Average edge order of normal $3$-pseudomanifolds
DOI:
https://doi.org/10.55016/1y7q3e57Abstract
In their work, Feng Luo and Richard Stong introduced the concept of the average edge order, denoted as $\mu_0$. They demonstrated that if $\mu_0(K)\leq \frac{9}{2}$ for a closed triangulated $3$-manifold $K$, then $K$ must be a sphere. Building upon this foundation, Makoto Tamura extended similar results to compact triangulated $3$-manifolds with nonempty boundaries in \cite{Tamura1, Tamura2}. In our present study, we extend these findings to normal $3$-pseudomanifolds. Specifically, we establish that for a normal $3$-pseudomanifold $K$ with singularities, $\mu_0(K)\geq\frac{30}{7}$. 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 $\frac{30}{7}\leq\mu_0(K)\leq\frac{9}{2}$, 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.Downloads
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Copyright (c) 2026 Biplab Basak, Raju Kumar Gupta
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