Quantitative index, Novikov conjecture and coarse decomposability

TL;DR

This paper introduces quantitative assembly map estimates for families of finite metric spaces, linking them to the Novikov conjecture and demonstrating their compatibility with coarse decompositions, leading to a geometric proof for groups with finite decomposition complexity.

Contribution

It defines new quantitative estimates for assembly maps, connects them to the Novikov conjecture, and proves the conjecture for groups with finite decomposition complexity using coarse decompositions.

Findings

Established a relationship between assembly map estimates and the Novikov conjecture.

Showed that these estimates are compatible with coarse decompositions.

Provided a geometric proof of the Novikov conjecture for groups with finite decomposition complexity.

Abstract

We define for families of finite metric spaces quantitative assembly map estimates that take into account propagation phenomena for pseudo-differential calculus. We relate these estimates to the Novikov conjecture and we show that they fit nicely under coarse decompositions. As an application, we provide a geometric proof of Novikov conjecture for groups with finite decomposition complexity.