Crystallizations of small covers over the $n$-simplex $Δ^n$ and the prism $Δ^{n-1} \times I$
Anshu Agarwal, Biplab Basak
TL;DR
This work develops a combinatorial approach to small covers over the $n$-simplex and the prism $Δ^{n-1}×I$ via crystallizations, relating edge-colored graphs to the topology of $\mathbb{RP}^n$ and $\mathbb{RP}^3$-bundles over $\mathbb{S}^1$. It proves a unique $2^n$-vertex crystallization for $\mathbb{RP}^n$ (for all $n≥2$) and enumerates $1+2^{n-1}$ Davis–Januszkiewicz classes of small covers over $Δ^{n-1}×I$ (for $n≥3$). For each $\mathbb{Z}_2$-characteristic function, the authors construct a $2^{n-1}(n+1)$-vertex crystallization with regular genus $1+2^{n-4}(n^2-2n-3)$ ($n≥4$), obtaining in dimension $4$ eight genus-6 crystallizations yielding four orientable and four non-orientable $\mathbb{RP}^3$-bundles over $\mathbb{S}^1$, classified up to Davis–Januszkiewicz equivalence. The results contribute a concrete, computable framework for analyzing regular genus and fundamental groups of these manifolds via crystallizations, and underscore weak semi-simple crystallizations as a mechanism to attain lower bounds on regular genus. Overall, the paper advances discrete combinatorial tools for toric-like manifolds and enriches the catalog of crystallizations with explicit constructions and classifications.
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 $Δ^n$ 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 $n\geq 4$. In particular, we construct four orientable and four non-orientable $\mathbb{RP}^3$-bundles over $\mathbb{S}^1$ up to D-J equivalence with regular genus 6.