Geometric simple connectivity in dimension four

Every simply connected closed smooth manifold of dimension other than four admits a handle decomposition without 1-handles; this follows from the celebrated work of Smale in higher dimensions and of Perelman in dimension three. Whether the same holds in dimension four remains a wide open question. I’ll survey the state of the art and explain our recent (partial) progress, which includes settling a 30-year-old problem from Kirby’s problem list.