小覆盖物在 $n$-单纯形 $Δ^n$ 和棱柱 $Δ^{n-1} \times I$ 上的晶格化
Crystallizations of small covers over the $n$-simplex $Δ^n$ and the prism $Δ^{n-1} \times I$
AI总结：
本文研究了小覆盖物在 $n$-单纯形和棱柱上的晶格化，证明了 $\mathbb{RP}^n$ 有唯一的 $2^n$ 顶点晶格化，并确定了棱柱上小覆盖物的 D-J 等价类数量。
摘要：
一个 PL 流形的晶格化是一种边着色图，对应于流形的收缩三角剖分，有助于研究其拓扑和组合性质。一个小覆盖物是闭合的 $n$-维流形，具有局部标准的 $\mathbb{Z}_2^n$-作用，其轨道空间与 $P^n$ 拓扑同胚。本文研究了小覆盖物在 $n$-单纯形 $Δ^n$ 和棱柱 $Δ^{n-1} \times I$ 上的晶格化。已知小覆盖物在 $n$-单纯形 $Δ^n$ 上是 $\mathbb{RP}^n$。对于每个 $n\geq 2$，证明 $\mathbb{RP}^n$ 有唯一的 $2^n$ 顶点晶格化。我们还证明，对于棱柱 $Δ^{n-1} \times I$ 上的小覆盖物，存在恰好 $1 + 2^{n-1}$ 个 D-J 等价类，其中 $n\geq 3$。对于每个 $Δ^{n-1} \times I$ 的 $\mathbb{Z}_2$-特征函数，我们构造了一个 $2^{n-1}(n+1)$ 顶点的晶格化，其正则亏格为 $1 + 2^{n-4}(n^2 - 2n - 3)$，其中 $n\geq 4$。正则亏格是闭合 PL $n$-维流形的扩展概念，扩展了曲面的亏格和 3 维流形的 Heegaard 果冻到更高维度。本文还构造了四个可定向和四个不可定向的 $\mathbb{RP}^3$-纤维丛，每个的正则亏格为 6。尽管四个可定向（或不可定向）的小覆盖物不 D-J 等价，但它们是 PL 同胚的。
英文摘要：
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 $Δ^n$ and the prism $Δ^{n-1} \times I$. It is known that the small cover over the $n$-simplex $Δ^n$ 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 $Δ^{n-1} \times I$, where $n\geq 3$. For each $\mathbb{Z}_2$-characteristic function of $Δ^{n-1} \times I$, we construct a $2^{n-1}(n+1)$-vertex crystallization of the small cover $M^n(λ)$ 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.
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