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The relationship of generalized manifolds to Poincaré duality complexes and topological manifoldsHegenbarth, Friedrich, 1940- ; Repovš, Dušan, 1954-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.Source: Topology and its Applications. - ISSN 0166-8641 (Vol. 239, April 2018, str. 126-141)Type of material - article, component part ; adult, seriousPublish date - 2018Language - englishCOBISS.SI-ID - 18273369
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Hegenbarth, Friedrich, 1940- | 28253 |
Repovš, Dušan, 1954- | 07083 |
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