  We introduce new three-dimensional topological phases of two-band models
using the Pontryagin-Thom construction. In symmetry class A these are the
infinitely many Hopf-Chern topological insulators, which are hybrids of layered
Chern insulators and the Hopf insulator. In symmetry class C, there is a
$\mathbb{Z}_2$-classification with the non-trivial topological phase, the Hopf
superconductor, being realized by a construction that doubles the usual Hopf
insulator in momentum space. For these new topological phases we investigate
the energy spectrum in the presence of a boundary and find gapless surface
modes, confirming the validity of the bulk-boundary correspondence.
