We apply the Gromov–Hausdorff metric
d
G
for characterization of certain generalized manifolds. Previously, we have proven that with respect to the metric
d
G
,
generalized
n
-manifolds are limits of ...spaces which are obtained by gluing two topological
n
-manifolds by a controlled homotopy equivalence (the so-called 2-patch spaces). In the present paper, we consider the so-called
manifold-like
generalized
n
-manifolds
X
n
,
introduced in 1966 by Mardeić and Segal, which are characterized by the existence of
δ
-mappings
f
δ
of
X
n
onto closed manifolds
M
δ
n
,
for arbitrary small
δ
>
0
, i.e., there exist onto maps
f
δ
:
X
n
→
M
δ
n
such that for every
u
∈
M
δ
n
,
f
δ
-
1
(
u
)
has diameter less than
δ
. We prove that with respect to the metric
d
G
,
manifold-like generalized
n
-manifolds
X
n
are limits of topological
n
-manifolds
M
i
n
. Moreover, if topological
n
-manifolds
M
i
n
satisfy a certain local contractibility condition
M
(
ϱ
,
n
)
, we prove that generalized
n
-manifold
X
n
is resolvable.
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 Λ (Λ-complexes). Standard homology theory implies that X is a 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,∂N) of a mapping cylinder neighborhood of X in some Euclidean space. Then X satisfies the Λ-Poincaré duality if this class induces an isomorphism with Λ-coefficients. Unfortunately, the proof of Browder's theorem gives only isomorphisms with Z-coefficients. It is also not very helpful that X is homotopy equivalent to a finite complex K, because K is not automatically a Λ-PD complex. Therefore it is convenient to introduce Λ-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 Λ-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.
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