  Layers of transition metal dichalcogenides (TMDs) combine the enhanced
effects of correlations associated with the two-dimensional limit with
electrostatic control over their phase transitions by means of an electric
field. Several semiconducting TMDs, such as MoS$_2$, develop superconductivity
(SC) at their surface when doped with an electrostatic field, but the mechanism
is still debated. It is often assumed that Cooper pairs reside only in the two
electron pockets at the K/K' points of the Brillouin Zone. However,
experimental and theoretical results suggest that a multi-valley Fermi surface
(FS) is associated with the SC state, involving 6 electron pockets at the Q/Q'
points. Here, we perform low-temperature transport measurements in ion-gated
MoS$_2$ flakes. We show that a fully multi-valley FS is associated with the SC
onset. The Q/Q' valleys fill for doping$\gtrsim2\cdot10^{13}$cm$^{-2}$, and the
SC transition does not appear until the Fermi level crosses both spin-orbit
split sub-bands Q$_1$ and Q$_2$. The SC state is associated with the FS
connectivity and promoted by a Lifshitz transition due to the simultaneous
population of multiple electron pockets. This FS topology will serve as a
guideline in the quest for new superconductors.
