  The Hopf insulator is a weak topological insulator characterized by an
insulating bulk with conducting edge states protected by an integer-valued
linking number invariant. The state exists in three-dimensional two-band
models. We demonstrate that the Hopf insulator can be naturally realized in
lattices of dipolar-interacting spins, where spin exchange plays the role of
particle hopping. The long-ranged, anisotropic nature of the dipole-dipole
interactions allows for the precise detail required in the momentum-space
structure, while different spin orientations ensure the necessary structure of
the complex phases of the hoppings. Our model features robust gapless edge
states at both smooth edges, as well as sharp edges obeying a certain
crystalline symmetry, despite the breakdown of the two-band picture at the
latter. In a companion manuscript [2105.10504], we provide a specific
experimental blueprint for implementing our proposal using ultracold polar
molecules of $^{40}$K$^{87}$Rb.
