The Casson-Sullivan invariant for homeomorphisms of 4-manifolds

We investigate the realisability of the Casson-Sullivan invariant for homeomorphisms of smooth 4-manifolds, which is the obstruction to a homeomorphism being stably pseudo-isotopic to a diffeomorphism, valued in the third cohomology of the source manifold with Z/2-coefficients. We prove that for all orientable pairs of homeomorphic, smooth 4-manifolds this invariant can be realised fully after stabilising with a single S² S². As an application, we obtain that topologically isotopic surfaces in a smooth, simply-connected 4-manifold become smoothly isotopic after sufficient external stabilisations. We further demonstrate cases where this invariant can be realised fully without stabilisation for self-homeomorphisms, which includes for manifolds with finite cyclic fundamental group. This method allows us to produce many examples of homeomorphisms which are not stably pseudo-isotopic to any diffeomorphism but are homotopic to the identity. Finally, we reinterpret these results in terms of finding examples of smooth structures on 4-manifolds which are diffeomorphic but not stably pseudo-isotopic.