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Algebraic normal complexes 47 N with ΩP −→K from nn (K) (resp. Ωn (K)) the bordism group of maps X− dimensional geometric Poincar´e (resp. normal) complexes, with π = π1 (K) the fundamental group of K and n ≥ 5. 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) .

11 Given a ring with involution R and q = p (resp. g. projective (resp. g. g. 11, Bq (R) = B (A)q (R) the category of finite chain complexes in Aq (R), and Cq (R) ⊆ Bq (R) the subcategory of contractible complexes C, such that τ (C) = 0 ∈ K1 (R) for q = s. The quadratic L-groups of Λq (R) are the type q quadratic L-groups of R L∗ (Λq (R)) = Lq∗ (R) . Let ∗ : K0 (R) −−→ K0 (R) ; [P ] −−→ [P ∗ ] ∗ : K1 (R) −−→ K1 (R) ; τ (f : Rn−−→Rn ) −−→ τ (f ∗ : Rn −−→Rn ) be the induced involution of the reduced projective class group of 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 ∗ ) .

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