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An element of Qn (C, γ) is an equivalence class of pairs ( φ ∈ (W % C)n , χ ∈ (W % C)n+1 ) such that d(φ) = 0 ∈ (W % C)n−1 , J(φ) − (φ0 )% (S n γ) = d(χ) ∈ (W % C)n , 40 Algebraic L-theory and topological manifolds with 1 + T : Qn (C, γ) −−→ Qn (C) ; (φ, χ) −−→ φ , H : Qn+1 (C) −−→ Qn (C, γ) ; χ −−→ (0, χ) . The addition in Qn (C, γ) is by (φ, χ) + (φ , χ ) = (φ + φ , χ + χ + ξ) , with ξs = φ0 (γs−n+1 )φ0 : C r −−→ Cn−r+s+1 (r, s ∈ Z) . Jγ is induced by a morphism of the simplicial abelian groups K(W % C) −−→K(W % C) associated to the abelian group chain complexes W % C, W % C by the Kan–Dold theorem, rather than by a chain map W % C−−→W % C.

Ii) For stable Λ the double skew-suspension functor defines an isomorphism of categories 2 S : {n-dimensional symmetric complexes in Λ} −−→ {(n + 4)-dimensional symmetric complexes in Λ} for all n ∈ Z by virtue of the stability of B and C. ) Similarly for quadratic and normal complexes, and also for pairs. 3 is stable. The L-groups of Λ(A) are the quadratic quadratic L-groups of the additive category with chain duality A L∗ (Λ(A)) = L∗ (A) L∗ (Λ(A)) = L∗ (A) . 5 the normal L-groups of Λ(A) are the symmetric L-groups of A N L∗ (Λ(A)) = L∗ (A) , since Q∗ (C) = 0 for any C (A)-contractible (= contractible) finite chain complex in A .

Use the standard complete (Tate) free Z[Z2 ]-module resolution of Z 1−T 1+T 1−T W : . . −−→ Z[Z2 ] −−→ Z[Z2 ] −−→ Z[Z2 ] −−→ Z[Z2 ] −−→ . . to define for any finite chain complex C in A the Z-module chain complex W % C = HomZ[Z2 ] (W , C ⊗A C) = HomZ[Z2 ] (W , HomA (T C, C)) . A chain θ ∈ (W % C)n is a collection of morphisms θ = {θs ∈ HomA (C n−r+s , Cr ) | r, s ∈ Z} , with the boundary d(θ) ∈ (W % C)n−1 given by d(θ)s = dθs + (−)r θs d∗ + (−)n+s−1 (θs−1 + (−)s T θs−1 ) : C n−r+s−1 −−→ Cr (r, s ∈ Z) .