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]

Strong Arnold conjecture

Let ${\displaystyle (M,\omega )}$ be a closed (compact without boundary) symplectic manifold. For any smooth function ${\displaystyle H:M\to {\mathbb {R} }}$, the symplectic form ${\displaystyle \omega }$ induces a Hamiltonian vector field ${\displaystyle X_{H}}$ on ${\displaystyle M}$ defined by the formula

${\displaystyle \omega (X_{H},\cdot )=dH.}$

The function ${\displaystyle H}$ is called a Hamiltonian function.

Suppose there is a smooth 1-parameter family of Hamiltonian functions ${\displaystyle H_{t}\in C^{\infty }(M)}$, ${\displaystyle t\in [0,1]}$. This family induces a 1-parameter family of Hamiltonian vector fields ${\displaystyle X_{H_{t}}}$ on ${\displaystyle M}$. The family of vector fields integrates to a 1-parameter family of diffeomorphisms ${\displaystyle \varphi _{t}:M\to M}$. Each individual ${\displaystyle \varphi _{t}}$ is a called a Hamiltonian diffeomorphism of ${\displaystyle M}$.

The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of ${\displaystyle M}$ is greater than or equal to the number of critical points of a smooth function on ${\displaystyle M}$.^[2][3]

Weak Arnold conjecture

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

In view of the Morse inequality, the Morse number is also greater than or equal to a homological invariant of ${\displaystyle M}$, for example, the sum of Betti numbers over a field ${\displaystyle {\mathbb {F} }}$,

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

The weak Arnold conjecture says that the above integer is a lower bound on the number of fixed points of a nondegenerate Hamiltonian diffeomorphism of ${\displaystyle M}$.^[2][3] The weak Arnold conjecture for ${\displaystyle \mathbb {F} =\mathbb {Z} /2\mathbb {Z} }$ is a special case of the Arnold-Givental conjecture.

See also

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

References