The total surgery obstruction revisited

Abstract

The total surgery obstruction of a finite n-dimensional Poincare complex X is an element s(X) of a certain abelian group S_n (X) with the property that for n >= 5 we have s(X) = 0 if and only if X is homotopy equivalent to a closed n-dimensional topological manifold. The definitions of S_n (X) and s(X) and the property are due to Ranicki in a combination of results of two books and several papers. In this paper we present these definitions and a detailed proof of the main result so that they are in one place and we also add some of the details not explicitly written down in the original sources.

71 pages, revised version, section 14 substantially expanded

References in corpus (3)

• Foundations of algebraic surgery
• The structure set of an arbitrary space, the algebraic surgery exact sequence and the total surgery obstruction
• On hyperbolic groups with spheres as boundary

Cited by in corpus (8)

• Commutativity properties of Quinn spectra
• The family signature theorem
• On the stable Cannon Conjecture
• On Steenrod $\mathbb{L}$-homology, generalized manifolds, and surgery
• Generalized manifolds, normal invariants, and $\mathbb{L}$-homology
• Manifolds homotopy equivalent to certain torus bundles over lens spaces
• L-homology on ball complexes and products
• Chain duality for categories over complexes