Secondary Stiefel-Whitney class and diffeomorphisms of rational and ruled symplectic 4-manifolds

We introduce the secondary Stiefel-Whitney class ${\tilde{w}}_2$ of homotopically trivial diffeomorphisms and show that a homotopically trivial symplectomorphism of a ruled 4-manifold is isotopic to identity if and only if the class ${\tilde{w}}_2$ vanishes. Using this, we give a detailed description of the combinatorial structure of the diffeotopy group of ruled symplectic 4-manifolds $X$, either minimal or blown-up, and its action on the homology and homotopy groups $H_2(X,\text{zz})$, ${\pi }_1(X)$, and ${\pi }_2(X)$.