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.

Statement

Let $(M, \omega)$ be a compact symplectic manifold. For any smooth function $H: M \to$, the symplectic form $\omega$ induces a Hamiltonian vector field $X_H$ on $M$, defined by the identity

$\omega( X_H, \cdot) = dH.$

The function $H$ is called a Hamiltonian function.

Suppose there is a 1-parameter family of Hamiltonian functions $H_t: M \to , 0 \leq t \leq 1$, inducing a 1-parameter family of Hamiltonian vector fields $X_$ on $M$. The family of vector fields integrates to a 1-parameter family of diffeomorphisms $\varphi_t: M \to M$. Each individual of $\varphi_t$ is a Hamiltonian diffeomorphism of $M$.

The Arnold conjecture says that for each Hamiltonian diffeomorphism of $M$, it possesses at least as many fixed points as a smooth function on $M$ possesses critical points.

Nondegenerate Hamiltonian and weak Arnold conjecture

A Hamiltonian diffeomorphism $\varphi:M \to M$ is called nondegenerate if its graph intersects the diagonal of $M\times M$ 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 $M$, called the Morse number of $M$.

In view of the Morse inequality, the Morse number is also greater than or equal to a homological invariant of $M$, for example, the sum of Betti numbers over a field $$:

$\sum_^ H_i (M; ).$

The weak Arnold conjecture says that for a nondegenerate Hamiltonian diffeomorphism on $M$ the above integer is a lower bound of its number of fixed points.

References

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Symplectic geometry
Conjectures