  We employ the EAGLE hydrodynamical simulation to uncover the relationship
between cluster environment and $\rm H_2$ content of star-forming galaxies at
redshifts spanning $0\leq z\leq 1$. To do so, we divide the star-forming sample
into those that are bound to clusters and those that are not. We find that, at
any given redshift, the galaxies in clusters generally have less $\rm H_2$ than
their non-cluster counterparts with the same stellar mass (corresponding to an
offset of $\lesssim 0.5$ dex), but this offset varies with stellar mass and is
virtually absent at $M_\star\lesssim10^{9.3}~{\rm M}_\odot$. The $\rm H_2$
deficit in star-forming cluster galaxies can be traced back to a decline in
their $\rm H_2$ content that commenced after first infall into a cluster, which
occurred later than a typical cluster galaxy. Evolution of the full cluster
population after infall is generally consistent with `slow-then-rapid'
quenching, but galaxies with $M_\star\lesssim 10^{9.5}~{\rm M}_\odot$ exhibit
rapid quenching. Unlike most cluster galaxies, star-forming ones were not
pre-processed in groups prior to being accreted by clusters. For both of these
cluster samples, the star formation efficiency remained oblivious to the
infall. We track the particles associated with star-forming cluster galaxies
and attribute the drop in $\rm H_2$ mass after infall to poor replenishment,
depletion due to star formation, and stripping of $\rm H_2$ in cluster
environments. These results provide predictions for future surveys, along with
support and theoretical insights for existing molecular gas observations that
suggest there is less $\rm H_2$ in cluster galaxies.
