Light bulb smoothing for topological surfaces in 4-manifolds

Abstract

We present new smoothing techniques for topologically embedded surfaces in smooth $4$-manifolds, which give topological isotopy to a smooth surface. As applications, we prove “topological = smooth” results in dimension 4 for certain disks and spheres modulo isotopy. A key step in our approach is to link Quinn’s smoothing theory with ideas in Gabai’s 4-dimensional light bulb theorem and succeeding developments of Schneiderman–Teichner and Kosanović–Teichner. As another application of our smoothing technique, we obtain a topological version of the Dax invariant which gives topological isotopy obstructions for topological disks in $4$-manifolds.