  Let $G$ be a connected Lie group of rank one. In this paper the existence of
free actions of group $G$ on spheres, real projective spaces and lens spaces
has been studied. Most of the results have been obtained for finitistic spaces
with cohomology ring isomorphic to the cohomology ring of these spaces. The
cohomology algebra of the orbit space has been determined in the case of free
$G$-action. The non-existence of $G$-equivariant maps from spheres to
cohomology spheres has also been established.
