The strong Novikov conjecture for low degree cohomology

TL;DR

This paper proves the injectivity of a rational assembly map for groups, focusing on classes dual to low-degree cohomology, and establishes homotopy invariance of higher signatures for these classes.

Contribution

It introduces a new proof of the Novikov conjecture for low-degree cohomology using flat twisting bundles, independent of previous methods.

Findings

Injectivity of the rational assembly map for classes dual to low-degree cohomology.

Homotopy invariance of higher signatures associated with these classes.

A novel approach based on constructing flat twisting bundles from almost flat bundles.

Abstract

We show that for each discrete group G, the rational assembly map K_*(BG) \otimes Q \to K_*(C*_{max} G) \otimes \Q is injective on classes dual to the subring generated by cohomology classes of degree at most 2 (identifying rational K-homology and homology via the Chern character). Our result implies homotopy invariance of higher signatures associated to these cohomology classes. This consequence was first established by Connes-Gromov-Moscovici and Mathai. Our approach is based on the construction of flat twisting bundles out of sequences of almost flat bundles as first described in our previous work. In contrast to the argument of Mathai, our approach is independent of (and indeed gives a new proof of) the result of Hilsum-Skandalis on the homotopy invariance of the index of the signature operator twisted with bundles of small curvature.