$G$-homotopy invariance of the analytic signature of proper co-compact $G$-manifolds and equivariant Novikov conjecture

Abstract

The main result of this paper is the $G$-homotopy invariance of the $G$-index of the 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 $C^{\ast }$-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 $G$.