  We investigate the effect of a putative X-ray emitting halo surrounding the
Local Group of galaxies, and specifically the possible temperature anisotropies
induced in the COBE-DMR four-year sky maps by an associated Sunyaev-Zel'dovich
effect. By fitting the isothermal spherical halo model proposed by Suto et.al.
(1996) to the coadded four-year COBE-DMR 53 and 90 GHz sky maps in Galactic
coordinates, we find no significant evidence of a contribution. We therefore
reject the claim that such a halo can affect the estimation of the primordial
spectral index and amplitude of density perturbations as inferred from the DMR
data. We find that correlation with the DMR data imposes constraints on the
plausible contribution of such an X-ray emitting halo to a distortion in the
CMB spectrum (as specified by the Compton-y parameter), up to a value for R --
the ratio of the core radius of the isothermal halo gas distribution to the
distance to the Local Group centroid -- of 0.68. For larger values of R, the
recent cosmological upper limit derived by COBE-FIRAS provides stronger
constraints on the model parameters. Over the entire parameter space for R, we
find an upper limit to the inferred sky-RMS anisotropy signal of 14 microKelvin
(95% c.l.), a negligible amount relative to the 35 microKelvin signal observed
in the COBE-DMR data.
