Suppose that $X$ is 4-dimensional handlebody (meaning a union of 4-dimensional 0 and 1-handles) and $S\subset \partial X$ is an embedded 2-sphere.
The author wants to prove that $S$ bounds an embedded ball in $X$.
The proof is as follows: if $S$ does not bound a 3-ball then it is an essential sphere, which can be separating (then surgering $\partial X$ along $S$ produces two components both of which are a connected sums of $S^1\times S^2$) or not (then surgering along $ S$ reduces by 1 the genus of $\partial X$). In both cases we can attach a 3-handle along $S$ and then continue by attaching 3-handles along spheres contained in $\partial X$ and finally a single 4-handle to obtain a 4-dimensional handlebody $Y$. By construction $S$ bounds a 3-ball in $Y$.
If i understood correctly the conclusion follows from the fact that this process of attaching 3-handles and a 4-handle is unique, so $X=Y$. Is this true? The closest thing I know is that attaching 3-handles and a 4-handle to a union of 0,1,2-handles is unique IF the resulting 4-manifold is closed.