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 
be a compact symplectic manifold. For any smooth function 
, the symplectic form 
induces a Hamiltonian vector field 
on 
, defined by the identity:
The function 
is called a Hamiltonian function.
Suppose there is a 1-parameter family of Hamiltonian functions 
, inducing a 1-parameter family of Hamiltonian vector fields 
on . The family of vector fields integrates to a 1-parameter family of diffeomorphisms 
. Each individual 
is a Hamiltonian diffeomorphism of .
The Arnold conjecture says that for each Hamiltonian diffeomorphism of , it possesses at least as many fixed points as a smooth function on possesses critical points.
Nondegenerate Hamiltonian and weak Arnold conjecture
A Hamiltonian diffeomorphism 
is called nondegenerate if its graph intersects the diagonal of 
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 , called the Morse number of .
In view of the Morse inequality, the Morse number is also greater than or equal to a homological invariant of , for example, the sum of Betti numbers over a field 
:
The weak Arnold conjecture says that for a nondegenerate Hamiltonian diffeomorphism on the above integer is a lower bound of its number of fixed points.
See also