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

Simple crystallizations are edge-coloured graphs representing PL 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In the present 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 \(\mathbb S^3\), with exactly \(\beta_2(M)\) components (\(\beta_2(M)\) being the second Betti number of \(M\)). As a consequence, the regular genus of \(M\) is proved to be the double of \(\beta_2(M)\). Moreover, the characterization of any such PL \(4\)-manifold by \(k(M)= 3 \beta_2(M)\), 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).