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- W1952314710 abstract "The algebraic theory of surgery on chain complexes C with Poincare duality H∗(C) ∼= Hn−∗(C) describes geometric surgeries on the chain level. The algebraic effect of a geometric surgery on an n-dimensional manifold M is an algebraic surgery on the n-dimensional symmetric Poincare complex (C,φ) over Z[π1(M)] with the homology of the universal cover M H∗(C) = H∗(M) . The algebraic effect of a geometric surgery on an n-dimensional normal map (f, b) : M → X is an algebraic surgery on the kernel n-dimensional quadratic Poincare complex (C,ψ) over Z[π1(X)] with homology H∗(C) = K∗(M) = ker(f∗ : H∗(M)→ H∗(X)) . For n > 4 and i-connected (f, b) with 2i ≤ n there is a one-one correspondence between geometric surgeries on (f, b) killing elements x ∈ Ki(M) and algebraic surgeries on (C,ψ) killing x ∈ Hi(C). The Wall surgery obstruction of an n-dimensional normal map (f, b) : M → X σ∗(f, b) ∈ Ln(Z[π1(X)]) was originally defined by first making (f, b) [n/2]-connected by geometric surgery below the middle dimension, using forms for even n and automorphisms of forms for odd n. The algebraic theory of surgery identifies σ∗(f, b) with the cobordism class of the kernel quadratic Poincare complex (C,ψ), so the algebraic surgery obstruction has the same formulation for odd and even n. The identification is used for n = 2i (resp. 2i + 1) to find a representative form (resp. automorphism) without preliminary geometric surgeries." @default.
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- W1952314710 date "2001-11-30" @default.
- W1952314710 modified "2023-09-25" @default.
- W1952314710 title "Foundations of algebraic surgery" @default.
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