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

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 $n\ge 2$, 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 $n\ge 3$. 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 $n\ge 4$. 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.

16 pages, 4 figures

References in corpus (1)

• Classifying compact 4-manifolds via generalized regular genus and G-degree