## Algebraic & Geometric Topology

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

#### 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 $Nil˜$–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.

#### Article information

Source
Algebr. Geom. Topol., Volume 11, Number 4 (2011), 2391-2436.

Dates
Revised: 28 June 2011
Accepted: 26 July 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715273

Digital Object Identifier
doi:10.2140/agt.2011.11.2391

Mathematical Reviews number (MathSciNet)
MR2835234

Zentralblatt MATH identifier
1236.19002

Subjects
Primary: 19D35: Negative $K$-theory, NK and Nil
Secondary: 57R19: Algebraic topology on manifolds

#### Citation

Davis, James F; Khan, Qayum; Ranicki, Andrew. Algebraic $K$–theory over the infinite dihedral group: an algebraic approach. Algebr. Geom. Topol. 11 (2011), no. 4, 2391--2436. doi:10.2140/agt.2011.11.2391. https://projecteuclid.org/euclid.agt/1513715273

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