On the Strong Novikov Conjecture of Locally Compact Groups for Low Degree Cohomology Classes

Abstract

The main result of this paper is non-vanishing of the image of the index map from the $G$-equivariant $K$-homology of a proper $G$-compact $G$-manifold $X$ to the $K$-theory of the $C^{*}$-algebra of the group $G$. Under the assumption that the Kronecker pairing of a $K$-homology class with a low-dimensional cohomology class is non-zero, we prove that the image of this class under the index map is non-zero. Neither discreteness of the locally compact group $G$ nor freeness of the action of $G$ on $X$ are required. The case of free actions of discrete groups was considered earlier by B. Hanke and T. Schick.