Open Access
2011 Algebraic $K$–theory over the infinite dihedral group: an algebraic approach
James F Davis, Qayum Khan, Andrew Ranicki
Algebr. Geom. Topol. 11(4): 2391-2436 (2011). DOI: 10.2140/agt.2011.11.2391


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.


Download Citation

James F Davis. Qayum Khan. Andrew Ranicki. "Algebraic $K$–theory over the infinite dihedral group: an algebraic approach." Algebr. Geom. Topol. 11 (4) 2391 - 2436, 2011.


Received: 17 August 2010; Revised: 28 June 2011; Accepted: 26 July 2011; Published: 2011
First available in Project Euclid: 19 December 2017

zbMATH: 1236.19002
MathSciNet: MR2835234
Digital Object Identifier: 10.2140/agt.2011.11.2391

Primary: 19D35
Secondary: 57R19

Keywords: $K$–theory , Farrell–Jones conjecture , nil group

Rights: Copyright © 2011 Mathematical Sciences Publishers

Vol.11 • No. 4 • 2011
Back to Top