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G. free R-modules, with Whitehead torsion considerations. Given a finite chain complex C in A write C r = T (C)−r , Σn T (C) = C n−∗ . 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 .

9 (ii) . . −−→ Ln (Λ(R)) −−→ Ln (Γ(R, S)) −−→ Ln−1 (Λ(R, S)) −−→ Ln−1 (Λ(R)) −−→ . . The quadratic L-group Ln (R, S) is isomorphic to the cobordism group of n-dimensional quadratic Poincar´e complexes in the category of S-torsion R-modules of homological dimension 1. In particular, the boundary map for n = 0 ∂ : L0 (S −1 R) = L0 (Γ(R, S)) −−→ L0 (R, S) = L−1 (Λ(R, S)) sends the Witt class of a nonsingular quadratic form S −1 (M, λ, µ) over S −1 R induced from a quadratic form (M, λ, µ) over R to the Witt class of a nonsingular S −1 R/R-valued quadratic linking form ∂S −1 (M, λ, µ) = (∂M, ∂λ, ∂µ) , with ∂M = coker(λ: M −−→M ∗ ) , ∂λ : ∂M × ∂M −−→ S −1 R/R ; x −−→ (y −−→ x(z)/s) (x, y ∈ M ∗ , z ∈ M , s ∈ S , λ(z) = sy ∈ M ∗ ) .

The symmetric signature of Mishchenko [115] and Ranicki [145, §1] defines a map from geometric to symmetric Poincar´e bordism σ ∗ : ΩP −→ Ln (Z[π]) ; X −−→ σ ∗ (X) = (C(X), φ) . n (K) − The hyperquadratic signature of Ranicki [146, p. 619] defines a map from geometric to algebraic normal bordism σ ∗ : ΩN −→ Ln (Z[π]) ; X −−→ σ ∗ (X) = (C(X), φ, γ, χ) . n (K) − The signature maps fit together to define a map of exact sequences ... wΩ N n+1 (K) σ ˆ∗ ... wL n+1 u (Z[π]) σ∗ w L (Z[π]) wΩ σ∗ ∂ w L (Z[π]) n wΩ P n (K) n 1+T N n (K) u σ ˆ∗ w L (Z[π]) n J u w L (Z[π]) n w ...

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