On the diffeomorphism groups of rational and ruled 4-manifolds (Q2467635)

On a smooth 4-manifold \(M\), each diffeomorphism induces an automorphism of the lattice of the second integral cohomology. Hence there is a natural map from the group of diffeomorphisms \(\text{Diff}(M)\) to the automorphism group of the lattice \(A(M)\). Let \(D(M)\) be the image of this natural map. Let \(M=\mathbb CP^2\sharp n\overline{\mathbb CP^2}\). If \(n\leq 9\), there is the classical result of Wall that \(D(M)\) coincides with \(A(M)\). While for \(n>9\), Friedman and Morgan showed that \(D(M)\) is a subgroup of \(A(M)\) with infinite index. However, while there is neither description of a generating set of \(D(M)\), nor the coset space when \(n>9\). In this paper authors give explicit generators of \(D(M)\) for rational and ruled 4-manifolds. The authors also prove the uniqueness of reduced forms for classes with minimal genus 0 and non-negative square.