Two types of Nil-groups arise in the
codimension
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
–theory
relating the two types of Nil-groups.
The infinite dihedral group is a free product and has an
index
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
–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.