Algebraic L-theory and Topological Manifolds

The Browder-Novikov-Sullivan-Wall surgery theory emerged in the 1960s as the main technique for classifying high-dimensional topological manifolds, using the algebraic L-theory of quadratic forms to relate the geometric properties of manifolds and the Poincare duality between homology and cohomology. The abstract theory of quadratic forms on chain complexes developed by the author provides a comprehensive framework for understanding the connection between quadratic forms and manifolds. This book presents the definitive account of the applications of this algebra to the surgery classification of topological manifolds. The central result is the identification of manifold structure in the homotopy type of a Poincare duality space with a local quadratic structure in the chain homotopy type of the universal cover. The difference between the homotopy types of manifolds and Poincare duality spaces is identified with the fibre of the algebraic L-theory assembly map, which passes from local to global quadratic duality structures on chain complexes. The algebraic L-theory assembly map is used to give a purely algebraic formulation of the Novikov conjectures on the homotopy invariance of the higher signatures; any other formulation necessarily factors through this one. The book is designed as an introduction to the subject, accessible to graduate students in topology; no previous acquaintance with surgery theory is assumed, and every algebraic concept is justified by its occurrence in topology. However, research mathematicians applying ideas from algebraic and geometric topology in areas such as number theory or algebra will also benefit from this authoritative account.