Relative geometric assembly and mapping cones, Part II: Chern characters and the Novikov property

TL;DR

This paper explores the relationships between Chern characters, assembly maps, and Novikov properties within geometric K-homology, establishing new invariance results for manifolds with boundary under certain group homomorphisms.

Contribution

It introduces new relative Novikov properties for hyperbolic and polynomially bounded groups, linking algebraic group properties to geometric invariants.

Findings

Proves a relative strong Novikov property for hyperbolic group homomorphisms.

Establishes a relative strong ℓ^1-Novikov property for polynomially bounded group homomorphisms.

Shows homotopy invariance of higher signatures on manifolds with boundary under specified conditions.

Abstract

We study Chern characters and the assembly mapping for free actions using the framework of geometric $K$-homology. The focus is on the relative groups associated with a group homomorphism $\phi:\Gamma_1\to \Gamma_2$ along with applications to Novikov type properties. In particular, we prove a relative strong Novikov property for homomorphisms of hyperbolic groups and a relative strong $\ell^1$-Novikov property for polynomially bounded homomorphisms of groups with polynomially bounded cohomology in $\C$. As a corollary, relative higher signatures on a manifold with boundary $W$, with $\pi_1(\partial W)\to \pi_1(W)$ belonging to the class above, are homotopy invariant.