A 4-dimensional light bulb theorem for disks

Abstract

We give a 4-dimensional light bulb theorem for properly embedded disks, generalizing recent work of Gabai and Kosanovic-Teichner in certain contexts, and extending the 4-dimensional light bulb theorem for 2-spheres due to Gabai and Schneiderman-Teichner. In particular, we provide conditions under which homotopic disks properly embedded in a compact 4-manifold X with a common dual in the interior of X are smoothly isotopic rel boundary. We also provide a new geometric interpretation of the Dax invariant, to aid in its computation.