  Three-wave instability is a fundamental process that has important
applications in many branches of physics. It is widely accepted that the
resonant condition $\omega_{z}\approx\omega_{x}+\omega_{y}$ for participating
waves is the criteria for the onset of the instability. We show that this
condition is neither sufficient nor necessary, instead, the exact criteria for
the onset of the instability is that a positive-action mode resonates with a
negative-action mode. This mechanism is imposed by the topology and geometry of
the spectral space. Guided by this new theory, additional instability bands
previously unknown are discovered.
