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.
Keywords
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;
Data
Language: |
English |
Year of publishing: |
2018 |
Typology: |
1.01 - Original Scientific Article |
Organization: |
UL FMF - Faculty of Mathematics and Physics |
UDC: |
515.16 |
COBISS: |
18273369
|
ISSN: |
0166-8641 |
Views: |
539 |
Downloads: |
342 |
Average score: |
0 (0 votes) |
Metadata: |
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Other data
Type (COBISS): |
Article |
Pages: |
str. 126-141 |
Issue: |
ǂVol. ǂ239 |
Chronology: |
April 2018 |
DOI: |
10.1016/j.topol.2018.02.024 |
ID: |
11206334 |