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K −−→ K −−→ K −−→ K −−→ . . , β = 1) is universal. The quadratic Witt group L2∗ (K) is detected by the Arf invariant, and the symmetric Witt group L2∗ (K) is detected by the rank (mod 2), with isomorphisms Q2∗+1 (B, β) = K/{x + x2 | x ∈ K} −−→ L2∗ (K) ; a −−→ K ⊕ K , a 1 0 1 , Q2∗ (B, β) = {x ∈ K | x + x2 = 0} = Z2 −−→ N L2∗ (K) = L2∗ (K) ; 1 −−→ (K, 1) and L2∗+1 (K) = L2∗+1 (K) = 0 . In particular, this applies to K = F2 . 15 The algebraic mapping cylinder of a map of n-dimensional normal complexes in A (f, b) : (C , φ , γ , χ ) −−→ (C, φ, γ, χ) is an (n + 1)-dimensional normal pair in A M (f, b) = ( (f 1): C ⊕ C− →C , ((δφ, γ, δχ), (φ ⊕ −φ, γ ⊕ −γ, χ ⊕ −χ)) , b ⊕ 0 ) , 2.

It is immediate from the identity S −1 C(φ0 : C n−∗ −−→C) ∂C = S −1 C((1 + T )ψ0 : C n−∗ −−→C) symmetric complex quadratic ∂(C, φ) is contractible. 15 The homotopy equivalence classes of n-dimensional symmetric complexes in A are in one–one correspondence with the homoquadratic symmetric topy equivalence classes of n-dimensional Poincar´e pairs in quadratic A. 4], the special case A = Ap (R). (C, φ) symmetric Given an n-dimensional complex in A define the quadratic (C, ψ) symmetric n-dimensional Poincar´e pair quadratic δ∂(C, φ) δ∂(C, ψ) = pC = projection : ∂C −−→ C n−∗ , Conversely, given an n-dimensional (0, ∂φ) (0, ∂ψ) .

10 for (iii). e. the pair is B-contractible, C-Poincar´e and the boundary is D-Poincar´e). Define inverse isomorphisms Ln−1 (A, C, D) −−→ Ln (F ) ; (C, φ) −−→ ((C, φ), (C−−→0, (0, φ))) , Ln (F ) −−→ Ln−1 (A, C, D) ; (f : C−−→D, (δφ, φ)) −−→ (C , φ ) with (C , φ ) the (n − 1)-dimensional symmetric complex in (A, C, D) obtained from (C, φ) by algebraic surgery on the n-dimensional symmetric pair (f : C−−→D, (δφ, φ)) in (A, B, C). (ii) As for (i), with symmetric replaced by quadratic. 9 (ii) to obtain a quadratic structure on the effect of surgery on a normal pair.

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Algebraic L theory and topological manifolds by Ranicki


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