  Cosmological hydrodynamic simulations predict that the low redshift universe
comprises a web of warm-hot intergalactic gas and galaxies, with groups of
galaxies and clusters forming at dense knots in these filaments. Our own Galaxy
being no exception is also expected to be surrounded by the warm-hot
intergalactic medium, filling the Local Group. Some theoretical models also
predict the existence of a hot Galactic corona. With X-ray and FUV observations
of extragalactic sources, we can probe the warm-hot gas through absorption
lines of highly ionized elements. Indeed, Chandra, XMM and FUSE observations
have detected z=0 absorption lines toward many sightlines. The debate that has
emerged is over the interpretation of these observations: are the z=0
absorption systems from the halo of our Galaxy or from the extended Local Group
environment? This has important implications for our understanding of the mass
of the Local Group, the physical conditions in the intergalactic medium, the
structure of the Galaxy and galaxy formation in general. We will present the
current status of the debate and discuss our ongoing observing program aimed at
understanding the z=0 absorption systems, with an emphasis on the high quality
Chandra spectra of the Mrk 421 and Mrk 279 sightlines.
