Crystallizations of small covers over the n-simplex ⁿ and the prism ^n-1 I

A small cover is a closed manifold Mⁿ with a locally standard Z₂ⁿ-action such that its orbit space is a simple convex polytope Pⁿ. In this article, we study the crystallizations of small covers over the n-simplex ⁿ and the prism ^n-1 I. It is known that the small cover over the n-simplex ⁿ is RPⁿ. For every n 2, we prove that RPⁿ has a unique 2ⁿ-vertex crystallization. We also demonstrate that there are exactly 1 + 2^n-1 D-J equivalence classes of small covers over the prism ^n-1 I, where n 3. For each Z₂-characteristic function of ^n-1 I, we construct a 2^n-1 (n+1) -vertex crystallization of the small cover Mⁿ () with regular genus 1 + 2^n-4 (n² - 2n - 3), where n 4. In particular, we construct four orientable and four non-orientable RP³-bundles over S¹ up to D-J equivalence with the regular genus 6.