  The long-standing Alekseevskii conjecture states that a connected homogeneous
Einstein space G/K of negative scalar curvature must be diffeomorphic to R^n.
This was known to be true only in dimensions up to 5, and in dimension 6 for
non-semisimple G. In this work we prove that this is also the case in
dimensions up to 10 when G is not semisimple. For arbitrary G, besides 5
possible exceptions, we show that the conjecture holds up to dimension 8.
