  The main result of this paper is the $G$-homotopy invariance of the $G$-index
of signature operator of proper co-compact $G$-manifolds. If proper co-compact
$G$ manifolds $X$ and $Y$ are $G$-homotopy equivalent, then we prove that the
images of their signature operators by the $G$-index map are the same in the
$K$-theory of the $C^{*}$-algebra of the group $G$. Neither discreteness of the
locally compact group $G$ nor freeness of the action of $G$ on $X$ are
required, so this is a generalization of the classical case of closed
manifolds. Using this result we can deduce the equivariant version of Novikov
conjecture for proper co-compact $G$-manifolds from the Strong Novikov
conjecture for $G$.
