+elliptic+curve'>level structure</a>)</p> </blockquote> </li> </ul> <p>The self-<a class='existingWikiWord' href='/nlab/show/diff/Anderson+duality'>Anderson duality</a> of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_219' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math> is discussed in (<a href='#Stojanoska11'>Stojanoska 11</a>).</p> <p>On equivariant topological modular forms (on <a class='existingWikiWord' href='/nlab/show/diff/equivariant+elliptic+cohomology'>equivariant elliptic cohomology</a>):</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/David+Gepner'>David Gepner</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lennart+Meier'>Lennart Meier</a>, <em>On equivariant topological modular forms</em>, (<a href='https://arxiv.org/abs/2004.10254'>arXiv:2004.10254</a>)</li> </ul> <p>On the <a class='existingWikiWord' href='/nlab/show/diff/Boardman+homomorphism'>Boardman homomorphism</a> (generalized <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+theorem'>Hurewicz homomorphism</a>) to <a class='existingWikiWord' href='/nlab/show/diff/tmf'>tmf</a>:</p> <del class='diffmod'><ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Mark+Behrens'>Mark Behrens</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mark+Mahowald'>Mark Mahowald</a>, J.D. Quigley, <em>The 2-primary Hurewicz image of <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_220' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math></em> (<a href='https://arxiv.org/abs/2011.08956'>arXiv:2011.08956</a>)</li> </ul></del><ins class='diffmod'><p>On the <a class='existingWikiWord' href='/nlab/show/diff/Boardman+homomorphism'>Boardman homomorphism</a> (generalized <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+theorem'>Hurewicz homomorphism</a>) to <a class='existingWikiWord' href='/nlab/show/diff/tmf'>tmf</a>:</p></ins> <ins class='diffins'><ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Mark+Behrens'>Mark Behrens</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mark+Mahowald'>Mark Mahowald</a>, <a class='existingWikiWord' href='/nlab/show/diff/J.+D.+Quigley'>J. D. Quigley</a>: <em>The 2-primary Hurewicz image of <math class='maruku-mathml' display='inline' id='mathml_52f282000f4cf7a872bb4b0357e1308b615dcdb1_220' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>tmf</mi></mrow><annotation encoding='application/x-tex'>tmf</annotation></semantics></math></em>, Geometry & Topology <strong>27</strong> (2023) 2763–2831 [[arXiv:2011.08956](https://arxiv.org/abs/2011.08956), <a href='https://doi.org/10.2140/gt.2023.27.2763'>doi:10.2140/gt.2023.27.2763</a>]</li> </ul></ins><ins class='diffins'> </ins><p>More on the role of lifting the <a class='existingWikiWord' href='/nlab/show/diff/Witten+genus'>Witten genus</a> to tmf from the point of view of <a class='existingWikiWord' href='/nlab/show/diff/heterotic+string+theory'>heterotic string theory</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yuji+Tachikawa'>Yuji Tachikawa</a>, <em>Topological modular forms and the absence of a heterotic global anomaly</em>, Progress of Theoretical and Experimental Physics, <strong>2022</strong> 4 (2022) 04A107 <math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_221' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://arxiv.org/abs/2103.12211'>arXiv:2103.12211</a>, <a href='https://doi.org/10.1093/ptep/ptab060'>doi:10.1093/ptep/ptab060</a><math class='maruku-mathml' display='inline' id='mathml_ea9ad0cf19f1b27177d7250a7fc031c433ab6c84_222' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yuji+Tachikawa'>Yuji Tachikawa</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mayuko+Yamashita'>Mayuko Yamashita</a>, <em>Topological modular forms and the absence of all heterotic global anomalies</em>, Comm. Math. Phys. <strong>402</strong> (2023) 1585-1620 [[arXiv:2108.13542](https://arxiv.org/abs/2108.13542), <a href='https://doi.org/10.1007/s00220-023-04761-2'>doi:10.1007/s00220-023-04761-2</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yuji+Tachikawa'>Yuji Tachikawa</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mayuko+Yamashita'>Mayuko Yamashita</a>, <em>Anderson self-duality of topological modular forms, its differential-geometric manifestations, and vertex operator algebras</em> [[arXiv:2305.06196](https://arxiv.org/abs/2305.06196)]</p> </li> </ul> <p>On <a class='existingWikiWord' href='/nlab/show/diff/Adams+operation'>Adams operations</a> for tmf:</p> <ul> <li>Jack Morgan Davies, <em>Constructing and calculating Adams operations on dualisable topological modular forms</em> [[arXiv:2104.13407](https://arxiv.org/abs/2104.13407)]</li> </ul> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on December 21, 2024 at 15:51:32. See the <a href="/nlab/history/tmf" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/tmf" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/2211/#Item_12">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/tmf/71" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/tmf" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/tmf" accesskey="S" class="navlink" id="history" rel="nofollow">History (71 revisions)</a> <a href="/nlab/show/tmf/cite" style="color: black">Cite</a> <a href="/nlab/print/tmf" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/tmf" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>