  We report topological nonlinear optics with spin-orbit coupled Bose-Einstein
condensate in a cavity. The cavity is driven by a pump laser and weak probe
laser which excite Bose-Einstein condensate to an intermediate storage level,
where the standard Raman process engineers spin-orbit coupling. We show that
the nonlinear photonic interactions at the transitional pathways of dressed
states result in new type of optical transparencies, which get completely
inverted with atom induced gain. These nonlinear interactions also implant
topological sort of features in probe transmission modes by inducing gapless
Dirac-like cones, which become gaped in presence of Raman detuning. The
topological features get interestingly enhanced in gain regime where the
gapless topological edge-like states emerge among the probe modes, which can
cause non-trivial phase transition. We show that spin-orbit coupling and Zeeman
field effects also impressively revamp fast and slow probe light. The
manipulation of dressed states for quantum nonlinear optics with topological
characteristics in our findings could be a crucial step towards topological
quantum computation.
