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

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*-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.