  This paper is the first in a series of three devoted to the smooth
classification of simply connected elliptic surfaces. The method is to compute
some coefficients of Donaldson polynomials of $SO(3)$ invariants whose second
Stiefel-Whitney class is transverse to the unique primitive class $\kappa$ such
that a positive multiple of $\kappa$ is the class of a general fiber on the
surface. In this paper, we collect preliminary results on elliptic surfaces and
vector bundles and give the general outline of the argument.
