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

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