论文标题
pl $ d $ -manifolds的半e equivelar宝石
Semi-equivelar gems of PL $d$-manifolds
论文作者
论文摘要
我们定义了$（p_0，p_1，\ dots，p_d）$ - 键入封闭连接的pl $ d $ - manifolds的type semi-equivelar宝石，与代表$ s $ s $ s $ s $ fertices cycycys fertices $γ$ quam $γ$的$ s $ s y ny y yes s y n yes s facecycles的$ quam $γ$的定期嵌入有关的$ d $ manifolds。该术语的灵感来自表面的半等级图。鉴于具有非负欧拉特征的表面$ s $，我们在$ s $上找到了所有常规嵌入类型，然后构建一种嵌入$ s $的每种类型的属属半e e eekeivelar Gem（如果存在）。此外，我们介绍了以下半等级宝石的结构： （1）对于每个封闭的连接表面$ s $，我们构建了一个代表$ s $的属的半e e e e e e夫属。特别是，对于$ s = \ #_ n（\ mathbb {s}^1 \ times \ times \ mathbb {s}^1）$（$ \ #_ n（\＃__ n（\ sathbb {rp}^2）$），半equivelar gem of类型$ $（4n+2）^3）$（questect $（soundited）$（questeed）$（2n）。 （2）对于封闭的连接定向PL $ D $ -Manifold $ m $（其中$ d \ geq 3 $）的常规属最多$ 1 $，我们表明$ m $允许当$ m $是镜头空间时，只有$ m $，只有$ m $。 此外，如果我们考虑使用$ 2 $ - g的半eKeeivelar宝石，那么对于封闭的连接定向$ d $ -d $ -manifold $ m $（其中$ d \ geq 3 $），带有$ \ mathcal {g}（g}（m）\ leq 1 $，$ m $，$ m $，允许属属属属于Minimal semimimal semimimal Equivelar Gem（with bigons）。
We define the notion of $(p_0,p_1,\dots,p_d)$-type semi-equivelar gems for closed connected PL $d$-manifolds, related to the regular embedding of gems $Γ$ representing $M$ on a surface $S$ such that the face-cycles at all the vertices of $Γ$ 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 $S=\#_n (\mathbb{S}^1 \times \mathbb{S}^1)$ (resp., $\#_n(\mathbb{RP}^2)$), the semi-equivelar gem of type $((4n+2)^3)$ (resp., $((2n+2)^3)$) is constructed. (2) For a closed connected orientable PL $d$-manifold $M$ (where $d \geq 3$) 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 $d \geq 3$) with $\mathcal{G}(M)\leq 1$, $M$ admits a genus-minimal semi-equivelar gem (with bigons).
