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The mapping torus T (ζ) of the generating covering translation ζ : X−−→X of the infinite cyclic cover X of a space X is such that the projection q + : T (ζ) −−→ X ; (x, s) −−→ p(x) is a homotopy equivalence, with p : X−−→X the covering projection. Define similarly the homotopy equivalence q − : T (ζ −1 ) −−→ X ; (x, s) −−→ p(x) . 6 The fibering obstructions of a CW band X with respect to a choice of generating covering translation ζ : X−−→X are the Whitehead torsions Φ+ (X) = τ (q + : T (ζ)−−→X) , Φ− (X) = τ (q − : T (ζ −1 )−−→X) ∈ W h(π1 (X)) defined using the canonical finite structures on T (ζ), T (ζ −1 ).

9 A chain complex band C fibres if it is simple chain equivalent to the algebraic mapping torus T + (h) = C(1 − zh : C[z, z −1 ]−−→C[z, z −1 ]) of a simple chain equivalence h : C−−→C. 18 3. 10 The fibering obstructions of an A[z, z −1 ]-module chain complex band C are the Whitehead torsions Φ+ (C) = τ (q + : T + (ζ −1 )−−→C) , Φ− (C) = τ (q − : T − (ζ)−−→C) ∈ W h1 (A[z, z −1 ]) , defined using the canonical finite structures on T + (ζ −1 ), T − (ζ −1 ) with ζ : C −−→ C ; x −−→ zx , ζ −1 : C −−→ C ; x −−→ z −1 x .

G. g. g. free A-module. g. projective A-modules P, Q, with 4 1. g. projective A-module R. The reduced projective class group is the quotient of K0 (A) K0 (A) = coker(K0 (Z)−−→K0 (A)) with K0 (Z) = Z −−→ K0 (A) ; n −−→ [An ] . g. g. projective A-module Q. g. free Amodule chain complex D with chain maps f : C−−→D, g : D−−→C and a chain homotopy gf 1 : C−−→C. An A-module chain complex C is finitely dominated if it admits a finite domination. g. projective A-modules P : . . −−→ 0 −−→ . . −−→ 0 −−→ Pn −−→ Pn−1 −−→ .

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