Splitting the relative assembly map, Nil-terms and involutions

TL;DR

This paper proves split injectivity of the relative Farrell-Jones assembly map for algebraic K-theory with additive category coefficients, describes the relative term explicitly, and shows its rational vanishing and Tate cohomology vanishing under certain conditions.

Contribution

It generalizes previous results to additive categories, provides an explicit description of the relative term, and establishes conditions for its vanishing and cohomology properties.

Findings

Split injectivity of the assembly map in this setting.

Explicit description of the relative term.

Vanishing of the relative term rationally and Tate cohomology under orientability conditions.

Abstract

We show that the relative Farrell-Jones assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for algebraic K-theory is split injective in the setting where the coefficients are additive categories with group action. This generalizes a result of Bartels for rings as coefficients. We give an explicit description of the relative term. This enables us to show that it vanishes rationally if we take coefficients in a regular ring. Moreover, it is, considered as a Z[Z/2]-module by the involution coming from taking dual modules, an extended module and in particular all its Tate cohomology groups vanish, provided that the infinite virtually cyclic subgroups of type I of G are orientable. The latter condition is for instance satisfied for torsionfree hyperbolic groups.