The relationship of generalized manifolds to Poincar\'{e} duality complexes and topological manifolds

The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincar\'{e} 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 Poincar\'{e} duality with coefficients in the group ring $\Lambda$ ($\Lambda$-complexes). Standard homology theory implies that $X$ is a $\mathbb{Z}$-PD complex. Therefore by Browder's theorem, $X$ has a Spivak normal fibration which in turn, determines a Thom class of the pair $(N,\partial N)$ of a mapping cylinder neighborhood of $X$ in some Euclidean space. Then $X$ satisfies the $\Lambda$-Poincar\'{e} duality if this class induces an isomorphism with $\Lambda$-coefficients. Unfortunately, the proof of Browder's theorem gives only isomorphisms with $\mathbb{Z}$-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