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

By A. A. Ranicki

This publication provides the definitive account of the functions of this algebra to the surgical procedure class of topological manifolds. The important result's the identity of a manifold constitution within the homotopy form of a Poincaré duality house with a neighborhood quadratic constitution within the chain homotopy form of the common disguise. the variation 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 only algebraic formula of the Novikov conjectures at the homotopy invariance of the better signatures; the other formula inevitably components via this one.

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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. 13 below), and of the relative L-theory exact sequences of Vogel [174]. 3. 9 (iii) for the triple (B, B, C) can be written as 1+T J ∂ . . −−→ Ln (Λ) −−→ N Ln (Λ) −−→ N Ln (Λ) −−→ Ln−1 (Λ) −−→ . . with Λ = (A, B, B). 5 then N L∗ (Λ) can be replaced by L∗ (Λ). 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.

Symmetric (ii) A map of n-dimensional complexes in A quadratic { f : (C, ϕ) −−→ (C ′ , ϕ′ ) f : (C, ψ) −−→ (C ′ , ψ ′ ) is a chain map f : C−−→C ′ such that { % f (ϕ) = ϕ′ ∈ Qn (C ′ ) f% (ψ) = ψ ′ ∈ Qn (C ′ ) . The map is a homotopy equivalence if f : C−−→C ′ is a chain equivalence. ´ complexes 1. 6 is only required to be finite, and not n-dimensional as in Ranicki [144]. Let f : C−−→D be a chain map of finite chain complexes in A . An (n + 1)cycle { (δϕ, ϕ) ∈ C(f % : W % C−−→W % D)n+1 (δψ, ψ) ∈ C(f% : W% C−−→W% D)n+1 { ϕ ∈ (W % C)n is an n-cycle together with a collection ψ ∈ (W% C)n { δϕ = {δϕs ∈ (D ⊗A D)n+1+s | s ≥ 0} δψ = {δψs ∈ (D ⊗A D)n+1−s | s ≥ 0} such  that dD⊗A D (δϕs ) + (−)n+s (δϕs−1 + (−)s T δϕs−1 )     + (−)n (f ⊗A f )(ϕs ) = 0 ∈ (D ⊗A D)n+s  dD⊗A D (δψs ) + (−)n−s (δψs+1 + (−)s+1 T δψs+1 )    + (−)n (f ⊗A f )(ψs ) = 0 ∈ (D ⊗A D)n−s The (n + 1)-cycle { (δϕ0 , ϕ0 ) ∈ C(f ⊗A f : C ⊗A C−−→D ⊗A D)n+1 ((1 + T )δψ0 , (1 + T )ψ0 ) ∈ C(f ⊗A f : C ⊗A C−−→D ⊗A D)n+1 determines a chain map { (δϕ0 , ϕ0 ) : Dn+1−∗ −−→ C(f ) (1 + T )(δψ0 , ψ0 ) : Dn+1−∗ −−→ C(f ) with ) (  δϕ0   : Dn+1−r −−→ C(f )r = Dr ⊕ Cr−1 (δϕ0 , ϕ0 ) =  ∗  ϕ f 0  ( ) (1 + T )δψ0  ((1 + T )δψ0 , (1 + T )ψ0 ) =   (1 + T )ψ0 f ∗   : Dn+1−r −−→ C(f )r = Dr ⊕ Cr−1 .

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