The structure set of an arbitrary space, the algebraic surgery exact sequence and the total surgery obstruction

Abstract

The algebraic theory of surgery gives a necessary and suffcient chain level condition for a space with n-dimensional Poincare duality to be homotopy equivalent to an n- dimensional topological manifold. A relative version gives a necessary and suffcient chain level condition for a simple homotopy equivalence of n-dimensional topological manifolds to be homotopic to a homeomorphism. The chain level obstructions come from a chain level interpretation of the fibre of the assembly map in surgery. The assembly map A : Hn(X;L.) -> Ln(Z[Pi 1 | (X)]) is a natural transformation from the generalized homology groups of a space X with coefficients in the 1-connective simply-connected surgery spectrum L. to the non-simply-connected surgery obstruc- tion groups L.(Z[Pi 1 | (X)]). The (Z;X)-category has objects based f.g. free Z-modules with an X-local structure. The assembly maps A are induced by a functor from the (Z;X)-category to the category of based f.g. free Z[Pi 1 | (X)]-modules. The generalized homology groups H.(X;L.) are the cobordism groups of quadratic Poincare complexes over (Z;X). The relative groups S.(X) in the algebraic surgery exact sequence of X ... -> Hn(X;L.) A