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 108 theorem. As an application we construct examples of continuous Zp–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.David Baragli