On four-manifolds without $1$– and $3$–handles

Abstract:

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 $\#_m (S^2 \times S^2)$ and $\#_n ({\mathbb {CP}}{}^{2}\# \overline {\mathbb {CP}}{}^{2})$, respectively, which answers Problem 4.91 on Kirby’s 1997 list.