Volume 11, issue 4 (2011)

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Algebraic $K$–theory over the infinite dihedral group: an algebraic approach

James F Davis, Qayum Khan and Andrew Ranicki

Algebraic & Geometric Topology 11 (2011) 2391–2436
Abstract

Two types of Nil-groups arise in the codimension $1$ splitting obstruction theory for homotopy equivalences of finite CW–complexes: the Farrell–Bass Nil-groups in the nonseparating case when the fundamental group is an HNN extension and the Waldhausen Nil-groups in the separating case when the fundamental group is an amalgamated free product. We obtain a general Nil-Nil theorem in algebraic $K$–theory relating the two types of Nil-groups.

The infinite dihedral group is a free product and has an index $2$ subgroup which is an HNN extension, so both cases arise if the fundamental group surjects onto the infinite dihedral group. The Nil-Nil theorem implies that the two types of the reduced $\stackrel{˜}{Nil}$–groups arising from such a fundamental group are isomorphic. There is also a topological application: in the finite-index case of an amalgamated free product, a homotopy equivalence of finite CW–complexes is semisplit along a separating subcomplex.

Keywords
Nil group, $K$–theory, Farrell–Jones Conjecture
Primary: 19D35
Secondary: 57R19
Publication
Received: 17 August 2010
Revised: 28 June 2011
Accepted: 26 July 2011
Published: 5 September 2011
Authors
 James F Davis Department of Mathematics Indiana University Bloomington IN 47405 USA Qayum Khan Department of Mathematics University of Notre Dame Notre Dame IN 46556 USA Andrew Ranicki School of Mathematics University of Edinburgh Edinburgh EH9 3JZ UK