Many manifolds can be obtained from gluing the boundary of a ball. For example, $\mathbb{RP}(2)$ is obtained from gluing the two edges of a bi-gon (2-ball). Or, lens spaces are obtained from a 3-cell whose boundary consists of two $p$-gons, which are then glued under a rotation by $q$ edges.

Question: Is there a similar construction for the complex projective plane $\mathbb{CP}(2)$? Ideally, is there a way to divide the boundary of the 4-ball into two identical 3-cells, which are then glued under some symmetry? Or is there at least such a gluing construction with $4$ or more 3-cells?