Average edge order of normal $3$-pseudomanifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912976770957312 |
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| author | Basak, Biplab Gupta, Raju Kumar |
| author_facet | Basak, Biplab Gupta, Raju Kumar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14010 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Average edge order of normal $3$-pseudomanifolds Basak, Biplab Gupta, Raju Kumar Combinatorics Geometric Topology 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. |
| title | Average edge order of normal $3$-pseudomanifolds |
| topic | Combinatorics Geometric Topology |
| url | https://arxiv.org/abs/2406.14010 |