Math 2.0

Differential geometry Symplectic geometry Conjecture

Arnold conjecture

Builds on Symplectic manifoldHamiltonian vector fieldSymplectomorphismBetti numberMorse function

Statement

Let (M,ω)(M, \omega) be a closed symplectic manifold. Let H ⁣:[0,1]×M→ℝH \colon [0, 1] \times M \to \R be smooth, let XHtX_{H_t} be the Hamiltonian vector field of Ht=H(t,⋅)H_t = H(t, \cdot), and let ϕ ⁣:M→M\phi \colon M \to M be the time-11 map of the flow of XHtX_{H_t}. Such a symplectomorphism ϕ\phi is called a Hamiltonian diffeomorphism. Assume that every fixed point xx of ϕ\phi is nondegenerate: 11 is not an eigenvalue of dϕxd\phi_x. Then the number of fixed points of ϕ\phi is at least the minimal number of critical points of a Morse function on MM.

Remarks

The homological version of the conjecture replaces the right side by the sum of the Betti numbers ∑kdim⁡Hk(M;𝔽)\sum_k \dim H_k(M; \F) for a field 𝔽\F. By the Morse inequalities this sum is at most the number of critical points of any Morse function, so the homological version is weaker. A degenerate version, without the nondegeneracy assumption, asks for at least as many fixed points as the minimal number of critical points of an arbitrary smooth function.

For ϕ=id⁡\phi = \id generated by H=0H = 0 the hypothesis fails, and indeed every point is fixed. If HH does not depend on tt and is a C2C^2-small Morse function, the fixed points of ϕ\phi are exactly the critical points of HH. Taking HH with the minimal number of critical points shows that the bound is sharp.

Conley and Zehnder proved the conjecture for the standard torus 𝕋2n\mathbb{T}^{2n} in 1983. Floer proved the homological version for monotone symplectic manifolds by building Floer homology, whose chain complex is generated by the fixed points of ϕ\phi. The homological version with rational coefficients was proved for all closed symplectic manifolds in the late 1990s, by Fukaya and Ono and by Liu and Tian, among others. The version with the minimal number of critical points of a Morse function is open in general.

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Betti number, concept, no evidence levelBetti numberconceptHamiltonian vector field, concept, no evidence levelHamiltonian vector fi…conceptMorse function, concept, no evidence levelMorse functionconceptSymplectic manifold, concept, no evidence levelSymplectic manifoldconceptSymplectomorphism, concept, no evidence levelSymplectomorphismconceptArnold conjecture, conjecture, no evidence levelArnold conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Betti number concept
  2. Hamiltonian vector field concept
  3. Morse function concept
  4. Symplectic manifold concept
  5. Symplectomorphism concept
  6. Arnold conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Arnold_conjecture
About this page
TagHY7VRFBAP9
URIm2://HY7VRFBAP9
Kindconjecture
WritingW0 stub
MSC53D40, 37J11, 37J12
StatusThis copy of the site has no build data. Run m2 status HY7VRFBAP9.
AboutSymplectomorphism, Symplectic manifold
UsesSymplectic manifold, Hamiltonian vector field, Symplectomorphism, Betti number, Morse function
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