Get Algebraic L-theory and Topological Manifolds PDF

By A. A. Ranicki

ISBN-10: 0521055210

ISBN-13: 9780521055215

ISBN-10: 0521420245

ISBN-13: 9780521420242

This e-book offers the definitive account of the purposes of this algebra to the surgical procedure category of topological manifolds. The imperative result's the identity of a manifold constitution within the homotopy kind of a Poincaré duality house with a neighborhood quadratic constitution within the chain homotopy kind of the common hide. 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 worldwide quadratic duality buildings on chain complexes. The algebraic L-theory meeting map is used to provide a simply algebraic formula of the Novikov conjectures at the homotopy invariance of the better signatures; the other formula unavoidably components via this one.

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Extra info for Algebraic L-theory and Topological Manifolds

Sample text

An embedding X ⊂ S n+k (k large) determines the normal structure (νX , ρX ) of Spivak [164], so that X is an n-dimensional geometric normal complex. The n-dimensional symmetric Poincar´e complex in A (Z[π]w ) σ ∗ (X) = (C(X), ϕ) is such that Jσ ∗ (X) = σ ∗ (X, νX , ρX ). 6 (i) An n-dimensional symmetric complex (C, ϕ) in A has a normal structure (γ, χ) if and only if the boundary (n − 1)-dimensional symmetric Poincar´e complex ∂(C, ϕ) admits a quadratic refinement. (ii) There is a natural one–one correspondence between the homotopy equivalence classes of n-dimensional symmetric Poincar´e complexes (C, ϕ) in A and those of n-dimensional normal complexes (C, ϕ, γ, χ) with ϕ0 : C n−∗ −−→ C a chain equivalence.

As in §§1,2 let A be an additive category with chain duality, and let B (A) be the additive category of finite chain complexes in A and chain maps. 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.

R ∈ Z) is finite if Cr = 0 for all but a finite number of r ∈ Z. C is n-dimensional if Cr = 0 unless 0 ≤ r ≤ n. The algebraic mapping cone of a chain map f : C−−→D in A is the chain complex C(f ) in A defined by ( ) dD (−)r−1 f dC(f ) = : 0 dC C(f )r = Dr ⊕ Cr−1 −−→ C(f )r−1 = Dr−1 ⊕ Cr−2 . Inclusion and projection define chain maps D −−→ C(f ) , C(f ) −−→ SC with SC the suspension chain complex defined by dSC = dC : SCr = Cr−1 −−→ SCr−1 = Cr−2 . The total complex of a double complex C∗,∗ in A with differentials d′ : Cp,q −−→ Cp−1,q , d′′ : Cp,q −−→ Cp,q−1 (p, q ∈ Z) such that d′ d′ = 0, d′′ d′′ = 0, d′ d′′ = d′′ d′ is the chain complex C in A defined by ∑ ∑ dC = (d′′ + (−)q d′ ) : Cr = Cp,q −−→ Cr−1 .

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


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