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

Abstract

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 $\frac{10}{8}$ theorem. As an application we construct examples of continuous ${\mathbb{Z}}_p$–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 $8$.