  We report the results of the lattice simulation of the ${\mathbb C} P^{N-1}$
sigma model on $S_{s}^{1}$(large) $\times$ $S_{\tau}^{1}$(small). We take a
sufficiently large ratio of the circumferences to approximate the model on
${\mathbb R} \times S^1$. For periodic boundary condition imposed in the
$S_{\tau}^{1}$ direction, we show that the expectation value of the Polyakov
loop undergoes a deconfinement crossover as the compactified circumference is
decreased, where the peak of the associated susceptibility gets sharper for
larger $N$. For ${\mathbb Z}_{N}$ twisted boundary condition, we find that,
even at relatively high $\beta$ (small circumference), the regular $N$-sided
polygon-shaped distributions of Polyakov loop leads to small expectation values
of Polyakov loop, which implies unbroken ${\mathbb Z}_{N}$ symmetry if
sufficient statistics and large volumes are adopted. We also argue the
existence of fractional instantons and bions by investigating the dependence of
the Polyakov loop on $S_{s}^{1}$ direction, which causes transition between
${\mathbb Z}_{N}$ vacua.
