I am aware of the classical work by Smale and Barden computing the diffemorsphism type of smooth simply connected 5-manifolds in Barden. Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365–385 and S. Smale, On the structure of 5-manifolds, Ann. of Math. (2) 75 (1962), 38–46.

¿Is there been any effort to give invariants for 5-dimensional smooth manifolds fibering over a fixed hyperbolic surface?