Crystallizations of small covers over the $n$-simplex $\Delta^n$ and the prism $\Delta^{n-1} \times I$

TL;DR

This paper investigates the crystallizations of small covers over the $n$-simplex and prism, proving uniqueness and classifying equivalence classes, and constructing specific examples with calculated regular genus, advancing the understanding of their topological and combinatorial properties.

Contribution

It provides the first classification of crystallizations of small covers over simple polytopes like the simplex and prism, including explicit constructions and genus calculations.

Findings

Unique $2^n$-vertex crystallization for $ ext{RP}^n$.

Exactly $1 + 2^{n-1}$ D-J equivalence classes over the prism.

Constructed small covers with regular genus formulas.

Abstract

A crystallization of a PL manifold is an edge-colored graph that corresponds to a contracted triangulation of the manifold, facilitating the study of its topological and combinatorial properties. A small cover over a simple convex $n$-polytope $P^n$ is a closed $n$-manifold with a locally standard $\mathbb{Z}_2^n$-action such that its orbit space is homeomorphic to $P^n$. In this article, we study the crystallizations of small covers over the $n$-simplex $\Delta^n$ and the prism $\Delta^{n-1} \times I$. It is known that the small cover over the $n$-simplex $\Delta^n$ is $\mathbb{RP}^n$. For every $n\geq 2$, we prove that $\mathbb{RP}^n$ has a unique $2^n$-vertex crystallization. We also demonstrate that there are exactly $1 + 2^{n-1}$ D-J equivalence classes of small covers over the prism $\Delta^{n-1} \times I$, where $n\geq 3$. For each $\mathbb{Z}_2$-characteristic function of $\Delta^{n-1} \times I$, we construct a $2^{n-1}(n+1)$-vertex crystallization of the small cover $M^n(\lambda)$ with regular genus $1 + 2^{n-4}(n^2 - 2n - 3)$, where $n\geq 4$. The regular genus of closed PL \(n\)-manifolds extends the notions of the genus of surfaces and the Heegaard genus of 3-manifolds to higher dimensions. In this article, we construct four orientable and four non-orientable $\mathbb{RP}^3$-bundles over $\mathbb{S}^1$ up to D-J equivalence, each with regular genus $6$. Although the four orientable (resp. non-orientable) small covers are not D-J equivalent, we show that they are PL homeomorphic.