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 $\mathbb{Z}/2$-coefficients.

We prove that for all pairs of orientable, homeomorphic, smooth $4$-manifolds this invariant can be realised fully after stabilising with a single $S^2\times S^2$.

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.