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arXiv · 1501.02602

Splitting the relative assembly map, Nil-terms and involutions

Wolfgang Lueck·Wolfgang Steimle
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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.

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Wolfgang Lueck, Wolfgang Steimle. 2015-10-06. Splitting the relative assembly map, Nil-terms and involutions. https://doi.org/10.2140/akt.2016.1.339

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