Semi-equivelar gems of PL d-manifolds

期刊论文
发表日期：

2024年3月11日

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Biplab Basak,
Manisha Binjola

摘要

We define the notion of $(p_0,p_1,\ldots ,p_d)$ $(p_{0}, p_{1}, \dots, p_{d})$ -type semi-equivelar gems for closed connected PL d-manifolds, related to the regular embedding of gems $\Gamma $ $Γ$ representing M on a surface S such that the face-cycles at all the vertices of $\Gamma $ $Γ$ on S are of the same type. The term is inspired by semi-equivelar maps of surfaces. Given a surface S having non-negative Euler characteristic, we find all regular embedding types on S and then construct a genus-minimal semi-equivelar gem (if it exists) of each such type embedded on S. Moreover, we present constructions of the following semi-equivelar gems: (1)

For each closed connected surface S, we construct a genus-minimal semi-equivelar gem that represents S. In particular, for $S=\#_n (\mathbb {S}^1 \times \mathbb {S}^1)$ $S = #_{n} (S^{1} \times S^{1})$ (resp., $\#_n(\mathbb{R}\mathbb{P}^2)$ $#_{n} (R P^{2})$ ), the semi-equivelar gem of type $((4n+2)^3)$ $({(4 n + 2)}^{3})$ (resp., $((2n+2)^3)$ $({(2 n + 2)}^{3})$ ) is constructed.

(2)

For a closed connected orientable PL d-manifold M (where $d \ge 3$ $d \geq 3$ ) of regular genus at most 1, we show that M admits a genus-minimal semi-equivelar gem if and only if M is a lens space.

Moreover, if we consider semi-equivelar gems with 2-gons then for a closed connected orientable d-manifold M (where $d \ge 3$

d \geq 3

) with $\mathcal {G}(M)\le 1$

G (M) \leq 1

, M admits a genus-minimal semi-equivelar gem (with bigons).