Compact 4-manifolds admitting special handle decompositions

In this paper we study colored triangulations of compact PL $4$-manifolds with empty or connected boundary which induce handle decompositions lacking in 1-handles or in 1- and 3-handles, thus facing also the problem, posed by Kirby, of the existence of {em special handlebody decompositions} for any simply-connected closed PL $4$-manifold. In particular, we detect a class of compact simply-connected PL $4$-manifolds with empty or connected boundary, which admit such decompositions and, therefore, can be represented by (undotted) framed links. Moreover, this class includes any compact simply-connected PL $4$-manifold with empty or connected boundary having colored triangulations that minimize the combinatorially defined PL invariants {em regular genus, gem-complexity} or {em G-degree} among all such manifolds with the same second Betti number.