Minimal crystallizations of 3-manifolds with boundary (Q2094274)

A well-established theory exists, making use of a special type of edge-colored graphs, called \textit{crystallizations}, in order to combinatorially represent and study PL-manifolds in arbitrary dimension, with or without boundary (see [\textit{M. Ferri} et al., Aequationes Math. 31, 121--141 (1986; Zbl 0623.57012); \textit{M. R. Casali} et al., Atti Semin. Mat. Fis. Univ. Modena 49, 283--337 (2001; Zbl 1420.57066)] and references herein). Within this theory, the PL invariants \textit{gem-complexity} \(k(M)\) and \textit{regular genus} \(\mathcal G(M)\) have been defined for each \(n\)-manifold \(M\): they are respectively related to the minimal order of a crystallization of \(M\) and to the minimal genus of a surface in which a crystallization of \(M\) regularly embeds (see for example [\textit{C. Gagliardi}, Proc. Am. Math. Soc. 81, 473--481 (1981; Zbl 0467.57004); \textit{M. R. Casali} and \textit{P. Cristofori}, Electron. J. Comb. 22, No. 4, Research Paper P4.25, 25 p. (2015; Zbl 1332.57020)]). The present paper assumes \(M\) to be a 3-dimensional manifold with boundary, and establishes the following upper bounds for the gem-complexity of \(M\), in terms of the regular genus of \(M\), of the regular genus of its boundary \(\partial M\) and of the number \(h\) of boundary components: \[ k(M) \ge 3 (\mathcal G (M) + h-1) \ge 3 (\mathcal G (\partial M) + h-1) \]