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 $H:M\to {\mathbb {R} }$, 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 $H_{t}:M\to {\mathbb {R} },0\leq t\leq 1$, inducing a 1-parameter family of Hamiltonian vector fields $X_{H_{t}}$ on $M$. The family of vector fields integrates to a 1-parameter family of diffeomorphisms $\varphi _{t}:M\to M$. Each individual $\varphi _{t}$ 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 ${\mathbb {F} }$:

$\sum _{i=0}^{2n}{\rm {dim}}H_{i}(M;{\mathbb {F} }).$

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.

See also

• Arnold–Givental conjecture
• Symplectomorphism#Arnold conjecture
• Floer homology
• Spectral invariants
• Conley–Zehnder theorem