$K$-theory of relative group $C^*$-algebras and the relative Novikov conjecture

TL;DR

This paper investigates the relative Novikov conjecture by analyzing the relative Baum-Connes assembly map for group pairs, providing solutions under specific geometric conditions.

Contribution

It introduces new methods to verify the relative Novikov conjecture using the relative Baum-Connes assembly map for certain group pairs.

Findings

Proves the relative Novikov conjecture for groups satisfying specific geometric conditions.

Develops a framework for the relative Baum-Connes assembly map in the context of group pairs.

Establishes invariance of relative higher signatures under homotopy equivalences for certain groups.

Abstract

The relative Novikov conjecture states that the relative higher signatures of manifolds with boundary are invariant under orientation-preserving homotopy equivalences of pairs. In this paper, we study the relative Baum-Connes assembly map for any pair of groups and apply it to solve the relative Novikov conjecture when the groups satisfy certain geometric conditions.