Algebraic L theory and topological manifolds by Ranicki

By Ranicki

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1 (i) A subcategory C ⊆ B (A) is closed if it is a full additive subcategory such that the algebraic mapping cone C(f ) of any chain map f : C−−→D in C is an object in C. (ii) A chain complex C in A is C-contractible if it belongs to C. A chain map f : C−−→D in A is a C-equivalence if the algebraic mapping cone C(f ) is C-contractible. symmetric (C, φ) (iii) An n-dimensional complex in A is C-contractible quadratic (C, ψ) if the chain complexes C and C n−∗ are C-contractible. symmetric (C, φ) (iv) An n-dimensional complex in A is C-Poincar´e if quadratic (C, ψ) the chain complex ∂C = S −1 C(φ0 : C n−∗ −−→C) ∂C = S −1 C((1 + T )ψ0 : C n−∗ −−→C) is C-contractible.

N n (K) − −→ k The normal signature determines the quadratic signature σ∗ = ∂σ ∗ : ΩN −→ lim Ln+4k−1 (Z[π]) = Ln−1 (Z[π]) . n (K) − −→ k There is also a twisted version for a double covering K w −−→K, with the w-twisted involution on Z[π], and the bordism groups Ω∗ (K, w) of maps X−−→K such that the pullback X w −−→X is the orientation double cover. 14 (i) Let R be a ring with involution, and let (B, β) be a chain bundle over R, with B a free R-module chain complex (not necessarily finite or finitely generated).

Ii) An (n+1)-dimensional normal pair (f : C−−→D, (δθ, θ)) in A is an (n+1)dimensional symmetric pair (f : C−−→D, (δφ, φ)) in A together with a map of chain bundles (f, b): (C, γ)−−→(D, δγ) and chains χ ∈ (W % C)n+1 , δχ ∈ (W % D)n+2 such that J(φ) − (φ0 )% (S n γ) = d(χ) ∈ (W % C)n , J(δφ) − (δφ0 , φ0 )% (S n δγ) + f % (χ − (φ0 )% (S n b)) = d(δχ) ∈ (W % D)n+1 , with (δθ, θ) short for ((δφ, δγ, δχ), (φ, γ, χ)). (iii) A map of n-dimensional normal complexes in A (f, b) : (C, φ, γ, χ) −−→ (C , φ , γ , χ ) is a bundle map (f, b): (C, γ)−−→(C , γ ) such that (f, b)% (φ, χ) = (φ , χ ) ∈ Qn (C , γ ) .

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