Smooth stable isotopy of topologically isotopic surfaces

Abstract

A stabilisation of a $4$-manifold $X$ is the connected sum of $X$ with some number of copies of $S^2\times S^2$. If two smooth surfaces in a $4$-manifold are topologically isotopic, we investigate whether they must moreover be smoothly isotopic in some stabilisation of $X$. We prove this result holds whenever the surfaces are trivial in the $\mathbb{Z}/2$-homology of $X$. We also produce a large class of fundamental groups of the ambient $4$-manifold for which the result holds; this class includes free products of classical knot groups and, in particular, free groups.