  Topology of isofrequency surfaces plays a crucial role in characterizing the
interaction of an electromagnetic wave with a medium. Thus, engineering the
topology in complex media is leading to novel applications, ranging from
super-resolution microscopy with hyperbolic metamaterials to sub-wavelength
waveguiding structures. Here, we investigate the spin-governed nature of
isofrequency surfaces in a general gyromagnetic medium. We show that gyrotropy
also plays an important role in the topological properties of a medium, along
with the anisotropic permeability and permittivity. Even though the topology
primarily depends on permeability, gyrotropy can suppress or support the
existence of certain topological surfaces. We reveal the connection between the
gyrotropy imposed constraints and the photonic spin-profile of the topological
surfaces. The spin-profile along the isofrequency surface is locked to the
material, resulting in the non-reciprocity and breaking of the spin-momentum
locking in the gyromagnetic medium. Further, we show that the conflict between
spin-momentum locking and material locked spin leads to asymmetric mode profile
and gyrotropy-induced cutoff in guided wave structures. Our work provides
important insights into the underlying link between topology, spin, and
non-reciprocity in gyrotropic media.
