On graph-theoretical invariants of combinatorial manifolds
The notion of \textit{average edge order} for a triangulation of a closed 3-manifolds has been introduced by \textit{F. Luo} and \textit{R. Stong} [Trans. Am. Math. Soc. 337, No. 2, 891--906 (1993; Zbl 0785.57007)] and extended to the boundary case by \textit{M. Tamura} [ibid. 350, No. 5, 2129--2140 (1998; Zbl 0892.57016)]; in both cases, interesting connections between its value and the topology of the manifold arose. Then, in [\textit{M. R. Casali}, Can. Math. Bull. 37, No. 2, 154--161 (1994; Zbl 0819.57003); Osaka J. Math. 35, No. 1, 249--262 (1998; Zbl 0906.57009)], the similar notion of \textit{average \((n-2)\)-simplex order} has been defined and studied in the \(n\)-dimensional context of colored triangulations of compact PL-manifolds \(M^n\) (\(n \ge 3\)), i.e. pseudocomplexes \(K\) triangulating \(M^n\), whose vertices are labelled by ``colors'' \(0,1,\dots,n\), so that the coloring is injective on each \(n\)-simplex of \(K\). The present paper continues the study of the average \((n-2)\)-simplex order for colored triangulations of closed PL \(n\)-manifolds, in dimension \(3 \le n \le 5\). In particular, in both cases \(n=3\) and \(n=4\), the authors obtain connections with the so called \textit{reduced complexity}; hence, new classifying results with respect to average \((n-2)\)-simplex order follow from existing catalogues and/or classifications of colored triangulation of PL \(n\)-manifolds via reduced complexity (see [\textit{S. Lins}, Gems, computers and attractors for 3-manifolds. Singapore: World Scientific (1995; Zbl 0868.57002); \textit{M. R. Casali} and \textit{P. Cristofori}, J. Knot Theory Ramifications 17, No. 5, 579--599 (2008; Zbl 1163.57017)] for \(n=3\), [\textit{A. Cavicchioli} and \textit{F. Spaggiari}, Bol. Soc. Mat. Mex., III. Ser. 14, No. 2, 303--319 (2008; Zbl 1222.57019)] and [\textit{M. R. Casali} and \textit{P. Cristofori}, Electron. J. Comb. 22, No. 4, Research Paper P4.25, 25 p. (2015; Zbl 1332.57020)] for \(n=4\)). A further result of the paper answers, in the special context of average 2-simplex order for 4-dimensional colored triangulations, to a problem suggested by Trout in 2013 for average order of \(n\)-manifold triangulations (see \url{http://mathoverflow.net/questions/103384}).
scientific article; zbMATH DE number 7079135