Average edge order of normal $3$-pseudomanifolds

TL;DR

This paper extends the concept of average edge order to normal 3-pseudomanifolds, establishing bounds and characterizing structures based on this invariant, thus generalizing previous results from 3-manifolds.

Contribution

It introduces bounds for the average edge order in normal 3-pseudomanifolds and characterizes the structures achieving these bounds, extending prior work on 3-manifolds.

Findings

For a normal 3-pseudomanifold with singularities, μ₀(K) ≥ 30/7.

Equality μ₀(K) = 30/7 holds iff K is a one-vertex suspension of a triangulation of RP².

When 30/7 ≤ μ₀(K) ≤ 9/2, K can be obtained from boundary complexes of 4-simplices via specific operations.

Abstract

In their work [10], Feng Luo and Richard Stong introduced the concept of the average edge order, denoted as $\mu_0(K)$. They demonstrated that if $\mu_0(K)\leq \frac{9}{2}$ 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, $\mu_0(K)\geq\frac{30}{7}$. 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.