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.
In this article the new properties of relative retracts in the context of relative homotopy are studied. The results of the studies are particularly applied to the characterization of connected and ...locally connected spaces and absolute neighborhood retracts.
In this paper, we present relative retracts and we can say that these are multilevel retracts which either retain given properties depending on the level or not. Some properties are constant and are ...present on every level. These properties are especially important in regard to the theory of coincidence. The class of relative retracts consists of retracts in the sense of Borsuk, multiretracts and many fundamental retracts.
In this paper we generalize the concept of absolute neighborhood retract by introducing the notion of absolute neighborhood multi-retract. Furthermore, the Lefschetz fixed point theorem for ...admissible maps defined on absolute neighborhood multi-retracts is proved.
In the paper titled “Bockstein basis and resolution theorems in extension theory” (Tonić, 2010 10), we stated a theorem that we claimed to be a generalization of the Edwards–Walsh resolution theorem. ...The goal of this note is to show that the main theorem from Tonić (2010) 10 is in fact equivalent to the Edwards–Walsh resolution theorem, and also that it can be proven without using Edwards–Walsh complexes. We conclude that the Edwards–Walsh resolution theorem can be proven without using Edwards–Walsh complexes.
We prove the following theorem.
Theorem. Let X be a nonempty compact metrizable space, let l1≤ l2≤ ⋅⋅⋅ be a sequence in N, and let X1 ⊂ X2⊂ ⋅⋅⋅ be a sequence of nonempty closed subspaces of X such ...that for each kN, dimZ/p Xk≤ lk. Then there exists a compact metrizable space Z, having closed subspaces Z1⊂ Z2⊂ ⋅⋅⋅, and a (surjective) cell-like map π:Z → X, such that for each kN,
(a) dim Zk≤ lk,
(b) π(Zk)=Xk, and
(c) π|Zk:Zk→ Xk is a Z/p-acyclic map.
Moreover, there is a sequence A1⊂ A2⊂⋅⋅⋅ of closed subspaces of Z such that for each k, dim Ak≤ lk, π|Ak:Ak → X is surjective, and for kN, Zk⊂ Ak and π|Ak:Ak→ X is a UVlk-1-map.
It is not required that X=∪∞k=1 Xk or that Z=∪∞k=1 Zk. This result generalizes the Z/p-resolution theorem of A. Dranishnikov and runs parallel to a similar theorem of S. Ageev, R. Jiménez, and the first author, who studied the situation where the group was Z.
J. L. Taylor constructed a cell-like map of a compactum X onto the Hilbert cube IN such that X is not cell-like. In this note, we point out a defect in the construction and show how to fix it.
We prove a generalization of the Edwards–Walsh Resolution Theorem:
Theorem
Let G be an abelian group with
P
G
=
P
, where
P
G
=
{
p
∈
P
:
Z
(
p
)
∈
Bockstein basis
σ
(
G
)
}
. Let
n
∈
N
and let K be ...a connected CW
-complex with
π
n
(
K
)
≅
G
,
π
k
(
K
)
≅
0
for
0
⩽
k
<
n
. Then for every compact metrizable space X with XτK (
i.e., with K an absolute extensor for X)
, there exists a compact metrizable space Z and a surjective map
π
:
Z
→
X
such that
(a)
π is cell-like,
(b)
dim
Z
⩽
n
, and
(c)
ZτK.
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