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

Abstract

The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincaré 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 Poincaré 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 $\Lambda$-Poincaré 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