PL 4-manifolds admitting simple crystallizations: framed links and regular genus (Q2788557)

The number of \(i\)-faces in the associated triangulation of a crystallization of a PL \(d\)-manifold is at least \(\binom{d+1}{i+1}\). If the number of 1-faces is \(\binom{d+1}{2}\) then the crystallization is called \textit{simple}. The \textit{gem-complexity} of a PL manifold \(M\) is the number \(k(M) := p - 1\), where \(2p\) is the minimum order of a crystallization of \(M\). The regular genus \(\rho(\Gamma)\) of a bipartite graph \(\Gamma\) is the least genus of the orientable surface into which \(\Gamma\) embeds regularly. The \textit{regular genus} of an orientable PL manifold \(M\) is the number \({\mathcal G}(M) := \min\{\rho(\Gamma) : (\Gamma; \gamma)\) is a crystallization of \(M\}\).NEWLINENEWLINEAny PL manifold admitting simple crystallizations is simply-connected. In [Adv. Geom. 16, No. 1, 111--130 (2016; Zbl 1346.57022)], \textit{B. Basak} and \textit{J. Spreer} have shown that any ``standard'' simply-connected PL 4-manifold (i.e., \(S^4\), \(\mathbb{CP}^2\), \(S^2 \times S^2\) and the \(K3\)-surface, together with their connected sums) admits simple crystallizations. For a manifold \(M\), let \(\beta_2(M)\) be the second Betti number of \(M\). In this article, the authors have proved the following two main results. (i) If a PL 4-manifold \(M\) admits a simple crystallization then \({\mathcal G}(M) = 2 \beta_2(M)\). (ii) A closed simply-connected PL 4-manifold \(M\) admits a simple crystallization if and only if \(k(M) = 3\beta_2(M)\). To prove these results, the authors show that any PL 4-manifold admitting a simple crystallization admits a special handlebody decomposition.