Arnold conjecture
The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry.[1]
Statement
Let [math]\displaystyle{ (M, \omega) }[/math] be a compact symplectic manifold. For any smooth function [math]\displaystyle{ H: M \to {\mathbb R} }[/math], the symplectic form [math]\displaystyle{ \omega }[/math] induces a Hamiltonian vector field [math]\displaystyle{ X_H }[/math] on [math]\displaystyle{ M }[/math], defined by the identity:
[math]\displaystyle{ \omega( X_H, \cdot) = dH. }[/math]
The function [math]\displaystyle{ H }[/math] is called a Hamiltonian function.
Suppose there is a 1-parameter family of Hamiltonian functions [math]\displaystyle{ H_t: M \to {\mathbb R}, 0 \leq t \leq 1 }[/math], inducing a 1-parameter family of Hamiltonian vector fields [math]\displaystyle{ X_{H_t} }[/math] on [math]\displaystyle{ M }[/math]. The family of vector fields integrates to a 1-parameter family of diffeomorphisms [math]\displaystyle{ \varphi_t: M \to M }[/math]. Each individual [math]\displaystyle{ \varphi_t }[/math] is a Hamiltonian diffeomorphism of [math]\displaystyle{ M }[/math].
The Arnold conjecture says that for each Hamiltonian diffeomorphism of [math]\displaystyle{ M }[/math], it possesses at least as many fixed points as a smooth function on [math]\displaystyle{ M }[/math] possesses critical points.[2]
Nondegenerate Hamiltonian and weak Arnold conjecture
A Hamiltonian diffeomorphism [math]\displaystyle{ \varphi:M \to M }[/math] is called nondegenerate if its graph intersects the diagonal of [math]\displaystyle{ M\times M }[/math] transversely. For nondegenerate Hamiltonian diffeomorphisms, a variant of the Arnold conjecture says that the number of fixed points is at least equal to the minimal number of critical points of a Morse function on [math]\displaystyle{ M }[/math], called the Morse number of [math]\displaystyle{ M }[/math].
In view of the Morse inequality, the Morse number is also greater than or equal to a homological invariant of [math]\displaystyle{ M }[/math], for example, the sum of Betti numbers over a field [math]\displaystyle{ {\mathbb F} }[/math]:
[math]\displaystyle{ \sum_{i=0}^{2n} {\rm dim} H_i (M; {\mathbb F}). }[/math]
The weak Arnold conjecture says that for a nondegenerate Hamiltonian diffeomorphism on [math]\displaystyle{ M }[/math] the above integer is a lower bound of its number of fixed points.
See also
References
- ↑ Asselle, L.; Izydorek, M.; Starostka, M. (2022). "The Arnold conjecture in [math]\displaystyle{ \mathbb C\mathbb P^n }[/math] and the Conley index". arXiv:2202.00422 [math.DS].
- ↑ Buhovsky, Lev; Humilière, Vincent; Seyfaddini, Sobhan (2018-04-11). "A C0 counterexample to the Arnold conjecture". Inventiones Mathematicae (Springer Science and Business Media LLC) 213 (2): 759–809. doi:10.1007/s00222-018-0797-x. ISSN 0020-9910.
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