Semi-equivelar gems of PL $d$-manifolds

Abstract

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 $\Gamma$ representing $M$ on a surface $S$ such that the face-cycles at all the vertices of $\Gamma$ 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).