2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/149821This 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 $κ$ such that a positive multiple of $κ$ 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.19 pages, AMS-TeXAlgebraic Geometrytext