We prove a K-resolution theorem for simply connected CW-\linebreak complexes K in extension theory in the class of metrizable compacta X. This means that if K is a connected CW-complex, G is an ...abelian group, n\in \mathbb N _{\geq 2}, G=\pi _{n}(K), \pi _{k}(K)=0 for 0\leq k<n, and \operatorname{extdim} X\leq K (in the sense of extension theory, that is, K is an absolute extensor for X), then there exists a metrizable compactum Z and a surjective map \pi :Z\rightarrow X such that: (a) \pi is G-acyclic, (b) \dim Z\leq n+1, and (c) \operatorname{extdim} Z\leq K. \noindent This implies the G-resolution theorem for arbitrary abelian groups G for cohomological dimension \dim _{G} X\leq n when n\in \mathbb N_{\geq 2}. Thus, in case K is an Eilenberg-MacLane complex of type K(G,n), then (c) becomes \dim _{G} Z\leq n. If in addition \pi _{n+1}(K)=0, then (a) can be replaced by the stronger statement, (aa) \pi is K-acyclic. To say that a map \pi is K-acyclic means that for each x\in X, every map of the fiber \pi ^{-1}(x) to K is nullhomotopic.
Suppose that
X is a nonempty compact metrizable space and
X
1⊂
X
2⊂⋯ is a sequence of nonempty closed subspaces such that for each
k∈
N
,
dim
Z
X
k⩽k<∞
. We show that there exists a compact ...metrizable space
Z, having closed subspaces
Z
1⊂Z
2⊂⋯
, and a surjective cell-like map
π
:Z→X
, such that for each
k∈
N
,
(a)
dim
Z
k
⩽
k,
(b)
π(
Z
k
)=
X
k
, and
(c)
π|Z
k
:Z
k→X
k
is a cell-like map.
Moreover, there is a sequence
A
0⊂
A
1⊂⋯ of closed subspaces of
Z such that for each
k,
Z
k
⊂
A
k
, dim
A
k
⩽
k,
π|A
k
:A
k→X
is surjective, and for
k∈
N
,
π|A
k
:A
k→X
is a UV
k−1
-map.
All abstract reflection groups act geometrically on non-positively curved geodesic spaces. Their natural space at infinity, consisting of (bifurcating) infinite geodesic rays emanating from a fixed ...base point, is called a boundary of the group.
We will present a condition on right-angled Coxeter groups under which they have topologically homogeneous boundaries. The condition is that they have a nerve which is a connected closed orientable PL manifold.
In the event that the group is generated by the reflections of one of Davis’ exotic open contractible n-manifolds (n⩾4), the group will have a boundary which is a homogeneous cohomology manifold. This group boundary can then be used to equivariantly Z-compactify the Davis manifold.
If the compactified manifold is doubled along the group boundary, one obtains a sphere if n⩾5. The system of reflections extends naturally to this sphere and can be augmented by a reflection whose fixed point set is the group boundary. It will be shown that the fixed point set of each extended original reflection on the thus formed sphere is a tame codimension-one sphere.
We study the relation between (topological) inner metrics and Riemannian metrics on smoothable manifolds. We show that inner metrics on smoothable manifolds can be approximated by Riemannian metrics. ...More generally, if f: M → X is a continuous surjection from a smooth manifold to a compact metric space with f-1(x) connected for every x ∈ X, then there is a metric d on X and a sequence of Riemannian metrics {ψi} on M so that (M, ψi) converges to (X, d) in Gromov-Hausdorff space. This is used to obtain a (fixed) contractibility function ρ and a sequence of Riemannian manifolds with ρ as contractibility function so that$\lim(M, \psi_i)$is infinite dimensional. Using results of Dranishnikov and Ferry, this also gives examples of nonhomeomorphic manifolds M and N and a contractibility function ρ so that for every$\varepsilon > 0$there are Riemannian metrics φεand ψεon M and N so that (M, φε) and (N, ψε) have contractibility function ρ and$d_{GH}((M, \phi_\varepsilon), (N, \psi_\varepsilon)) < \varepsilon$.
We introduce new classes of compact metric spaces: Cannon—Štan'ko, Cainian, and nonabelian compacta. In particular, we investigate compacta of cohomological dimension one with respect to certain ...classes of nonabelian groups, e.g., perfect groups. We also present a new method of constructing compacta with certain extension properties.
This paper introduces a notion of strongly hereditarily aspherical compacta and gives a sufficient condition for an inverse limit to have this property. The main result shows that cell-like maps ...defined on strongly hereditarily aspherical compact metric spaces cannot raise dimension. It suggests why 2-dimensional examples of this sort are plentiful and then sets forth 3-dimensional and 4-dimensional examples.
Compact Hausdorff spaces $X$ of cohomological dimension $\dim_Z X \leq n$ are characterized as cell-like images of compact Hausdorff spaces $Z$ with covering dimension $\dim Z \leq n$. The proof ...essentially uses the newly developed techniques of approximate inverse systems.
Let f: X' → X be a cell-like map between metric spaces and set$N_f = \{x \in X: f^{-1}(x) \neq \operatorname{point} \}$. Even if$N_f \subset \bigcup^\infty_{n = 1} B_n$, where each Bnis closed and ...each f∣ f-1(Bn): f-1(Bn) → Bnis hereditary shape equivalence, f may not be a hereditary shape equivalence. Conditions are placed on the Bn's to assure that f is a hereditary shape equivalence. For example, if$N_f \subset \bigcup^\infty_{n = 1} B_n$, where Bnis closed for each n = 1, 2,..., f∣ f-1(Bn): f-1(Bn) → Bnis a hereditary shape equivalence, and Bnhas arbitrary small neighborhoods whose boundaries miss$\bigcup^\infty_{i = 1} B_i$, then f is a hereditary shape equivalence. An immediate consequence is that if$\{B_n \}^\infty_{n = 1}$is a pairwise disjoint null-sequence and each f∣ f-1(Bn) is a hereditary shape equivalence, then f is a hereditary shape equivalence. Previously G. Kozlowski showed that if { f-1(Bn)}∞
n = 1is a pairwise disjoint null-sequence and each f∣ f-1(Bn) is a hereditary shape equivalence, then f is a hereditary shape equivalence, which can be obtained as an immediate corollary of one of our results.
We consider the existence of cell-like maps f: In→ X such that no nonempty open subset of X is contractible in X. From the Taylor Example, it is easy to construct such a map for n = ∞. We show that ...there exists such a map for some finite n if (and only if) there exists a dimension raising cell-like map of a compactum.
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