A Fake Cusp and a Fishtail

We construct smooth 4-manifolds that are homeomorphic but not diffeomorphic to the “cusp” and the “fishtail”, which are thickened singular 2-spheres. Even though many fake smoothings of 4-manifolds are known to exists, we know little about the basic building blocks of exotic smooth manifolds. This is mainly due to the fact that we still don’t know if basic manifolds like S, S ×S, and S ×S could admit fake smooth structures. Unable to show this, we demonstrate fake smooth structures on manifolds that are in a way “small deformations” of S ×B (as in [A1],[A2]). Fishtail F is the tubular neighborhood of an immersed 2-sphere in S with one self intersection; as an handlebody it can be described as a 4-ball with 1 and 2-handles attached as in first picture of Figure 1. Recalling the 1-handle notation of [A0], F is obtained by removing a tubular neighborhood of the obvious disc B ⊂ S × B (which the “circle with dot” bounds). Cusp C is a 4-ball with a 2-handle attached along a trefoil knot with 0-framing as shown in the second picture of Figure 1.