Arnold conjecture
E651228
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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Arnold conjecture canonical | 3 |
| Arnold conjecture for aspherical symplectic manifolds | 1 |
| Arnold conjecture for complex projective spaces | 1 |
| Arnold conjecture for tori | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7251366 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Arnold conjecture Context triple: [Poincaré–Birkhoff fixed-point theorem, relatedTo, Arnold conjecture]
-
A.
Introduction to Symplectic Topology
Introduction to Symplectic Topology is a foundational graduate-level textbook that systematically develops the theory and applications of symplectic manifolds and symplectic geometry.
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B.
Morse Theory
Morse Theory is a branch of differential topology that studies the relationship between the topology of manifolds and the critical points of smooth real-valued functions defined on them.
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C.
McDuff–Salamon theory of J-holomorphic curves
The McDuff–Salamon theory of J-holomorphic curves is a foundational framework in symplectic geometry that systematically develops the analysis, topology, and applications of pseudoholomorphic curves in symplectic manifolds.
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D.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
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E.
Lefschetz fixed-point theorem
The Lefschetz fixed-point theorem is a fundamental result in algebraic topology that relates the number of fixed points of a continuous map on a topological space to traces of the induced maps on its homology groups.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Arnold conjecture Target entity description: 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.
-
A.
Introduction to Symplectic Topology
Introduction to Symplectic Topology is a foundational graduate-level textbook that systematically develops the theory and applications of symplectic manifolds and symplectic geometry.
-
B.
Morse Theory
Morse Theory is a branch of differential topology that studies the relationship between the topology of manifolds and the critical points of smooth real-valued functions defined on them.
-
C.
McDuff–Salamon theory of J-holomorphic curves
The McDuff–Salamon theory of J-holomorphic curves is a foundational framework in symplectic geometry that systematically develops the analysis, topology, and applications of pseudoholomorphic curves in symplectic manifolds.
-
D.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
-
E.
Lefschetz fixed-point theorem
The Lefschetz fixed-point theorem is a fundamental result in algebraic topology that relates the number of fixed points of a continuous map on a topological space to traces of the induced maps on its homology groups.
- F. None of above. chosen
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
ⓘ
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
ⓘ
linked to:
Arnold conjecture
Arnold conjecture for complex projective spaces ⓘ
linked to:
Arnold conjecture
Arnold conjecture for tori ⓘ
linked to:
Arnold conjecture
|
| implies | existence of many fixed points for Hamiltonian diffeomorphisms ⓘ |
| influenced |
Hamiltonian dynamics
ⓘ
low-dimensional topology ⓘ modern symplectic topology ⓘ |
| isCentralTo |
Hamiltonian Floer theory
ⓘ
linked to:
Floer theory
|
| knownFor | being a central problem in symplectic topology ⓘ |
| motivatedDevelopmentOf |
Floer homology
ⓘ
linked to:
Floer theory
symplectic fixed point theory ⓘ |
| namedAfter |
Vladimir Igorevich Arnold
ⓘ
linked to:
Vladimir Arnold
|
| partiallyProvedBy |
Andreas Floer
ⓘ
Dusa McDuff ⓘ Helmut Hofer ⓘ Leonid Polterovich ⓘ |
| predicts | lower bound on the number of fixed points of Hamiltonian diffeomorphisms ⓘ |
| proposedBy |
Vladimir Igorevich Arnold
ⓘ
linked to:
Vladimir Arnold
|
| relatedTo |
Conley conjecture
ⓘ
Weinstein conjecture ⓘ
linked to:
Gromov’s non-squeezing theorem
|
| 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
ⓘ
Hamiltonian Floer homology ⓘ
linked to:
Floer theory
Morse theory on loop spaces ⓘ fixed point theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Arnold conjecture Description of subject: 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.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Arnold conjecture
linked to: Arnold conjecture
linked to: Arnold conjecture