20.500.12556/RUL-109104 The relationship of generalized manifolds to Poincaré duality complexes and topological manifolds 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. generalized manifold Poincaré duality complex ENR 2-patch space resolution obstruction controlled surgery controlled structure set Lq-surgery Wall obstruction cell-like map Gromov-Hausdorff metric true false true Angleški jezik Ni določen Članek v reviji 2019-08-22 08:12:41 2019-08-22 08:12:44 2022-08-21 03:39:59 0000-00-00 00:00:00 2018 0 0 Str. 126-141 Vol. 239 April 2018 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 515.16 0166-8641 10.1016/j.topol.2018.02.024 18273369 26538752 RAZ_Hegenbarth_Friedrich_i2018.pdf RAZ_Hegenbarth_Friedrich_i2018.pdf 1 33E326AC31AC289ACBD7313E8B36CB62 8aade5e0353c93d0816f52dfd0811313f7cb8b2be3fbc00ac9a10a1e3cdf35a4 c6590daa-a1b6-11eb-a523-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=120548 Pedagoška fakulteta Fakulteta za matematiko in fiziko 0 0 0