By A. A. Ranicki

This booklet provides the definitive account of the functions of this algebra to the surgical procedure category of topological manifolds. The crucial result's the id of a manifold constitution within the homotopy kind of a Poincaré duality house with a neighborhood quadratic constitution within the chain homotopy form of the common disguise. the variation among the homotopy different types 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 unavoidably components via this one.

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40 Algebraic L-theory and topological manifolds The addition in Qn (C, γ) is by (ϕ, χ) + (ϕ′ , χ′ ) = (ϕ + ϕ′ , χ + χ′ + ξ) , with ξs = ϕ0 (γs−n+1 )ϕ′0 : C r −−→ Cn−r+s+1 (r, s ∈ Z) . Jγ is induced by a morphism of the simplicial abelian groups K(W % C) −−→K(W % C) associated to the abelian group chain complexes W % C, W % C by the Kan–Dold theorem, rather than by a chain map W % C−−→W % C. For γ = 0 Jγ = J is induced by the chain map J: W % C−−→W % C and Q∗ (C, 0) = Q∗ (C). A map of chain bundles (f, b): (C, γ)−−→(C ′ , γ ′ ) induces morphisms of the twisted quadratic Q-groups (f, b)% : Qn (C, γ) −−→ Qn (C ′ , γ ′ ) ; (ϕ, χ) −−→ (f % ϕ, f % χ + (f ϕ0 )% (S n b)) .

Algebraic Poincare ( d∂C = ∂ϕ0 ∂ϕs dC 0 (−)r ϕ0 (−)r d∗C 35 ) : ∂Cr = Cr+1 ⊕ C n−r −−→ ∂Cr−1 = Cr ⊕ C n−r−1 , ( ) (−)n−r T ϕ1 (−)r(n−r−1) e = : 1 0 ∂C n−r−1 = C n−r ⊕ (T 2 C)r+1 −−→ ∂Cr = Cr+1 ⊕ C n−r , ( ) (−)n−r+s T ϕs+1 0 = : 0 0 ∂C n−r+s−1 = C n−r+s ⊕ (T 2 C)r−s+1 −−→ ∂Cr = Cr+1 ⊕ C n−r (s ≥ 1) and in the quadratic case ) ( dC (−)r (1 + T )ψ0 : d∂C = 0 (−)r d∗C ∂ψ0 ∂ψs ∂Cr = Cr+1 ⊕ C n−r −−→ ∂Cr−1 = Cr ⊕ C n−r+1 , ( ) 0 0 = : 1 0 ∂C n−r−1 = C n−r ⊕ (T 2 C)r+1 −−→ ∂Cr = Cr+1 ⊕ C n−r , ( ) (−)n−r−s−1 T ψs−1 0 = : 0 0 ∂C n−r−s−1 = C n−r−s ⊕ (T 2 C)r+s+1 −−→ ∂Cr = Cr+1 ⊕ C n−r (s ≥ 1) .

A chain θ ∈ (W % C)n is a collection of morphisms θ = {θs ∈ HomA (C n−r+s , Cr ) | r, s ∈ Z} , with the boundary d(θ) ∈ (W % C)n−1 given by d(θ)s = dθs + (−)r θs d∗ + (−)n+s−1 (θs−1 + (−)s T θs−1 ) : C n−r+s−1 −−→ Cr (r, s ∈ Z) . 1 (i) The hyperquadratic Q-groups of a ﬁnite chain complex C in A are deﬁned by Qn (C) = Hn (W % C) (n ∈ Z) . (ii) A chain map f : C−−→D of ﬁnite chain complexes in A induces a Zmodule chain map f % : W % C −−→ W % D via the Z[Z2 ]-module chain map f ⊗ f : C ⊗A C−−→D ⊗A D .