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

TL;DR

This paper explores the relationship between generalized manifolds and Poincaré duality complexes, introducing Λ-PD structures to better understand their topological and homotopy properties.

Contribution

It develops the concept of Λ-PD structures on generalized manifolds and enlarges the class of complexes to recognize generalized manifolds via Gromov-Hausdorff metric.

Findings

Established conditions for Λ-PD structures on generalized manifolds

Constructed 2-patch spaces to analyze duality properties

Extended the class of PD complexes to include all generalized manifolds

Abstract

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