Download Algebraic L-theory and Topological Manifolds by A. A. Ranicki PDF

By A. A. Ranicki

This publication provides the definitive account of the functions of this algebra to the surgical procedure type of topological manifolds. The important result's the id of a manifold constitution within the homotopy form of a Poincaré duality area with an area quadratic constitution within the chain homotopy form of the common disguise. the adaptation among the homotopy forms of manifolds and Poincaré duality areas is pointed out with the fibre of the algebraic L-theory meeting map, which passes from neighborhood to international quadratic duality constructions on chain complexes. The algebraic L-theory meeting map is used to offer a merely algebraic formula of the Novikov conjectures at the homotopy invariance of the better signatures; the other formula unavoidably elements via this one.

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Ln (A) −−→ Ln (A) −−→ Ln (A) −−→ Ln−1 (A) −−→ . . 10 are just the hyperquadratic L-groups L∗ (R) of Ranicki [146, p. 137]. 11 The hyperquadratic L-groups L∗ (A) are isomorphic to the cobordism groups N L∗ (A) of normal complexes in A L∗ (A) ∼ = N L∗ (A) , so that there is defined an exact sequence 1+T J ∂ . . −−→ Ln (A) −−→ Ln (A) −−→ N Ln (A) −−→ Ln−1 (A) −−→ . . 9) of normal complexes. 8 (i) and its relative version relating (symmetric, quadratic) Poincar´e triads and normal pairs. 1] for algebraic Poincar´e triads.

3 (cf. 12 1+T J ∂ . . −−→ Ln (A) −−→ Ln (A) −−→ N Ln (A) −−→ Ln−1 (A) −−→ . . 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 ) { projective class be the induced involution of the reduced 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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