  We study a simple electron-phonon model on square and triangular versions of
the Lieb-lattice using an asymptotically exact strong coupling analysis. At
zero temperature and electron density $n = 1$ (one electron per unit cell), for
various ranges of parameters in the model, we exploit a mapping to the quantum
dimer model to establish the existence of a spin-liquid phase with
$\mathbb{Z}_2$ topological order (on the triangular lattice) and a
multi-critical line corresponding to a quantum critical spin liquid (on the
square lattice). In the remaining part of the phase diagram, we find a host of
charge-density-wave phases (e.g. valence-bond crystals), a conventional s-wave
superconducting phase, and with the addition of a small Hubbard $U$ to tip the
balance, a phonon-induced d-wave superconducting phase. Under a special
condition, we find a hidden pseudo-spin $SU(2)$ symmetry that implies an exact
constraint on the superconducting order parameters.
