The Casson-Sullivan invariant for homeomorphisms of 4-manifolds

TL;DR

This paper studies the Casson-Sullivan invariant in 4-manifold homeomorphisms, showing how it can be realized through stabilization and applying this to isotopy problems and smooth structures.

Contribution

It proves the full realization of the Casson-Sullivan invariant after stabilization and explores its implications for isotopy and smooth structures on 4-manifolds.

Findings

Invariant can be realized after stabilizing with S^2×S^2

Topologically isotopic surfaces become smoothly isotopic after stabilization

Examples of homeomorphisms not pseudo-isotopic to any diffeomorphism

Abstract

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 orientable pairs of 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.