PL 4-manifolds admitting simple crystallizations: framed links and regular genus

Abstract

Simple crystallizations are edge-colored graphs representing piecewise linear (PL) 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In this paper, we prove that any (simply-connected) PL 4-manifold [Formula: see text] admitting a simple crystallization admits a special handlebody decomposition, too; equivalently, [Formula: see text] may be represented by a framed link yielding [Formula: see text], with exactly [Formula: see text] components ([Formula: see text] being the second Betti number of [Formula: see text]). As a consequence, the regular genus of [Formula: see text] is proved to be the double of [Formula: see text]. Moreover, the characterization of any such PL 4-manifold by [Formula: see text], where [Formula: see text] is the gem-complexity of [Formula: see text] (i.e. the non-negative number [Formula: see text], [Formula: see text] being the minimum order of a crystallization of [Formula: see text]), implies that both PL invariants gem-complexity and regular genus turn out to be additive within the class of all PL 4-manifolds admitting simple crystallizations (in particular, within the class of all “standard” simply-connected PL 4-manifolds).