  Topological states of matter in equilibrium, as well as out of equilibrium,
have been thoroughly investigated during the last years in condensed-matter and
cold-atom systems. However, the geometric topology of the studied samples is
usually trivial, such as a ribbon or a cylinder. In this paper, we consider a
graphene M\"obius band irradiated with circularly polarised light.
Interestingly, due to the non-orientability of the M\"obius band, a homogeneous
quantum Hall effect cannot exist in this system, but the quantum spin Hall
effect can. To avoid this restriction, the irradiation is applied in a
longitudinal-domain-wall configuration. In this way, the periodic
time-dependent driving term tends to generate the quantum anomalous Hall
effect. On the other hand, due to the bent geometry of the M\"obius band, we
expect a strong spin-orbit coupling, which may lead to quantum spin Hall-like
topological states. Here, we investigate the competition between these two
phenomena upon varying the amplitude and the frequency of the light, for a
fixed value of the spin-orbit coupling strength. The topological properties are
analysed by identifying the edge states in the Floquet spectrum at intermediate
frequencies, when there are resonances between the light frequency and the
energy difference between the conduction and valence bands of the graphene
system.
