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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

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 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.

Nil group, $K$–theory, Farrell–Jones Conjecture
Mathematical Subject Classification 2000
Primary: 19D35
Secondary: 57R19
Received: 17 August 2010
Revised: 28 June 2011
Accepted: 26 July 2011
Published: 5 September 2011
James F Davis
Department of Mathematics
Indiana University
Bloomington IN 47405
Qayum Khan
Department of Mathematics
University of Notre Dame
Notre Dame IN 46556
Andrew Ranicki
School of Mathematics
University of Edinburgh