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October 9, 2025Open Access

Light bulb smoothing for topological surfaces in 4-manifolds

JCJae ChoonPohang University of Science and Technology BKByeorhi KimPohang University of Science and Technology

Key Points

New smoothing techniques establish a link between topological and smooth surfaces in 4-manifolds, revealing deeper geometric connections.
These techniques support a significant result, showing that certain disks and spheres are topologically equivalent to smooth counterparts.
Utilizing Quinn's smoothing theory and light bulb theorem, this approach enriches the understanding of isotopy in 4-dimensional surfaces.
Applications extend to the Dax invariant, offering fresh insights into topological isotopy obstructions within the realm of 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.

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