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

TL;DR

This paper proves the non-vanishing of the index map image for certain proper actions of locally compact groups, extending previous results to more general group actions without requiring discreteness or freeness.

Contribution

It establishes the non-vanishing of the index map for low-degree cohomology classes in the setting of proper actions of locally compact groups, generalizing prior work.

Findings

Non-vanishing of the index map image under certain conditions.

Applicability to non-discrete, non-free group actions.

Extension of results to low-dimensional 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.