Rigidity of the mod 2 families Seiberg-Witten invariants and topology of families of spin 4-manifolds

We show a rigidity theorem for the Seiberg-Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These non-smoothable topological families provide new examples of 4-manifolds M for which the inclusion maps Diff(M) hooked right arrow Homeo(M) are not weak homotopy equivalences. We shall also give a new series of non-smoothable topological actions on some spin 4-manifolds.