On the equivariant triangulation of some small covers

In this paper, we study certain properties of ${\mathbb{Z}}_2^n$-equivariant triangulations of small covers. We show that any ${\mathbb{Z}}_2^n$-equivariant triangulation of a small cover naturally induces a triangulation of the orbit space. Then, we explicitly construct the minimal ${\mathbb{Z}}_2^3$-equivariant triangulation of ${\mathbb{R}\mathbb{P}}^3$, which contains $11$ vertices and prove that this is the unique ${\mathbb{Z}}_2^3$-equivariant triangulation of ${\mathbb{R}\mathbb{P}}^3$ with $11$ vertices. For a finite group $G$, we give a method for constructing some $G$-equivariant triangulations of connected sums of manifolds from their respective $G$-equivariant triangulations. In particular, we construct a ${\mathbb{Z}}_2^3$-equivariant triangulation of ${\mathbb{R}\mathbb{P}}^3\mathrm{\#}{\mathbb{R}\mathbb{P}}^3$ with $17$ vertices, which is the best known yet. This triangulation of ${\mathbb{R}\mathbb{P}}^3\mathrm{\#}{\mathbb{R}\mathbb{P}}^3$ provides another minimal $g$-vector improving one of the result of Lutz in \cite{LS}. Moreover, we prove that a ${\text{ZZ}}_2^4$-equivariant triangulation of ${\mathbb{R}\mathbb{P}}^4$ requires at least $18$ vertices.