Average edge order of normal $3$-pseudomanifolds

Abstract

In their work [10], Feng Luo and Richard Stong introduced the concept of the average edge order, denoted as $\hat{I}{¼}_0(K)$. They demonstrated that if $\hat{I}{¼}_0(K)\le \frac{9}{2}$ for a closed $3$-manifold $K$, then $K$ must be a sphere. Building upon this foundation, Makoto Tamura extended similar results to $3$-manifolds with non-empty boundaries in [12,13]. In our present study, we extend these findings to normal $3$-pseudomanifolds. Specifically, we establish that for a normal $3$-pseudomanifold $K$ with singularities, $\hat{I}{¼}_0(K)\ge \frac{30}{7}$. Moreover, equality holds if and only if $K$ is a one-vertex suspension of a triangulation of ${\mathbb{RP}}^2$ with seven vertices. Furthermore, we establish that when $\frac{30}{7}\le \hat{I}{¼}_0(K)\le \frac{9}{2}$, the $3$-pseudomanifold $K$ can be derived from some boundary complexes of $4$-simplices by a sequence of possible operations, including connected sums, bistellar $1$-moves, edge contractions, edge expansions, vertex folding, and edge folding.

10 pages. To appear in Contributions to Discrete Mathematics