  We have carried out an HI stacking analysis of a volume-limited sample of
~5000 galaxies with imaging and spectroscopic data from GALEX and the Sloan
Digital Sky Survey, which lie within the current footprint of the Arecibo
Legacy Fast ALFA (ALFALFA) Survey. Our galaxies are selected to have stellar
masses greater than 10^10 Msun and redshifts in the range 0.025<z<0.05. We
extract a sub-sample of 1833 "early-type" galaxies with inclinations less than
70deg, with concentration indices C>2.6 and with light profiles that are well
fit by a De Vaucouleurs model. We then stack HI line spectra extracted from the
ALFALFA data cubes at the 3-D positions of the galaxies from these two samples
in bins of stellar mass, stellar mass surface density, central velocity
dispersion, and NUV-r colour. We use the stacked spectra to estimate the
average HI gas fractions M_HI/M_* of the galaxies in each bin.
  Our main result is that the HI content of a galaxy is not influenced by its
bulge. The average HI gas fractions of galaxies in both our samples correlate
most strongly with NUV-r colour and with stellar surface density. The relation
between average HI fraction and these two parameters is independent of
concentration index C. We have tested whether the average HI gas content of
bulge-dominated galaxies on the red sequence, differs from that of late-type
galaxies on the red sequence. We find no evidence that galaxies with a
significant bulge component are less efficient at turning their available gas
reservoirs into stars. This result is in contradiction with the "morphological
quenching" scenario proposed by Martig et al. (2009).
