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

TL;DR

This paper shows that simply-connected PL 4-manifolds with simple crystallizations have a specific handlebody structure, and their regular genus and gem-complexity are directly related to their second Betti number, revealing additive properties.

Contribution

It establishes a link between simple crystallizations, handlebody decompositions, and PL invariants like regular genus and gem-complexity for simply-connected PL 4-manifolds.

Findings

Regular genus equals twice the second Betti number.

Gem-complexity is three times the second Betti number.

Both invariants are additive within the class of simple crystallizations.

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).