Stein manifolds on which the strong Poincaré problem can be solved

Abstract:

Let M be a Stein manifold. Suppose every meromorphic function on M may be written as the quotient of two holomorphic functions which are pointwise relatively prime at every point of M. Then it will be shown that ${H^2}(M,Z) = 0$. Thus, the solvability of the Strong Poincaré Problem is equivalent to the vanishing of the second integral cohomology, which in turn is equivalent to the solvability of Cousin II, all on a Stein manifold M. This closes a gap in the classically known theory.