About: Topological modular forms

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In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer n there is a topological space , and these spaces are equipped with certain maps between them, so that for any topological space X, one obtains an abelian group structure on the set of homotopy classes of continuous maps from X to . One feature that distinguishes tmf is the fact that its coefficient ring, (point), is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of torsion information in the coefficient ring.

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rdf:type	yago:WikicatCohomologyTheories yago:Abstraction100002137 yago:Cognition100023271 yago:Explanation105793000 yago:HigherCognitiveProcess105770664 yago:Process105701363 yago:PsychologicalFeature100023100 yago:Theory105989479 yago:Thinking105770926
rdfs:label	Topological modular forms (en)
rdfs:comment	In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer n there is a topological space , and these spaces are equipped with certain maps between them, so that for any topological space X, one obtains an abelian group structure on the set of homotopy classes of continuous maps from X to . One feature that distinguishes tmf is the fact that its coefficient ring, (point), is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of torsion information in the coefficient ring. (en)
dcterms:subject	Cohomology theories Algebraic topology
Wikipage page ID	10274436 (xsd:integer)
Wikipage revision ID	1100061743 (xsd:integer)
Link from a Wikipage to another Wikipage	List of cohomology theories Euclidean field theory Algebraic stack Homotopy groups of spheres Charles Rezk Cusp form Elliptic curves Mathematics Elliptic cohomology Genus of a multiplicative sequence Obstruction theory Cohomology theories Edward Witten Graeme Segal Conformal field theory String theory Compactification (mathematics) Functor Haynes Miller K-theory Pontryagin class Algebraic topology Number theory Graded ring Atiyah–Singer index theorem Jacob Lurie Abelian group Eilenberg–Steenrod axioms Highly structured ring spectrum Homotopy group Dirac equation Manifold Mark Mahowald Grothendieck topology Michael J. Hopkins Loop space Sheaf (mathematics) Smooth manifold Étale topology Formal group Topological space Modular forms Moduli stack Ring spectrum Supersymmetric Equivariant index theory Topological Cohomology theories Cohomology theory Modular discriminant Spectrum (homotopy theory) dbr:Paul_Goerss Presheaf Spin manifold dbr:Deligne–Mumford_compactification
Link from a Wikipage to an external page	http://www.ams.org/bookstore-getitem/item=surv-201 http://www.math.purdue.edu/research/atopology/Hopkins-Mahowald/eo2homotopy.pdf http://www.math.uiuc.edu/~rezk/512-spr2001-notes.pdf https://web.archive.org/web/20060911084103/http:/www.math.purdue.edu/research/atopology/Hopkins-Mahowald/eo2homotopy.pdf http://math.berkeley.edu/~teichner/Papers/Survey.pdf http://www-math.mit.edu/~mbehrens/papers/buildTMF.pdf http://www.math.harvard.edu/~lurie/papers/survey.pdf http://www.math.northwestern.edu/~pgoerss/papers/sum.pdf
sameAs	Topological modular forms Topological modular forms Topological modular forms Topological modular forms
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:Cite_journal dbt:Webarchive
date	2006-09-11 (xsd:date)
url	https://web.archive.org/web/20060911084103/http:/www.math.purdue.edu/research/atopology/Hopkins-Mahowald/eo2homotopy.pdf
has abstract	In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer n there is a topological space , and these spaces are equipped with certain maps between them, so that for any topological space X, one obtains an abelian group structure on the set of homotopy classes of continuous maps from X to . One feature that distinguishes tmf is the fact that its coefficient ring, (point), is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of torsion information in the coefficient ring. The spectrum of topological modular forms is constructed as the global sections of a sheaf of E-infinity ring spectra on the moduli stack of (generalized) elliptic curves. This theory has relations to the theory of modular forms in number theory, the homotopy groups of spheres, and conjectural index theories on loop spaces of manifolds. tmf was first constructed by Michael Hopkins and Haynes Miller; many of the computations can be found in preprints and articles by , Hopkins, Mark Mahowald, Miller, Charles Rezk, and Tilman Bauer. (en)
prov:wasDerivedFrom	wikipedia-en:Topological_modular_forms?oldid=1100061743&ns=0
page length (characters) of wiki page	7308 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Topological_modular_forms
is Link from a Wikipage to another Wikipage of	List of cohomology theories Topological modular form Timeline of category theory and related mathematics Elliptic cohomology Timeline of manifolds Commutative ring spectrum TMF Landweber exact functor theorem Chromatic homotopy theory

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