Cobordisms of quadratic chain complexes

A fundamental problem in surgery theory is to decide whether a given finite CW-complex X of dimension n>=5 satisfying Poincare duality is homotopy equivalent to a topological manifold. In the classical surgery theory due to Browder-Novikov-Sullivan-Wall this question is answered by a two-stage obstruction method.

Ranicki came up with a one-stage method where the answer to the question is yes if and only if a certain element, which he called ``the total surgery obstruction’’ TSO(X), is zero in a certain abelian group S_n (X). This group is related to the so-called ``assembly map’’ which yields a new approach for answering the fundamental question in certain situations.

The method of Ranicki is called ``algebraic surgery’’ and its crucial component is the notion of a cobordism of quadratic chain complexes. In our project we aim at simplifying the proof of the main theorem about the TSO. The original proof makes heavy use of orientations of a Poincare complex with respect to various L-theory spectra. We want to replace this by employing certain Mayer-Vietoris type of argument.

In the talk I will present an overview, some background of Ranicki’s method and main ideas in the modified proof.