The chain complex approach to the K- and L-theory of the Laurent polynomial extension category A[z, z–1] developed in §10 and §16 will now be used to give an abstract algebraic treatment of the obstruction theory for fibering n-dimensional manifolds over S1 for n ≤ 6. Following the positive results of Stallings [79] for n = 3 and Browder and Levine [11] for n ≤ 6 and π1 = Z a general fibering obstruction theory for n ≤ 6 was developed by Farrell [21], [22] and Siebenmann [74], [76], with obstructions in the Whitehead group of an extension by an infinite cyclic group. See Kearton [38] and Weinberger [85] for examples of non-fibering manifolds in dimensions n = 4, 5 with vanishing Whitehead torsion fibering obstruction.

The mapping torus of a self-map h : F → F is defined by as usual. If F is a compact (n – 1)-dimensional manifold and h : F → F is a self homeomorphism then T(h) is a compact n-dimensional manifold such that is the projection of a fibre bundle over S1 with fibre F and monodromy h.

A CW complex band is a finite CW complex X with a finitely dominated infinite cyclic cover X. Let ζ : X → X be a generating covering translation. For the sake of simplicity we shall only consider CW complex bands with so that