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| In [[mathematics]], '''elliptic cohomology''' is a [[cohomology theory]] in the sense of [[algebraic topology]]. It is related to [[elliptic curves]] and [[modular forms]].
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| ==History and motivation==
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| Historically, elliptic cohomology arose from the study of [[elliptic genus|elliptic genera]]. It is known by Atiyah and Hirzebruch that if <math>S^1</math> acts smoothly and non-trivially on a spin manifold, then the index of the [[Dirac operator]] vanishes. In 1983, [[Edward Witten|Witten]] conjectured that in this situation the equivariant index of a certain twisted Dirac operator is at least constant. This led to certain other problems concerning <math>S^1</math>-actions on manifolds, which could be solved by Ochanine by the introduction of elliptic genera. These got in turn by Witten related to (conjectural) index theory on [[free loop]] spaces. Elliptic cohomology, invented in its original form by Landweber, Stong and [[Douglas Ravenel|Ravenel]] in the late 1980s, was introduced to clarify certain issues with elliptic genera and provide a context for (conjectural) index theory of families of differential operators on free loop spaces. In some sense it can be seen as an approximation to the [[topological K-theory|K-theory]] of the free loop space.
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| ==Definitions and constructions==
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| Call a cohomology theory <math>A^*</math> even periodic if <math>A^i = 0</math> for i odd and there is an invertible element <math>u\in A^2</math>. These theories possess a [[complex orientation]], which gives a [[formal group law]]. A particularly rich source for formal group laws are [[elliptic curve]]s. A cohomology theory A with <math>A^0 = R</math> is called ''elliptic'' if it is even periodic and its formal group law is isomorphic to a formal group law of an elliptic curve E over R. The usual construction of such elliptic cohomology theories uses the [[Landweber exact functor theorem]]. If the formal group laws of E is Landweber exact, one can define an elliptic cohomology theory (on finite complexes) by
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| : <math>A^*(X) = MU^*(X)\otimes_{MU^*}R[u,u^{-1}]. \, </math>
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| Franke has identified the condition needed to fulfill Landweber exactness:
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| # R needs to be flat over <math>\mathbb{Z}</math>
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| # There is no irreducible component ''X'' of <math>\text{Spec }R/pR</math>, where the fiber <math>E_x</math> is [[supersingular]] for every <math>x\in X</math>
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| These conditions can be checked in many cases related to elliptic genera. Moreover, the conditions are fulfilled in the universal case in the sense that the map from the [[moduli stack]] of elliptic curves to the moduli stack of [[formal group]]s <math>\mathcal{M}_{1,1}\to\mathcal{M}_{fg}</math> is [[flat morphism|flat]]. This gives then a [[presheaf]] of cohomology theories over the site of affine [[scheme (algebraic geometry)|schemes]] flat over the moduli stack of elliptic curves. The desire to get a universal elliptic cohomology theory by taking global sections has led to the construction of the [[topological modular forms]].
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| ==References==
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| *{{Citation |last=Franke |first=Jens |year=1992 |title=On the construction of elliptic cohomology |journal=[[Mathematische Nachrichten]] |volume=158 |issue=1 |pages=43–65 |doi=10.1002/mana.19921580104 }}.
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| *{{Citation |last=Landweber |first=Peter S. |chapter=Elliptic genera: An introductory overview |editor1-first=P. S. |editor1-last=Landweber |title=Elliptic Curves and Modular Forms in Algebraic Topology |series=Lecture Notes in Mathematics |volume=1326 |location=Berlin |publisher=Springer |year=1988 |pages=1–10 |isbn=3-540-19490-8 }}.
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| *{{Citation |last=Landweber |first=Peter S. |chapter=Elliptic cohomology and modular forms |editor1-first=P. S. |editor1-last=Landweber |title=Elliptic Curves and Modular Forms in Algebraic Topology |series=Lecture Notes in Mathematics |volume=1326 |location=Berlin |publisher=Springer |year=1988 |pages=55–68 |isbn=3-540-19490-8 }}.
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| *{{Citation |last=Landweber |first=P. S. |last2=Ravenel |first2=D. |lastauthoramp=yes |last3=Stong |first3=R. |chapter=Periodic cohomology theories defined by elliptic curves |title=The Čech Centennial 1993 |location=Boston |year=1995 |series=Contemp. Math. |volume=181 |publisher=Amer. Math. Soc. |isbn=0-8218-0296-8 |pages=317–338 |editor1-last=Cenkl |editor1-first=M. |editor2-last=Miller |editor2-first=H. }}.
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| *{{Citation |last=Lurie |first=Jacob |year=2009 |chapter=A Survey of Elliptic Cohomology |title=Algebraic Topology: The Abel Symposium 2007 |editor1-last=Baas |editor1-first=Nils |editor2-last=Friedlander |editor2-first=Eric M. |editor3-last=Jahren |editor3-first=Björn |editor4-last=Østvæ |editor4-first=Paul Arne |location=Berlin |publisher=Springer |pages=219–277 |isbn=978-3-642-01199-3 |doi=10.1007/978-3-642-01200-6 }}.
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| [[Category:Algebraic topology]]
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| [[Category:Cohomology theories]]
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