Local Poincare Duality

Abstract

This work is part of a project which aims to describe algebraic structures on the chains and cochains of closed manifolds that characterize those manifolds up to homeomorphism. Once knows from the rational homotopy theory of Quillen and Sullivan, and from the more recent work of Mandell, that the homotopy type of a simply connected space is determined by algebraic structure on the cochains of the space. There is an informational gap, however, between the homotopy type of a manifold and its homeomorphism type, as there are nonhomeomorphic manifolds which have the same homotopy type. Moreover, the surgery theory of Browder, Novikov, Sullivan, and Wall tells us that not every space satisfying Poincar'e duality has the homotopy type of a manifold.We represent a Poincare duality space as a chain complex with a fixed basis satisfying certain axioms. We use the combinatorial data of the basis to to define an algebraic notion of locality, which we use to describe manifold structures. Our main result is that in dimensions greater than 4, simply-connected closed topological manifold structures in the homotopy type of a suitable based chain complex are in one-to-one correspondence with choices of local inverse to the Poincare duality map up to algebraic bordism. The proof relies on Ranicki's algebraic formulation of surgery theory.We expect that the theory of based chain complexes and algebraic locality developed here can be extended to encode the E-infinity algebra structure on the cochains of a space.