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 ${\displaystyle (M,\omega )}$  be a compact 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 identity:

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

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

Suppose there is a 1-parameter family of Hamiltonian functions ${\displaystyle H_{t}:M\to {\mathbb {R} },0\leq t\leq 1}$ , inducing 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 Hamiltonian diffeomorphism of ${\displaystyle M}$ .

The Arnold conjecture says that for each Hamiltonian diffeomorphism of ${\displaystyle M}$ , it possesses at least as many fixed points as a smooth function on ${\displaystyle M}$  possesses critical points.^[2]

Nondegenerate Hamiltonian and 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 ${\displaystyle {\mathbb {F} }}$ :

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

The weak Arnold conjecture says that for a nondegenerate Hamiltonian diffeomorphism on ${\displaystyle M}$  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

References

1. Asselle, L.; Izydorek, M.; Starostka, M. (2022). "The Arnold conjecture in ${\displaystyle \mathbb {C} \mathbb {P} ^{n}}$  and the Conley index". arXiv:2202.00422 [math.DS].
2. Buhovsky, Lev; Humilière, Vincent; Seyfaddini, Sobhan (2018-04-11). "A C⁰ counterexample to the Arnold conjecture". Inventiones Mathematicae. 213 (2). Springer Science and Business Media LLC: 759–809. arXiv:1609.09192. doi:10.1007/s00222-018-0797-x. ISSN 0020-9910. S2CID 46900145.