Characteristic cohomotopy classes for families of 4-manifolds

Families of smooth closed oriented 4-manifolds with a complex spin structure are studied by means of a family version of the Bauer–Furuta invariants in the context of parametrised stable homotopy theory, leading to a definition of characteristic cohomotopy classes on Thom spectra associated to the classifying spaces of their complex spin diffeomorphism groups. This is illustrated with mapping tori of such diffeomorphisms and related to the equivariant invariants.