  We investigate the evolution of the primordial gas surrounding the first
massive black holes formed by the collapse of Population III stars at redshifts
z > 20. Carrying out three-dimensional hydrodynamical simulations using GADGET,
we study the dynamical, thermal and chemical evolution of the first relic H II
regions. We also carry out simulations of the mergers of relic H II regions
with neighboring neutral minihaloes, which contain high density primordial gas
that can accrete onto a Pop III remnant black hole. We find that there may have
been a significant time delay, of order ~10^8 yr, between black hole formation
and the onset of efficient accretion. The build-up of supermassive black holes,
believed to power the z > 6 quasars observed in the Sloan Digital Sky Survey,
therefore faces a crucial early bottleneck. More massive seed black holes may
thus be required, such as those formed by the direct collapse of a primordial
gas cloud facilitated by atomic line cooling. The high optical depth to
Lyman-Werner (LW) photons that results from the high fraction of H_2 molecules
that form in relic H II regions, combined with the continued formation of H_2
inside the dynamically expanding relic H II region, leads to shielding of the
molecules inside these regions at least until a critical background LW flux of
\~10^{-24} ergs s^{-1} cm^{-2} Hz^{-1} sr^{-1}, is established. Furthermore, we
find that a high fraction of HD molecules, X_{HD} > 10^{-7}, is formed,
potentially enabling the formation of Pop II.5 stars during later stages of
structure formation when the relic H II region gas is assembled into a
sufficiently deep potential well to gravitationally confine the gas again.
