Lerman's symplectic cut construction applied on 4-ball by collapsing its boundary 3-sphere along the $\mathbb{S}^1$ orbits of Hopf fibration gives a closed 4-dimensional symplectic manifold. Topologically, this manifold is the cone of the Hopf fibration. Is there some other (topological) description of it?



What is symplectic cut of a 4-ball?
The answer is actually written in p.249 of Lerman paper Symplectic cuts. It says that, more generally, symplectic cut of 2n-ball along its boundary is symplectomorphic to $\mathbb{C}P^n.$ In particular, here we get $\mathbb{C}P^2.$


