The relationship of generalized manifolds to Poincare duality complexes and topological manifolds

The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincare duality complexes (PD complexes). The problem is that an arbitrary generalized manifold X is always an ENR space, but it is not necessarily a complex. Moreover, finite PD complexes require the Poincare duality with coefficients in the group ring Lambda (Lambda-complexes). Standard homology theory implies that X is a Z-PD complex. Therefore by Browder's theorem, X has a Spivak normal fibration which in turn, determines a Thom class of the pair (N, partial derivative N) of a mapping cylinder neighborhood of X in some Euclidean space. Then X satisfies the Lambda-Poincare duality if this class induces an isomorphism with Lambda-coefficients. Unfortunately, the proof of Browder's theorem gives only isomorphisms with Z-coefficients. It is also not very helpful that X is homotopy equivalent to a finite complex K, because K is not automatically a Lambda-PD complex. Therefore it is convenient to introduce Lambda-PD structures. To prove their existence on X, we use the construction of 2-patch spaces and some fundamental results of Bryant, Ferry, Mio, and Weinberger. Since the class of all Lambda-PD complexes does not contain all generalized manifolds, we appropriately enlarge this class and then describe (i.e. recognize) generalized manifolds within this enlarged class in terms of the Gromov-Hausdorff metric. (C) 2018 Elsevier B.V. All rights reserved.