The Novikov conjecture, the group of diffeomorphisms and continuous fields of Hilbert-Hadamard spaces

TL;DR

This paper proves the Novikov conjecture for certain non-linear groups, specifically discrete subgroups of diffeomorphism groups, by analyzing operator K-theory and actions on continuous fields of infinite-dimensional non-positively curved spaces.

Contribution

It extends previous results by removing the volume-preserving condition and applies to a broader class of groups.

Findings

Proves the Novikov conjecture for discrete subgroups of diffeomorphism groups.

Develops methods involving operator K-theory and continuous fields of Hilbert-Hadamard spaces.

Removes the volume-preserving restriction from earlier work.

Abstract

In this paper, we prove the Novikov conjecture for a class of highly non-linear groups, namely discrete subgroups of the diffeomorphism group of a compact smooth manifold. This removes the volume-preserving condition in a previous work. This result is proved by studying operator $K$-theory and group actions on continuous fields of infinite dimensional non-positively curved spaces.