  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 (S^2 x S^2) and # n (CP^2 #
-CP^2), respectively, which answers Problem 4.91 on Kirby's 1997 list.
