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Algebraic L theory and topological manifolds by Ranicki

By Ranicki

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Extra resources for Algebraic L theory and topological manifolds

Example text

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. 9 are generalizations of the localization exact sequence of Ranicki [146] (cf.

L (Λ)   F . . −−→ Ln (Λ) −−→ Ln (Λ ) −−→ Ln (F ) −−→ Ln−1 (Λ) −−→ . .    F  . . −−→ N Ln (Λ) −−→ N Ln (Λ ) −−→ N Ln (F ) −−→ N Ln−1 (Λ) −−→ . . Proof For any objects M, N in A define a chain map of abelian group chain complexes F (M, N ) : M ⊗A N −−→ F (M ) ⊗A F (N ) ; (φ: T (M )−−→N ) −−→ (F (φ)G(M ): T F (M )−−→F T (M )−−→F (N )) which is compatible with the duality equivalences. An n-dimensional symmetric complex (C, φ) in Λ induces an n-dimensional symmetric complex (F (C), F (φ)) in Λ .

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) .

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