  Path-integral Monte Carlo calculations have been performed to study the
$^4$He adsorption on $\gamma$-graphyne, a planar network of benzene rings
connected by acetylene bonds. Assuming the $^4$He-substrate interaction
described by a pairwise sum of empirical $^4$He-carbon interatomic potentials,
we find that unlike $\alpha$-graphyne, a single sheet of $\gamma$-graphyne is
not permeable to $^4$He atoms in spite of its large surface area.
One-dimensional density distributions computed as a function of the distance
from the graphyne surface reveal a layer-by-layer growth of $^4$He atoms. A
partially-filled $^4$He monolayer is found to exhibit different commensurate
solid structures depending on the helium coverage; it shows a C$_{2/3}$
commensurate structure at an areal density of 0.0491\AA$^{-2}$, a C$_{3/3}$
structure at 0.0736\AA$^{-2}$, and a C$_{4/3}$ structure at 0.0982\AA$^{-2}$.
While the promotion to the second layer starts beyond the C$_{4/3}$ helium
coverage, the first $^4$He layer is found to form an incommensurate triangular
solid when compressed with the development of the second layer.
