Foundations of algebraic surgery

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 eect of a geometric surgery on an n-dimensional manifold M is an algebraic surgery on the n-dimensional symmetric Poincare complex (C;) over Z[1(M)] with the homology of the universal cover fM H(C) = H(fM): The algebraic eect 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; ) over Z[1(X)] with homology H(C) = K(M) = ker(f : H(fM)! H ( eX)): 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 2 Ki(M) and algebraic surgeries on (C; ) killing x 2 Hi(C). The Wall surgery obstruction of an n-dimensional norma