On the asymptotic extension dimension
We introduce an asymptotic counterpart of the extension dimension defined by Dranishnikov. The main result establishes the relationship between the asymptotic extensional dimension of a proper metric space and the extension dimension of its Higson corona.
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Інститут математики НАН України
2010
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irk-123456789-1662922020-02-20T01:26:10Z On the asymptotic extension dimension Repovs, D. Zarichnyi, М. Статті We introduce an asymptotic counterpart of the extension dimension defined by Dranishnikov. The main result establishes the relationship between the asymptotic extensional dimension of a proper metric space and the extension dimension of its Higson corona. Метою статті є введення асимптотичного аналога розширеної вимірності, що визначена Дранішніковим. Основний результат полягає у встановленні співвідношення між асимптотичною розширеною вимірністю власного метричного простору та розширеною вимірністю його корони Хігсона. 2010 Article On the asymptotic extension dimension / D. Repovs, M. Zarichnyi // Український математичний журнал. — 2010. — Т. 62, № 11. — С. 1523–1530. — Бібліогр.: 9 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166292 515.12 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Repovs, D. Zarichnyi, М. On the asymptotic extension dimension Український математичний журнал |
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We introduce an asymptotic counterpart of the extension dimension defined by Dranishnikov. The main result establishes the relationship between the asymptotic extensional dimension of a proper metric space and the extension dimension of its Higson corona. |
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Repovs, D. Zarichnyi, М. |
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Repovs, D. Zarichnyi, М. |
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Repovs, D. |
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On the asymptotic extension dimension |
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On the asymptotic extension dimension |
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On the asymptotic extension dimension |
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On the asymptotic extension dimension |
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On the asymptotic extension dimension |
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on the asymptotic extension dimension |
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Інститут математики НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/166292 |
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On the asymptotic extension dimension / D. Repovs, M. Zarichnyi // Український математичний журнал. — 2010. — Т. 62, № 11. — С. 1523–1530. — Бібліогр.: 9 назв. — англ. |
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Український математичний журнал |
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AT repovsd ontheasymptoticextensiondimension AT zarichnyim ontheasymptoticextensiondimension |
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2025-07-14T21:07:22Z |
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| fulltext |
UDC 515.12
D. Repovš (Univ. Ljubljana, Slovenia),
M. Zarichnyi (Lviv Nat. Univ., Ukraine)
ON ASYMPTOTIC EXTENSION DIMENSION*
ПРО АСИМПТОТИЧНУ РОЗШИРЕНУ ВИМIРНIСТЬ
The aim of this paper is to introduce an asymptotic counterpart of the extension dimension defined by
Dranishnikov. The main result establishes a relation between the asymptotic extensional dimension of a proper
metric space and extension dimension of its Higson corona.
Метою статтi є введення асимптотичного аналога розширеної вимiрностi, що визначена Дранiшнiковим.
Основний результат полягає у встановленнi спiввiдношення мiж асимптотичною розширеною вимiрнi-
стю власного метричного простору та розширеною вимiрнiстю його корони Хiгсона.
1. Introduction. Asymptotic dimension of metric spaces was first defined by Gromov
[1] for finitely generated groups. Since then, this dimension is an object of study in
numerous publications (see an expository paper [2]).
A metric space (X, d) is of asymptotic dimension ≤ n (written asdimX ≤ n)
if for every D > 0 there exists a uniformly bounded cover U of X such that U =
= U0 ∪ . . . ∪ Un, where every family U i is D-disjoint, i = 0, 1, . . . , n. Recall that a
family A of subsets of X is uniformly bounded if
meshA = sup{diamA | A ∈ A} <∞
(as usual, diamA = sup{d(x, y) | x, y ∈ A} is the diameter of a subset A in a metric
space (X, d)) and is called D-disjoint if inf {d(a, a′) | a ∈ A, a′ ∈ A′} > D, for every
distinct A,A′ ∈ A.
The asymptotic dimension can be characterized in different terms; in particular, in
terms of extension of maps into Euclidean spaces [3]: a proper metric space X is of
asymptotic dimension ≤ n if and only if any proper asymptotically Lipschitz map
f : A → Rn+1 (see the definition below) defined on a closed subset A of X admits a
proper asymptotically Lipschitz extension over X. In the classical dimension theory, to
this result there corresponds the Aleksandrov theorem: for any metric space X, dimX ≤
≤ n, where dim stands for the covering dimension, if and only if any continuous map
f : A→ Sn defined on a closed subset A of X admits a continuous extension over X.
In [3, 4] Dranishnikov introduced the notion of extension dimension. This dimension
takes its values in the so called dimension types of CW-complexes. The aim of this paper
is to develop an asymptotic counterpart of the extension dimension. Our main results
is a generalization of the well-known result due to Dranishnikov [3] on the equality,
for the spaces of finite asymptotic dimension, of the asymptotic dimension of a proper
metric space and the dimension of the Higson corona of this space.
2. Preliminaries. A typical metric is denoted by d. By Nr(x) we denote the open
ball of radius r centered at a point x of a metric space.
2.1. Asymptotic category. A map f : X → Y between metric spaces is called (λ, ε)-
Lipschitz for λ > 0, ε ≥ 0 if d(f(x), f(x′)) ≤ λd(x, x′) + ε for every x, x′ ∈ X. A
map is called asymptotically Lipschitz if it is (λ, ε)-Lipschitz for some λ, ε > 0.
*This research was supported by the Slovenian Research Agency, grants P1-0292-0101-04 and BI-UA/07-
08-001.
c© D. REPOVŠ, M. ZARICHNYI, 2010
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11 1523
1524 D. REPOVŠ, M. ZARICHNYI
The (λ, 0)-Lipschitz maps are also called λ-Lipschitz, (1, 0)-Lipschitz maps are also
called short.
A metric space X is called proper if every closed ball in X is compact.
The asymptotic category A is introduced by A. Dranishnikov [3]. The objects of A
are proper metric spaces and the morphisms are proper asymptotically Lipschitz maps.
Recall that a map is called proper if the preimage of every compact set is compact.
We also need the notion of a coarse map. A map between proper metric spaces is
called coarse uniform if for every C > 0 there is K > 0 such that for every x, x′ ∈ X
with d(x, x′) < C we have d(f(x), f(x′)) < K. A map f : X → Y is called metric
proper if the preimage f−1(B) is bounded for every bounded set B ⊂ Y. A map is
coarse if it is metric proper and coarse uniform.
2.2. Higson compactification and Higson corona. Let ϕ : X → R be a function
defined on a metric space X. For every x ∈ X and every r > 0 let
Varrϕ(x) = sup{|ϕ(y)− ϕ(x)| | y ∈ Nr(x)}.
A function ϕ is called slowly oscillating whenever for every r > 0 we have Varrϕ(x)→
→ 0 as x → ∞ (the latter means that for every ε > 0 there exists a compact subspace
K ⊂ X such that |Varrϕ(x)| < ε for all x ∈ X \K. Let X̄ be the compactification of
X that corresponds to the family of all continuous bounded slowly oscillation functions.
The Higson corona of X is the remainder νX = X̄ \X of this compactification.
It is known that the Higson corona is a functor from the category of proper metric
spaces and coarse maps into the category of compact Hausdorff spaces. In particular, if
X ⊂ Y, then νX ⊂ νY.
For any subset A of X we denote by A′ its trace on νX, i.e., the intersection of the
closure of A in X̄ with νX. Obviously, the set A′ coincides with the Higson corona νA.
2.3. Cone. Let X be a metric space of diameter ≤ 1. The open cone of X is the
set OX = (X × R+)/(X × {0}) endowed with the metric (by [x, t] we denote the
equivalence class of (x, t) ∈ X × R+):
d([x1, t1], [x2, t2]) = |t1 − t2|+ min{t1, t2}d(x1, x2).
For a map f : X → Y of metric spaces we denote by Of : OX → OY the map
defined as Of([x, t]) = [f(x), t].
Proposition 2.1. If f : X → Y is a Lipschitz map than Of is an asymptotically
Lipschitz map.
Proof. Suppose a map f : X → Y is λ-Lipschitz. Then for any [x1, t1], [x2, t2] ∈
∈ OX we have
d(Of([x1, t1]),Of([x2, t2])) = d([f(x1), t1], [f(x2), t2]) =
= |t1 − t2|+ min{t1, t2}d(f(x1), f(x2)) ≤
≤ λ′(|t1 − t2|+ min{t1, t2}d(x1, x2)),
where λ′ = max{λ, 1}.
Proposition 2.1 is proved.
The open cone of a finite CW-complex is a coarse CW-complex in the sense of [5].
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
ON ASYMPTOTIC EXTENSION DIMENSION 1525
Denote by αL : OL→ R the function defined by αL([x, t]) = t. Obviously, αL is a
short function.
Let ÕL = {[x, t] ∈ OL | t ≥ 1}. Denote by βL : ÕL→ L the map βL([x, t]) = x.
Lemma 2.1. The map βL is slowly oscillating.
Proof. For R > 0, the R-ball centered at [x, 0] is {[x, t] | t < R. If d([x, t], [x1, t1]) <
< K < R, then |t − t1| + min{t, t1}d(x, x1) < K, i.e., (t − R)d(x, x1) < R and
d(x, x1) < K/(t−K). Therefore, d(βL(x), βL(x1)) < K/(R−K)→ 0 as R→∞.
Lemma 2.1 is proved.
Let β̄L : ÕL→ L be the (unique) extension of the map βL. Denote by ηL : νÕL→
→ L the restriction of βL.
Proposition 2.2. Let f : A → OL be a proper asymptotically Lipschitz map de-
fined on a proper closed subset A of a proper metric space X. There exists a neighbor-
hood W of A in X, a proper asymptotically Lipschitz map g : W → OL with the fol-
lowing property: there exist constants λ, s > 0 such that αL(g(a)) ≤ λd(a,X \W )+s.
Proof. We may assume that L is a subset of In for some n and there exists a
Lipschitz retraction r : U → L of a neighborhood U of L in In. Since OIn is Lipschitz
equivalent to Rn+1
+ , there exists a (λ′, s′)-Lipschitz extension g̃ : X → OIn of g.
Put W = g̃−1(OU) and ḡ = g̃|W. For every a ∈ A and w ∈ X \W we have
d(g(a), g̃(w) ≤ λ′d(a,w) + s′ ≤ λ′d(a,X \W ) + s.
Suppose that d(L, In \ U) = c > 0, then, since g̃(w) /∈ CU,
d(g(a), g̃(w)) = |αL(g(a))− αL(g̃(w))|+
+ min{αL(g(a)), αL(g̃(w))}d(βL(g(a)), βL(g̃(w))) ≥
≥ |αL(g(a))− αL(g̃(w))|+ cmin{αL(g(a)), αL(g̃(w))} ≥ c′αL(g(a)),
where c′ = min{c, 1}. Then αL(g(a)) ≤ λd(a,X \W )+s, where λ = λ′/c′, s = s′/c′.
Proposition 2.2 is proved.
3. Auxiliary results. In this section we collect some results needed in the proof of
the main result. They are proved in [3] but it turned out that we have also cover the case
of functions with infinite values.
A map f : X → R+ ∪{∞} is said to be coarsely proper if the preimage f−1([0, c])
is bounded for every c ∈ R+.
Lemma 3.1. For any function ϕ : X → R+ with ϕ(x) → 0 as x → ∞ the
function 1/ϕ : X → R+ ∪ {∞} is coarsely proper.
Proposition 3.1. Let f : X → R+ ∪ {∞} be a coarsely proper function. There
exists an asymptotically Lipschitz proper function q : X → R+ with q ≤ f.
Proof. This was proved in [3] for the case of f : X → R+ (see Proposition 3.5).
This proof also works for our case.
Proposition 3.2. Let fn : X → R+ ∪ {∞} be a sequence of coarsely proper
functions. Then there exists a filtration X = ∪∞n=1An and a coarsely proper function
f : X → R+ with f |An ≤ n and f |(X \An) ≤ fn for every n.
Proof. Let Bn =
⋃n
i=1
f−1i ([0, n]). The sets Bi are bounded and B1 ⊂ B2 ⊂ . . . .
Therefore, there exist bounded subsets A1 ⊂ A2 ⊂ . . . such that An∩
(⋃∞
i=1
Bi
)
= Bn
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1526 D. REPOVŠ, M. ZARICHNYI
and
⋃∞
i=1
Ai = X. For x ∈ An \An−1, put f(x) = n. Obviously, f is coarsely proper
and f |An ≤ n. Now suppose that x /∈ An, then x /∈ Bn and therefore x /∈ f−1n ([0, n]),
i.e., fn(x) > n ≥ f |(X \An).
Proposition 3.2 is proved.
The following is an easy modification of Lemma 3.6 from [3] and the proof of it
works in our case as well.
Lemma 3.2. Suppose that f : A → R+ ∪ {∞} is a coarsely proper map de-
fined on a closed subset A of a proper metric space X and g : W → R+ is a proper
asymptotically Lipschitz map such that g ≤ f |W and there exist λ, s such that λd(a,
X \W ) + s ≥ g(a) for every a ∈ A. Then there exists a proper asymptotically Lipschitz
map ḡ : X → R+ for which ḡ ≤ f and ḡ|A = g.
3.1. Almost geodesic spaces. A metric space X is said to be almost geodesic if
there exists C > 0 such that for every two points x, y ∈ X there is a short map
f : [0, Cd(x, y)] → X with f(0) = x, f(Cd(x, y)) = y. If in this definition C = 1,
then we come to the well-known notion of geodesic space.
We are going to describe a construction of embedding of a discrete metric space X
into an almost geodesic space of the asymptotic dimension min{asdimX, 1}.
For an unbounded discrete metric space X with base point x0 define a function
f : X → [0,∞) by the formula f(x) = d(x, x0). Choose a sequence 0 = t0 < t1 <
< t2 < . . . in f(X) so that ti+1 > 2ti for every i. To every pair of points x, y ∈
∈ f−1([ti, ti+1]), for some i, attach the line segment [0, d(x, y)] along its endpoints.
Let X̂ be the union of X and all attached segments. We endow X̂ with the maximal
metric that agrees with the initial metric on X and the standard metric on every attached
segment.
Note that since X is discrete and proper, every set f−1([ti, ti+1]) is finite and there-
fore X̂ is a proper metric space.
Proposition 3.3. The space X̂ is almost geodesic.
Proof. Suppose that x, y ∈ X̂, then x ∈ [x1, x2], y ∈ [y1, y2], where x1, x2, y1, y2 ∈
∈ X and [x1, x2], [y1, y2] are attached segments. We may suppose that d(x, y) =
= d(x, x1) + d(x1, y1) + d(y1, y).
Case 1: There exists i such that x1, y1 ∈ f−1([ti, ti+1]). Then [x, x1] ∪ [x1, y1] ∪
∪ [y1, y] is a segment of diameter d(x, y) that connects x and y in X̂.
Case 2: f(x1) ∈ [ti, ti+1], f(y1) ∈ [tj , tj+1], where i 6= j. Without loss of genera-
lity, we may assume that i < j.
Obviously, d(x1, y1) ≤ d(x, y). Since |tj − tj−1| ≤ d(x1, y1), we see that |tj −
− tj−1| ≤ d(x, y). This implies that tj/2 ≤ d(x, y), or equivalently, tj ≤ d(x, y).
Besides, d(y1, f
−1([0, tj−1]))) ≤ d(x1, y1) ≤ d(a, b).
For every k = i, i+ 1, . . . , j1 choose zk ∈ f−1(tk). Then
d(y1, zj−1) ≤ d(y1, f
−1([0, tj−1]) + diam (f−1([0, tj−1])) ≤
≤ d(a, b) + 2tj−1 ≤ d(a, b) + tj ≤ 3d(a, b).
We connect x and y by the segment
J = [x, x1] ∪ [x1, z1] ∪
⋃j−1
k=i
[zk, zk+1]) ∪ [zj−1, y1] ∪ [y1, y].
Then
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
ON ASYMPTOTIC EXTENSION DIMENSION 1527
diam J ≤ d(x, x1) + d(x1, zi+1) +
(
j−1∑
k=i+1
d(zk, zk+1)
)
+ d(zj−1, y1) + d(y1, y) =
= d(x, y) + 2ti+1 +
j−1∑
k=i+1
2tk+1 + 5d(x, y) + d(x, y) ≤
≤ 7d(x, y) + 2(ti+1 + . . .+ tj) ≤ 7d(x, y) + 4tj ≤ 15d(x, y).
Proposition 3.3 is proved.
We need a version of the fact proved in [3] for geodesic spaces.
Proposition 3.4. Let f : X → Y be a coarse uniform map of an almost geodesic
space X. Then f is asymptotically Lipschitz.
Proof. Let C be a constant from the definition of almost geodesic space. Suppose
x, y ∈ X, then there exists a short map α : [0, Cd(x, y)] → X such that α(0) = x,
α(Cd(x, y)) = y. There exist points 0 = t0 < t1 < . . . < tk−1 < tk = Cd(x, y), where
k ≤ [d(x, y)] + 1, such that |ti − ti−1| ≤ C for every i = 1, . . . , k.
Since f is coarse uniform, there exists R > 0 such that d(f(x′), f(y′)) < R when-
ever d(x′, y′) ≤ C. Then
d(f(x), f(y)) ≤
k∑
i=1
d(f(α(ti)), f(α(ti−1))) ≤
≤ kR ≤ ([d(x, y)] + 1)R ≤ Rd(x, y) + 2R.
Proposition 3.4 is proved.
4. Asymptotic extension dimension. Let P be an object of the category A. For
any object X of A the Kuratowski notation XτP means the following: for every proper
asymptotically Lipschitz map f : A → P defined on a closed subset A of X there is a
proper asymptotically Lipschitz extension of f onto X.
Denote by L the class of compact absolute Lipschitz neighborhood Euclidean exten-
sors (ALNER). Following [4], we define a preorder relation ≤ on L. For L1, L2 ∈ L,
we have L1 ≤ L2 if and only if XτOL1 implies XτOL2 for all proper metric spaces
X. This preorder relation leads to the following equivalence relation ∼ on L: L1 ∼ L2
if and only if L1 ≤ L2 and L2 ≤ L1. We denote by [L] the equivalence class containing
L ∈ L. The class [L] is called the asymptotic extension dimension type of L. The men-
tioned preorder relation induces the partial order relation on all the asymptotic extension
dimension types.
For a proper metric space X, we say that its asymptotic extension dimension does
not exceed [OL] (briefly as-ext-dimX ≤ [OL] whenever XτOL.
If as-ext-dimX ≤ [OL], then the equality as-ext-dimX = [OL] means the follow-
ing. If we also have as-ext-dimX ≤ [OL′], then [OL] ≤ [OL′].
From the result on extension of asymptotically Lipschitz functions ([3]; see also [6]),
the element [∗] is maximal.
Theorem 4.1. Let L be a compact metric ALNER. The following conditions are
equivalent:
1) as-ext-dimX ≤ [OL];
2) ext-dim νX ≤ [L].
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1528 D. REPOVŠ, M. ZARICHNYI
Proof. 1)⇒ 2). Assume that as-ext-dimX ≤ [OL]. Let ϕ : C → L be a map defined
on a closed subset C of νX. Since L ∈ ANE, there exists an extension ϕ′ : V → L
of ϕ over a closed neighborhood V of C in X̄ = X ∪ νX. Then VarRϕ
′(x) → 0 as
x→∞, for any fixed R > 0. By Lemma 3.1, the function
fn : V ∩X → R+ ∪ {∞}, fn(x) =
1
VarRϕ′(x)
,
is coarsely proper, for every n ∈ N. By Proposition 3.2, there is a coarsely proper
function f : V ∩ X → R+ and a filtration V ∩ X =
⋃∞
n=1An such that f |An ≤ n
and f |(X \An) ≤ fn. By Proposition 3.5 from [3], there is an asymptotically Lipschitz
function q : V ∩X → R+ with q ≤ f. We suppose that q is (λ, s)-Lipschitz for some
λ, s > 0. Define the map g : V ∩X → OL by the formula g(x) = [ϕ′(x), q(x)].
We are going to check that the map g(x) is asymptotically Lipschitz. Let x, y ∈
∈ V ∩X and n− 1 ≤ d(x, y) ≤ n.
Suppose that x, y ∈ (V ∩X) \An, then q(x) ≤ fn(x), q(y) ≤ fn(y). We have
d(g(x), g(y)) = |q(x)− q(y)|+ min{q(x), q(y)}d(ϕ′(x), ϕ′(y)) ≤
≤ λd(x, y) + s+ min{q(x), q(y)}Varnϕ
′(x) ≤ λd(x, y) + s+ 1.
If x ∈ An, then q(x) ≤ n and we obtain
d(g(x), g(y)) ≤ λd(x, y) + s+ nd(ϕ′(x), ϕ′(y)) ≤
≤ λd(x, y) + s+ ndiam L ≤ λd(x, y) + s+ (d(x, y) + 1)diam L ≤
≤ (λ+ diam L)d(x, y) + (s+ diam L).
We argue similarly if y ∈ An.
Now, by the assumption, there is an asymptotically Lipschitz extension ḡ : X →
→ OL of g. Consider the composition ηLνḡ : νX → OL. Obviously, ηLνḡ|C = ϕ. We
conclude that ext-dimνX ≤ [L].
2)⇒ 1). Let f : A → OL be an asymptotically Lipschitz map defined on a proper
closed subset A of a proper metric space X. By Proposition 2.2, there is a proper
asymptotically Lipschitz map f̃ : W → OL defined on a neighborhood W of A and
constants λ, s such that αLf(a) ≤ λd(a,X \W )+s for all a ∈ A. Denote by ϕ : νX →
→ L an extension of the composition ηLνf̃ . Since L is an absolute neighborhood
extensor, there exists an extension ψ : V → L of ϕ onto a closed neighborhood of νX
in the Higson compactification X̄. Extend ψ to a map ψ̂ : (V ∩X )̂→ L as follows. Let
J be a segment attached to V with endpoints a and b. We require that ψ̂ linearly maps
J onto a geodesic segment in L with endpoints ψ(a) and ψ(b).
Show that ψ̂ is a slowly oscillating map. Since ψ is slowly oscillating, for every
ε > 0 and R > 0 there exists K > 0 such that VarRψ(x) < ε whenever d(x, x0) >
> K. Suppose that ψ̂ is not slowly oscillating, then there exist R > 0, C > 0, and
sequences (xi1), (xi2) in (V ∩ X )̂ such that d(xi1, x
i
2) < R, xi1 → ∞, xi2 → ∞ and
d(ψ̂(xi1), ψ̂(xi2)) > C for every i. We assume that xi1 ∈ [ai1, b
i
1], xi2 ∈ [ai2, b
i
2], for
every i, where ai1, b
i
1, a
i
2, b
i
2 ∈ X ∩ V. Without loss of generality we may assume that
ai1 → ∞ and there exists C1 > 0 such that d(ψ̂(xi1), ψ̂(ai1)) > C1 for every i. If
d(ai1, b
i
1) < K for all i and some K > 0, then d(ψ̂(xi1), ψ̂(ai1)) < d(ψ̂(ai1), ψ̂(bi1))→ 0,
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ON ASYMPTOTIC EXTENSION DIMENSION 1529
and we obtain a contradiction. Therefore, we may assume that d(ai1, b
i
1) → ∞. Then
d(ai1, x
i
1)/d(ai1, b
i
1) < R/d(ai1, b
i
1) → 0 and therefore, by the definition of the map ψ̂,
d(ψ̂(xi1), ψ̂(ai1))/d(ψ̂(ai1), ψ̂(bi1)) → 0. Then obviously d(ψ̂(xi1), ψ̂(ai1)) → 0 and we
obtain a contradiction.
Since the map f̃ is asymptotically Lipschitz, there exists K > 0 such that for any
a ∈W we have
diam (αLf̃(N1(a)) + αLf̃(a)diam (ψ(N1(a)) ≤ K.
Define the function r : (X ∩ V )̂ → R+ ∪ {∞} by the formula r(x) = K/(ψ(N1(x))).
We have f(a) ≤ r(a) for every a ∈ A. The function r is asymptotically proper and by
Proposition 3.1, there exists a (λ′, s′)-Lipschitz function f̄ : X → R+, for some λ′, s′,
with f̄ ≤ r and f̄ |A = αLf.
Define a map g : (X ∩ V )̂ → OL by the formula g(x) = (ψ(x), f̄(x)). Obviously,
g|A = f. We are going to show that g is a coarse uniform map.
Suppose x, y ∈ X, d(x, y) < 1, then
d(g(x), g(y)) ≤ |f̄(x)− f̄(y)|+ min{f̄(x), f̄(y)}d(ψ(x), ψ(y)) ≤ λ′ + s′ +K.
Note that, since f̄ is proper, g is also proper. Since g is coarse uniform, by Proposi-
tion 3.4, g is asymptotically Lipschitz. Therefore, as-ext-dimX ≤ [OL].
Theorem 4.1 is proved.
Corollary 4.1 (Finite Sum Theorem). Suppose X is a proper metric space, X =
= X1∪X2, where X1, X2 are closed subsets of X with as-ext-dimXi ≤ [OL], i = 1, 2,
for some L ∈ L. Then as-ext-dimX ≤ [OL].
Proof. Since νX = νX1 ∪ νX2, the result follows from Theorem 4.1 and the finite
sum theorem for extension dimension (see [7]).
5. Remarks and open questions.
Question 5.1. Does the equality as-ext-dimRn = Sn hold?
Question 5.2. Let L1, L2 be finite polyhedra in euclidean spaces endowed with
the induced metric. Is the inequality [L1] ≤ [L2] introduced in [4] equivalent to the
inequality [L1] ≤ [L2] as in Section 4?
One can define a counterpart of the asymptotic extension dimension by using warped
cones instead of open cones. Following [8] we review this construction briefly. Let F be
a foliation on a compact smooth manifold V. Let N be any complementary subbundle
to TF in TM. Choose Euclidean metrics gN in N and gF in TF . The foliated warped
cone OF is the manifold V ×[0,∞)/V ×{0} equipped with the metric induced for t ≥ 1
by the Riemannian metric gR + gF + t2gN . Since we are interested by the asymptotic
properties of the warped cones, the metric structure on any bounded neighborhood of
the vertex of the cone is irrelevant.
Question 5.3. Is the obtained warped cone an absolute neighborhood extensor in
the asymptotic category?
An affirmative answer to this question would allow us to define asymptotic extension
dimension theory with the values in warped cones.
Question 5.4. Can one characterize the dimension of the sublinear corona (see [9])
in terms of the asymptotic extension dimension?
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1530 D. REPOVŠ, M. ZARICHNYI
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Received 30.09.09
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