  In this paper the Gromov-Witten invariants on a class of noncompact
symplectic manifolds are defined by combining Ruan-Tian's method with that of
McDuff-Salamon. The main point of the arguments is to introduce a method
dealing with the transversality problems in the case of noncompact manifolds.
Moreover, the techniques are also used to study the topological rigidity of
Hamiltonian loops with compact support on a class of noncompact symplectic
manifolds.
