Constraints on families of smooth 4-manifolds from Bauer-Furuta invariants (Q2664180)

Using finite-dimensional approximations to the Seiberg-Witten monopole maps, \textit{S. Bauer} and \textit{M. Furuta} [Invent. Math. 155, No. 1, 1--19 (2004; Zbl 1050.57024)] constructed invariants of smooth 4-manifolds. Among other advantages, this allowed us to bypass transversality considerations when generalising this construction to the equivariant [\textit{M. Szymik}, Q. J. Math. 63, No. 4, 1033--1054 (2012; Zbl 1275.57041)] and family settings [\textit{M. Szymik}, Forum Math. 22, No. 3, 509--523 (2010; Zbl 1200.57019)]. In this paper, the author builds on these ideas and obtains constraints on the topology of families of smooth 4-manifolds, see Theorems 1.1, 1.2, and 1.3 for detailed statements. As an application, the author constructs examples of odd-prime-order homeomorphisms that cannot be realised smoothly (see Theorem 1.7). He also shows that the inclusion of the group of diffeomorphisms into the group of homeomorphisms is not an equivalence for any indefinite, simply-connected 4-manifold with a sufficiently large signature (see Theorem 1.8).