Arnold conjecture

E651228

mathematical conjecture statement in symplectic geometry

The Arnold conjecture is a central statement in symplectic geometry predicting a lower bound on the number of fixed points of Hamiltonian diffeomorphisms in terms of the topology of the underlying manifold.

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Statements (47)

Predicate	Object
instanceOf	mathematical conjecture ⓘ statement in symplectic geometry ⓘ
appliesTo	closed symplectic manifolds ⓘ compact symplectic manifolds ⓘ
assumes	nondegenerate Hamiltonian fixed points in some formulations ⓘ
concerns	Hamiltonian diffeomorphisms ⓘ fixed points of Hamiltonian diffeomorphisms ⓘ symplectic manifolds ⓘ topology of symplectic manifolds ⓘ
connectedTo	Gromov’s theory of pseudo-holomorphic curves NERFINISHED ⓘ Morse inequalities on loop spaces ⓘ
era	late 20th century ⓘ
field	Hamiltonian dynamics ⓘ symplectic geometry ⓘ
formalismUses	action functional on loop space ⓘ gradient flow lines interpreted as pseudo-holomorphic curves ⓘ
givesLowerBoundInTermsOf	Lusternik–Schnirelmann category in some variants ⓘ sum of Betti numbers of the manifold ⓘ
hasConsequence	existence of periodic orbits for Hamiltonian systems on symplectic manifolds ⓘ
hasVersion	Arnold conjecture for aspherical symplectic manifolds NERFINISHED ⓘ Arnold conjecture for complex projective spaces NERFINISHED ⓘ Arnold conjecture for tori NERFINISHED ⓘ
implies	existence of many fixed points for Hamiltonian diffeomorphisms ⓘ
influenced	Hamiltonian dynamics ⓘ low-dimensional topology ⓘ modern symplectic topology ⓘ
isCentralTo	Hamiltonian Floer theory NERFINISHED ⓘ
knownFor	being a central problem in symplectic topology ⓘ
motivatedDevelopmentOf	Floer homology NERFINISHED ⓘ symplectic fixed point theory ⓘ
namedAfter	Vladimir Igorevich Arnold NERFINISHED ⓘ
partiallyProvedBy	Andreas Floer NERFINISHED ⓘ Dusa McDuff NERFINISHED ⓘ Helmut Hofer NERFINISHED ⓘ Leonid Polterovich NERFINISHED ⓘ
predicts	lower bound on the number of fixed points of Hamiltonian diffeomorphisms ⓘ
proposedBy	Vladimir Igorevich Arnold NERFINISHED ⓘ
relatedTo	Conley conjecture NERFINISHED ⓘ Weinstein conjecture NERFINISHED ⓘ
relates	Betti numbers of the underlying manifold ⓘ homology of the underlying manifold ⓘ number of fixed points of Hamiltonian diffeomorphisms ⓘ
status	open in full generality ⓘ
topic	Floer theory NERFINISHED ⓘ Hamiltonian Floer homology NERFINISHED ⓘ Morse theory on loop spaces ⓘ fixed point theory ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Vladimir Arnold → knownFor → Arnold conjecture ⓘ

Poincaré–Birkhoff fixed-point theorem → relatedTo → Arnold conjecture ⓘ