Crystallizations of small covers over the $n$-simplex $Δ^n$ and the prism $Δ^{n-1} imes I$

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.

💡 Research Summary

The paper investigates the crystallizations of small covers—closed manifolds equipped with a locally standard (\mathbb{Z}_2^n)-action whose orbit space is a simple convex polytope—over two families of polytopes: the (n)-simplex (\Delta^n) and the prism (\Delta^{,n-1}\times I). A crystallization is a properly edge‑colored ((n+1))-regular graph that encodes a contracted triangulation of a PL manifold; its regular genus extends the classical notions of surface genus and Heegaard genus to higher dimensions.

Real projective space (\mathbb{RP}^n).
It is well known that the unique small cover over the simplex (\Delta^n) is (\mathbb{RP}^n). The authors construct, from any (\mathbb{Z}_2)-characteristic function on the facets of (\Delta^n), an ((n+1))-regular colored graph with exactly (2^n) vertices. They prove that this graph is contracted, thus giving a minimal crystallization of (\mathbb{RP}^n). The main novelty is the proof of uniqueness: any ((n+1))-regular colored graph on (2^n) vertices representing (\mathbb{RP}^n) must be isomorphic to the constructed one. The argument proceeds by classifying all possible color‑pair subgraphs and showing that the only way to satisfy the required connectivity and bipartiteness constraints is the standard “binary” construction.

Small covers over the prism (\Delta^{,n-1}\times I).
The prism has (2n) facets: two copies of the ((n-1))-simplex and (n) side facets. A (\mathbb{Z}_2)-characteristic function (\lambda) assigns a non‑zero vector in (\mathbb{Z}_2^n) to each facet, subject to the condition that at each vertex the incident vectors form a basis. The authors enumerate all such functions up to Davis–Januszkiewicz (D‑J) equivalence, i.e. up to automorphisms of (\mathbb{Z}_2^n). They show that there are exactly (1+2^{,n-1}) D‑J classes when (n\ge 3): one class corresponds to the two trivial bundles (orientable and non‑orientable) and the remaining (2^{,n-1}) classes arise from non‑trivial assignments on the side facets.

For each characteristic function (\lambda) they explicitly construct a crystallization (\Gamma(\lambda)) with \