Average order of coloured triangulations : the general case

In [Combinatorics of triangulations of 3-manifolds, Trans. Amer. Math. Soc. 337 (2) (1993), 891-906], Luo and Stong introduced the notion of "average edge order" \mu_0(K) = \frac {3 F_0(K)}{E_0(K), K being a triangulation of a closed 3-manifold M with $E_0(K)$ edges and $F_0(K)$ triangles. The present paper extends the above notion to the "average (n-2)-simplex order" of a coloured triangulation K of a compact PL n-manifold $M^n$ with $\alpha_i(K)$ i-simplices: \mu(K) = \frac {n \alpha_{n-1}(K)}{\alpha_{n-2}(K).Main properties of $\mu(K)$ and its relations with the topology of $M^n$, both in the closed and bounded case, are investigated; the obtained results show the existence of strong analogies with the 3-dimensional simplicial case (see the quoted paper by Luo and Stong, together with [The average edge order of triangulations of 3-manifolds, Osaka J. Math. 33(1986), 761-773] by Tamura)

A note on the characterization of handlebodies

The work is devoted to extend to dimension five the following combinatorial characterization of (orientable and non-orientable) handlebodies, already proved for dimensions three and four by the same author: a compact connected 5-manifold $M^5$ is a handlebody (of genus g) iff G(M^5)= G(\partial M^5} (=g), $G(X) being the regular genus of the manifold X. Moreover, partial results in dimension n induce to conjecture that an analogous characterization also holds for handlebodies of arbitrary dimension