  We introduce topological phases in Photonic Time-Crystals. Photonic
Time-Crystals are materials in which the refractive index varies periodically
and abruptly in time. When the refractive index changes abruptly, the light
propagating in the material experiences time-refraction and time-reflection,
analogous to refraction and reflection in photonic crystals. Interference
between time-refracted and time-reflected waves gives rise to Floquet-Bloch
states and dispersion bands, which are gapped in the momentum. We show that
photonic time-crystals can be in a topologically non-trivial phase, and
calculate the topological invariant associated with the momentum bands of the
Photonic Time-Crystal. The topological invariants are related to the phase
between the forward and backward-propagating waves of the time-crystal, and to
localized edge states in time.
