By A. A. Ranicki

ISBN-10: 0521055210

ISBN-13: 9780521055215

This booklet offers the definitive account of the purposes of this algebra to the surgical procedure class of topological manifolds. The relevant result's the identity of a manifold constitution within the homotopy form of a Poincaré duality house with an area quadratic constitution within the chain homotopy form of the common conceal. 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 worldwide quadratic duality buildings on chain complexes. The algebraic L-theory meeting map is used to provide a merely algebraic formula of the Novikov conjectures at the homotopy invariance of the better signatures; the other formula inevitably elements via this one.

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**Example text**

It follows that γ0 is constant on the equivalence classes of B and therefore induces a unique map γ : Q → X such that γπ = γ0 . We have γτi = αi (i = 1, 2), so γ is a morphism in D from (Q, (λ1 , λ2 )) to (X, (α1 , α2 )): A λ1 λ2 B1 / B2 τ1 τ2 α2 /Q γ α1 ! X. Let γ be another such morphism, and let b ∈ B. Then b ∈ Bi for some i ∈ {1, 2} and γ (¯b) = γ π(b) = γ τi (b) = αi (b) = γ(¯b), whence γ = γ and the claim follows. 2 Example (Pushouts exist in Top) Let λi : A → Bi (i = 1, 2) be two continuous maps.

30 We need to show that (Q, π) is an initial object. Let (X, α) be an object of D. Define γ : Q → X by γ(b) = α(b). Note that for (x1 , x2 ) ∈ R we have (x1 , x2 ) = (λ1 (a), λ2 (a)) for some a ∈ A, implying α(x1 ) = αλ1 (a) = αλ2 (a) = α(x2 ). Therefore, if b = c, then b ∼ c and, with notation as above, α(b) = α(x0 ) = α(x1 ) = · · · = α(xn ) = α(c) and γ is well-defined. Also, γπ = α, so γ is a morphism in D from (Q, π) to (X, α): A λ1 λ2 // 6 XO α B γ π ' Q. 1, so γ is the unique such morphism and the claim follows.

Composition of morphisms is as defined above. In particular, identity morphisms exist as required. 1 Example In this example, we define the category of linear representations of a group and show that it can be regarded as a functor category. Let G be a group and let K be a field. A homomorphism ρ : G → GL(V ), with V a vector space over K, is called a K-linear representation of G. The collection of all such is the class of objects of a category RepK (G). A morphism ϕ : ρ → ρ from the object ρ : G → GL(V ) to the object ρ : G → GL(V ) in this category is a linear map ϕ : V → V such that ϕρ(g) = ρ (g)ϕ for all g ∈ G.

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