  We study a class of $3d$ and $4d$ topological insulators whose topological
nature is characterized by the Hopf map and its generalizations. We identify
the symmetry $\mathcal{C}^\prime$, a generalized particle-hole symmetry that
gives the Hopf insulator a $\mathbb{Z}_2$ classification. The $4d$ analogue of
the Hopf insulator with symmetry $\mathcal{C}^\prime$ has the same
$\mathbb{Z}_2$ classification. The minimal models for the $3d$ and $4d$ Hopf
insulator can be heuristically viewed as "Chern-insulator$ \rtimes S^1$" and
"Chern-insulator$\rtimes T^2$" respectively. We also discuss the relation
between the Hopf insulator and the Weyl semimetals, which points the direction
for its possible experimental realization.
