ABSTRACT
The purpose of this note is to describe some recent results concerning certain closed, smooth 4-manifolds. The basic ingredients needed in order to apply Donaldson’s invariant in our situation are the calculation of certain moduli spaces of stable holomorphic vector bundles over the algebraic surfaces in question, and an analysis of the geometry of certain reflection groups associated to quadratic forms of hyperbolic type. Analogously, the group of self-diffeomorphisms of such a manifold maps onto a subgroup of finite index in the group of integral automorphisms of the cohomology ring which preserve the Pontrjagin classes. The same techniques may be applied to study the group of self-diffeomorphisms of these elliptic surfaces.
