Average edge order of normal $3$-pseudomanifolds
Biplab Basak, Raju Kumar Gupta
TL;DR
This work extends the notion of average edge order $\mu_0(K)=\frac{3F(K)}{E(K)}$ from closed 3-manifolds to normal $3$-pseudomanifolds with singularities, deriving a tight lower bound $\mu_0(K)\ge\frac{30}{7}$ with equality exactly for the one-vertex suspension of $\mathbb{RP}^2$ on seven vertices. It further shows that whenever $\frac{30}{7}\le\mu_0(K)\le\frac{9}{2}$, the triangulation $K$ can be obtained from boundary complexes of $4$-simplices via a sequence of operations (connected sums, bistellar moves, edge contractions/expansions, vertex folding, edge folding), yielding either a handlebody with boundary coned off or a suspension of $\mathbb{RP}^2$. The authors also establish that $\mu_0(K)<6+n$ where $n$ is the number of singular vertices, and construct families $K_m$ with $\mu_0(K_m)\to 8$, illustrating upper-bound behavior; they discuss examples with larger $\mu_0$ arising from RP$^2$-link structures and neighborly configurations. Overall, the paper links low $\mu_0$ to simple topologies and provides constructive limits and open questions for neighborly normal $3$-pseudomanifolds.
Abstract
In their work [10], Feng Luo and Richard Stong introduced the concept of the average edge order, denoted as $μ_0(K)$. They demonstrated that if $μ_0(K)\leq \frac{9}{2}$ 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, $μ_0(K)\geq\frac{30}{7}$. 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 $\frac{30}{7}\leqμ_0(K)\leq\frac{9}{2}$, 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.