We obtain constraints on the topology of families of smooth –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 theorem. As an application we construct examples of continuous –actions, for any odd prime , 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 –manifold with signature of absolute value greater than .
"Constraints on families of smooth $4$–manifolds from Bauer–Furuta invariants." Algebr. Geom. Topol. 21 (1) 317 - 349, 2021. https://doi.org/10.2140/agt.2021.21.317