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

Abstract

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).

14 pages, no figures; this is a new version of the former paper "A characterization of PL 4-manifolds admitting simple crystallizations"

References in corpus (1)

• Simple crystallizations of 4-manifolds

Cited by in corpus (8)

• Topology in colored tensor models via crystallization theory
• The full Ward-Takahashi Identity for colored tensor models
• Simple crystallizations of 4-manifolds
• Lower bounds for regular genus and gem-complexity of PL 4-manifolds
• Combinatorial properties of the G-degree
• Kirby diagrams and 5-colored graphs representing compact 4-manifolds
• Crystallizations of compact 4-manifolds minimizing combinatorially defined PL-invariants
• Cataloguing PL 4-manifolds by gem-complexity