  In this paper we prove results on the existence and homotopy classification
of holomorphic submersions from Stein manifolds to other complex manifolds. We
say that a complex manifold Y satisfies Property S_n for some integer n bigger
or equal the dimension of Y if every holomorphic submersion from a compact
convex set in C^n of a certain special type to Y can be uniformly approximated
by holomorphic submersions from C^n to Y. Assuming this condition we prove the
following. A continuous map f from an n-dimensional Stein manifold X to Y is
homotopic to a holomorphic submersions of X to Y if and only if there exists a
fiberwise surjective complex vector bundle map from TX to TY covering f. We
also prove results on the homotopy classification of holomorphic submersions.
We show that Property S_n is satisfied when n>dim Y and Y is any of the
following manifolds: a complex Euclidean space, a complex projective space or
Grassmanian, a Zariski open set in any of the above whose complement does not
contain any complex hypersurfaces, a complex torus, a Hopf manifold, a
non-hyperbolic Riemann surface, etc. In the case when Y is a complex Euclidean
space the main result of this paper was obtained in [arXiv:math.CV/0211112].
