  The relationship between gas-rich galaxies and Ly-alpha absorbers is
addressed in this paper in the context of the baryonic content of galaxy halos.
Deep Arecibo HI observations are presented of two gas-rich spiral galaxies
within 125 kpc projected distance of a Ly-alpha absorber at a similar velocity.
The galaxies investigated are close to edge-on and the absorbers lie almost
along their major axes, allowing for a comparison of the Ly-alpha absorber
velocities with galactic rotation. This comparison is used to examine whether
the absorbers are diffuse gas rotating with the galaxies' halos, outflow
material from the galaxies, or intergalactic gas in the low redshift cosmic
web. The results indicate that if the gas resides in the galaxies' halos it is
not rotating with the system and possibly counter-rotating. In addition, simple
geometry indicates the gas was not ejected from the galaxies and there are no
gas-rich satellites detected down to 3.6 - 7.5 x 10^6 Msun, or remnants of
satellites to 5-6 x 10^{18} cm^{-2}. The gas could potentially be infalling
from large radii, but the velocities and distances are rather high compared to
the high velocity clouds around the Milky Way. The most likely explanation is
the galaxies and absorbers are not directly associated, despite the vicinity of
the spiral galaxies to the absorbers (58-77 kpc from the HI edge). The spiral
galaxies reside in a filament of intergalactic gas, and the gas detected by the
absorber has not yet come into equilibrium with the galaxy. These results also
indicate that the massive, extended dark matter halos of spiral galaxies do not
commonly have an associated diffuse baryonic component at large radii.
