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 $10\mathrm{/}8$ theorem. As an application we construct examples of continuous ${\mathbb{Z}}_p$-actions for any odd prime $p$, which can not 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 $8$.