We present the HI mass inventory for the RESOLVE survey, a volume-limited, multi-wavelength census of >1500 z=0 galaxies spanning diverse environments and complete in baryonic mass down to dwarfs of 10^9 Msun. This first 21cm data release provides robust detections or strong upper limits (1.4M_HI < 5 to 10% of stellar mass M_stars) for 94% of RESOLVE. We examine global atomic gas-to-stellar mass ratios (G/S) in relation to galaxy environment using several metrics: group dark matter halo mass M_h , central/satellite designation, relative mass density of the cosmic web, and distance to nearest massive group. We find that at fixed M_stars, satellites have decreasing G/S with increasing M_h starting clearly at M_h = 10^12 Msun, suggesting the presence of starvation and/or stripping mechanisms associated with halo gas heating in intermediate-mass groups. The analogous relationship for centrals is uncertain because halo abundance matching builds in relationships between central G/S, stellar mass, and halo mass, which depend on the integrated group property used as a proxy for halo mass (stellar or baryonic mass). On larger scales G/S trends are less sensitive to the abundance matching method. At fixed M_h < 10^12 Msun, the fraction of gas-poor centrals increases with large-scale structure density. In overdense regions, we identify a rare population of gas-poor centrals in low-mass (M_h < 10^11.4 Msun) halos primarily located within 1.5 times the virial radius of more massive (M_h > 10^12 Msun) halos, suggesting that gas stripping and/or starvation may be induced by interactions with larger halos or the surrounding cosmic web. We find that the detailed relationship between G/S and environment varies when we examine different subvolumes of RESOLVE independently, which we suggest may be a signature of assembly bias.
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