Semi-equivelar gems of PL $d$-manifolds

TL;DR

This paper defines semi-equivelar gems as regular embeddings of $d+1$-colored graphs representing closed PL $d$-manifolds, and analyzes their existence via regular embeddings on surfaces. It classifies regular embedding types on surfaces with $\chi(S) \ge 0$, providing genus-minimal semi-equivelar gems for each type and explicit constructions for connected sums of tori and projective planes. For orientable $d\ge 3$ with regular genus at most $1$, it proves that such a manifold admits a genus-minimal semi-equivelar gem if and only if it is a lens space, and extends the framework to semi-equivelar gems with bigons, showing existence results for manifolds with $\mathcal{G}(M) \le 1$. The results bridge crystallization theory with semi-equivelar map theory, delivering concrete constructions and obstructions for representing manifolds via gems and posing open questions about the broader existence of semi-equivelar gems.

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 $Γ$ 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).

Figures (6)

• Figure 1: A semi-equivelar gem of $(4^4)$-type embedded regularly on $\mathbb{S}^1 \times \mathbb{S}^1$, where the addition in the subscript of $u^1_{i}$ is of modulo $2s$, $1 \leq i \leq 2s$ and $0\leq r \leq s-1$.
• Figure 2: $(a)$ Embedding on $\#_n (\mathbb{S}^1\times \mathbb{S}^1)$ of gem representing $\#_n (\mathbb{S}^1\times \mathbb{S}^1)$ of type $((4n+2)^3)$, $(b)$ Embedding on $\#_n \mathbb{RP}^2$ of gem representing $\#_{n} \mathbb{RP}^2$ of type $((2n+2)^3)$.
• Figure 3: $(a)$ Embedding on $\mathbb{RP}^2$ of gem representing $\mathbb{RP}^2$ of type $(4,6,10)$ and $(b)$ Embedding on $\mathbb{S}^2$ of gem representing $\mathbb{S}^2$ of type $(4,6,10)$.
• Figure 4: Embedding on $\mathbb{S}^2$ of gem representing $\mathbb{S}^2$ of type $(6^2,4^1)$
• Figure 5: $(a)$ Embedding on $\mathbb{RP}^2$ of gem representing $\mathbb{RP}^2$ of type $(4^3)$, $(b)$ Embedding on $\mathbb{S}^2$ of gem representing $\mathbb{S}^2$ of type $(4^3)$, $(c)$ Embedding on $\mathbb{RP}^2$ of gem representing $\mathbb{RP}^2$ of type $(4,6,8)$, $(d)$ Embedding on $\mathbb{S}^2$ of gem representing $\mathbb{S}^2$ of type $(4,6,8)$, $(e)$ Embedding on $\mathbb{RP}^2$ of gem representing $\mathbb{RP}^2$ of type $(4^2,2p)$ and $(f)$ Embedding on $\mathbb{S}^2$ of gem representing $\mathbb{S}^2$ of type $(4^2,2p)$.

Theorems & Definitions (24)

• Proposition 1: dm17
• Definition 2
• Proposition 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Corollary 6
• Corollary 7