The relationship of generalized manifolds to Poincaré duality complexes and topological manifoldsHegenbarth, Friedrich (Avtor) Repovš, Dušan (Avtor) generalized manifoldPoincaré duality complexENR2-patch spaceresolution obstructioncontrolled surgerycontrolled structure setLq-surgeryWall obstructioncell-like mapGromov-Hausdorff metricThe 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.20182019-08-22 08:12:41Članek v reviji109104sl