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

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 M admitting a simple crystallization admits a special handlebody decomposition, too; equivalently, M may be represented by a framed link yielding S^3, with exactly β_2(M) components (β_2(M) being the second Betti number of M). As a consequence, the regular genus of M is proved to be the double of β_2(M). Moreover, the characterization of any such PL 4-manifold by k(M)=3β_2(M), where k(M) is the gem-complexity of M (i.e. the non-negative number p−1, 2p being the minimum order of a crystallization of M), 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).