On four-manifolds without 1- and 3-handles

We note that infinitely many irreducible, closed, simply connected 4-manifolds, with prescribed signature and spin type, admit perfect Morse functions, i. e. they can be given handle decompositions without 1- and 3-handles. In particular, there are many such 4-manifolds homeomorphic but not diffeomorphic to the standard 4-manifolds # m (S² x S²) and # n (CP² # -CP²), respectively, which answers Problem 4. 91 on Kirby's 1997 list.