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 $\widetilde{Nil}$–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.