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Iv) The normal L-groups N Ln (A) (n ∈ Z) are the cobordism groups of n-dimensional normal complexes in A . 5 Geometric normal (resp. Poincar´e) complexes and pairs determine algebraic normal (resp. Poincar´e) complexes and pairs. The methods of Ranicki [145] and Weiss [186] can be combined to associate to any (k − 1)-spherical fibration ν: X−−→BG(k) over a finite CW complex X a chain bundle in A (Z[π]w ) (cf. 4) σ ∗ (ν) = (C(X), γ) with X any regular covering of X such that the pullback ν˜: X−−→BG(k) is oriented, π the group of covering translations, C(X) the cellular Z[π]module chain complex of X, and w: π−−→{±1} a factorization of the orientation character w w1 (ν) : π1 (X) −−→ π −−→ {±1} .

Thus the free and projective hyperquadratic L-groups of R coincide L∗ (R) = L∗h (R) = L∗p (R) . Similarly, the hyperquadratic L-groups of the categories Ah (R) and Ap (R) coincide, being the 4-periodic versions of the hyperquadratic L-groups L∗ (R) Ln (Ah (R)) = Ln (Ap (R)) = lim Ln+4k (R) (n ∈ Z) , −→ k the direct limits being taken with respect to the double skew-suspension maps. 11 N L∗ (Aq (R)) ∼ = L∗ (Aq (R)) (q = h, p) to write N L∗ (R) = N L∗h (R) = N L∗p (R) = lim L∗+4k (R) . 11 for A = A (R) = Ah (R) is the algebraic analogue of the exact sequence of Levitt [92], Jones [80], Quinn [132] and Hausmann and Vogel [75] .

For a chain map f : C−−→C the components in each degree of the dual chain map T (f ): T (C )−−→T (C) are written f ∗ = T (f ) : C r = T (C )−r −−→ C r = T (C)−r . Given also a finite chain complex D in A define the abelian group chain complex C ⊗A D = HomA (T (C), D) . The duality isomorphism TC,D : C ⊗A D −−→ D ⊗A C is defined by TC,D = Σ(−)pq TCp ,Dq : (Cp ⊗A Dq )r −−→ (D ⊗A C)n , (C ⊗A D)n = p+q+r=n with inverse (TC,D )−1 = TD,C : D ⊗A C −−→ C ⊗A D . Hn (C ⊗A D) is the abelian group of chain homotopy classes of chain maps φ: C n−∗ −−→D in A .

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