  In 1983, Banchoff and Kuhnel constructed a minimal triangulation of $\CP^2$
with 9 vertices. $\CP^3$ was first triangulated by Bagchi and Datta in 2012
with 18 vertices. Known lower bound on number of vertices of a triangulation of
$\CP^n$ is $1 + \frac{(n + 1)^2}{2}$ for $n \geq 3$. We give explicit
construction of some triangulations of complex projective space $\CP^n$ with
$\frac{4^{n+1}-1}{3}$ vertices for all $n$. No explicit triangulation of
$\CP^n$ is known for $n \geq 4$.

Geometric Topology