Consider a complex surface given by homogeneous equation in $\mathbb{C}P^3$. Without loss of generality, take
\begin{equation} S = \{[x:y:z:w] \in \mathbb{C}P^3~ |~ x^d + y^d + z^d + w^d = 0\} \end{equation}
It is well known this surface is simply connected. We may compute the intersection form depending on whether $d$ is even or odd. My question is: is this surface admits handle decomposition with 2-handles only? If it is not, why?
More generally, given intersection form $Q$, if the 4-manifold is 2-handlebody, then the intersection form is represented by linking number of a link in $S^3$. However, given an intersection form and knowing that the 4-manifold is simply connected, can we deduce its handle decomposition?