The first systematic study of the warm gas (T=10^4-5 K) distribution across a galaxy cluster is presented using multiple background QSOs to the Virgo Cluster. We detect 25 Lya absorbers (N_HI = 10^13.1-15.4 cm^-2) in the Virgo velocity range toward 9 of 12 QSO sightlines observed with Cosmic Origin Spectrograph, with a cluster impact parameter range of 0.36-1.65 Mpc (0.23-1.05R_vir). Including 18 Lya absorbers previously detected by STIS or GHRS toward 7 of 11 background QSOs in and around the Virgo Cluster, we establish a sample of 43 absorbers towards a total of 23 background probes for studying the incidence of Lya absorbers in and around the Virgo Cluster. With these absorbers, we find: (1) warm gas is predominantly in the outskirts of the cluster and avoids the X-ray detected hot ICM. Also, Lya absorption strength increases with cluster impact parameter. (2) Lya absorbing warm gas traces cold HI emitting gas in the substructures of the Virgo Cluster. (3) Including the absorbers associated with the surrounding substructures, the warm gas covering fraction (100% for N_HI > 10^13.1 cm^-2) is in agreement with cosmological simulations. We speculate that the observed warm gas is part of large-scale gas flows feeding the cluster both the ICM and galaxies.
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