  The Holstein Hamiltonian describes fermions hopping on a lattice and
interacting locally with dispersionless phonon degrees of freedom. In the low
density limit, dressed quasiparticles, polarons and bipolarons, propagate with
an effective mass. At higher densities, pairs can condense into a low
temperature superconducting phase and, at or near commensurate filling on a
bipartite lattice, to charge density wave (CDW) order. CDW formation breaks a
discrete symmetry and hence occurs via a second order (Ising) transition, and
therefore at a finite $T_{\rm cdw}$ in two dimensions. Quantum Monte Carlo
calculations have determined $T_{\rm cdw}$ for a variety of geometries,
including square, honeycomb, and Lieb lattices. The superconducting transition,
on the other hand, in $d=2$ is in the Kosterlitz-Thouless (KT) universality
class, and is much less well characterized. In this paper we determine $T_{\rm
sc}$ for the square lattice, for several values of the density $\rho$ and
phonon frequency $\omega_0$. We find that quasi-long range order sets in at
$T_{\rm sc} \lesssim t/20$, where $t$ is the near neighbor hopping amplitude,
consistent with previous rough estimates from simulations which only
extrapolated to the temperatures we reach from considerably higher $T$. We also
show evidence for a discontinuous evolution of the density as the CDW
transition is approached at half-filling.
