Splitting With Continuous Control in Algebraic K-theory

Details Splitting With Continuous Control in Algebraic K-theory Splitting With Continuous Control in Algebraic K-theory

2003-09-05

arxiv.org/abs/math/0309106v1

K-Theory 32 (2004), no. 2, 139--166

Mathematics

Algebraic Topology

22 pages

Scientific paper

In this work, the continuously controlled assembly map in algebraic $K$-theory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups $\Gamma$ that satisfy certain geometric conditions. The group $\Gamma$ is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, $K_0(k\Gamma)$ is proved to be isomorphic to the colimit of $K_0(kH)$ over the finite subgroups $H$ of $\Gamma$, when $\Gamma$ is a virtually polycyclic group and $k$ is a field of characteristic zero.

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