  We study the Cauchy problem of the incompressible micropolar fluid system in
$\mathbb{R}^{3}$. In a recent work of the first author and Jihong Zhao
\cite{ZhuZ18}, it is proved that the Cauchy problem of the incompressible
micropolar fluid system is locally well-posed in the Fourier--Besov spaces
$\F^{2-\frac{3}{p}}_{p,r}$ for $1<p\leq\infty$ and $1\leq r<\infty$, and
globally well-posed in these spaces with small initial data. In this work we
consider the critical case $p=1$. We show that this problem is locally
well-posed in $\F^{-1}_{1,r}$ for $1\leq r\leq 2$, and is globally well-posed
in these spaces with small initial data. Furthermore, we prove that such
problem is ill-posed in $\F^{-1}_{1,r}$ for $2< r\leq \infty$, which implies
that the function space $\F^{-1}_{1,2}$ is sharp for well-posedness. In
addition, using a similar argument we also prove that this problem is ill-posed
in the Besov space $\B^{-1}_{\infty,r}$ with $2<r\leq\infty$.
