  The Geometric Shafarevich Conjecture and the Theorem of de Franchis state the
finiteness of the number of certain holomorphic objects on closed or punctured
Riemann surfaces. The analog for Riemann surfaces of second kind is an estimate
of the number of irreducible holomorphic objects up to homotopy (or isotopy,
respectively). This analog addresses the problem of the validity of Gromov's
Oka principle.
  For any finite open Riemann surface $X$ (maybe, of second kind) we give an
effective upper bound for the number of irreducible holomorphic mappings up to
homomotopy from $X$ to the twice punctured complex plane, and an effective
upper bound for the number of irreducible holomorphic torus bundles up to
isotopy on such Riemann surface.
  If $X_{\sigma}$ is the $\sigma$-neighbourhood of a skeleton of a torus with a
hole, then the number of irreducible holomorphic mappings up to homomotopy from
$X_{\sigma}$ to the twice punctured complex plane grows exponentially in
$\frac{1}{\sigma}$. This can be interpreted as a quantitative statement of
Gromov's Oka principle.
