There used to be many candidates for an exotic 4-sphere, but a lot of them are now known to be the standard smooth $S^4$. The ones of Cappell-Shaneson (maybe not all of them?) were described in terms of handlebody decompositions where there are no 3-handles, but I think the proofs that these were diffeomorphic to $S^4$ involved altering the decompositions with a cancellation trick by introducing some 3-handles.

I am not sure what is left over, but I seek the following:

**1) Is there a potentially exotic 4-sphere which is prescribed by an** explicit **4-handlebody decomposition with only 0,1,2-handles and a single 4-handle?**

**2) Of the** standard **Cappell-Shanelson spheres with a given handlebody decomposition involving no 3-handles, is it known that we can prove** [edit: some of them] **they're standard without the trick involving 3-handles?**

The reason I ask is because I'd like to study certain differential 2-forms on these spheres which "play well" with the handles, and to understand the deformation of these 2-forms as we change the handlebody decompositions.