Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants

We obtain constraints on the topology of families of smooth 4-manifolds arising from a finite-dimensional approximation of the families Seiberg-Witten monopole map. Amongst other results these constraints include a families generalisation of Donaldson's diagonalisation theorem and Furuta's 10/8 theorem. As an application we construct examples of continuous Z(p)-actions, for any odd prime p, which cannot be realised smoothly. As a second application we show that the inclusion of the group of diffeomorphisms into the group of homeomorphisms is not a weak homotopy equivalence for any compact, smooth, simply connected, indefinite 4-manifold with signature of absolute value greater than 8.