Coarse dimensions and partitions of unity

Abstract

Gromov \cite{Gr$_1$} and Dranishnikov \cite{Dr$_1$} introduced asymptotic and coarse dimensions of proper metric spaces via quite different ways. We define coarse and asymptotic dimension of all metric spaces in a unified manner and we investigate relationships between them generalizing results of Dranishnikov \cite{Dr$_1$} and Dranishnikov-Keesling-Uspienskij \cite{DKU}.

Full text

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arXiv:math/0506547v3 [math.GT] 15 Sep 2005
COARSE DIMENSIONS AND PARTITIONS OF UNITY
N. BRODSKIY AND J. DYDAK
Abstract. Gromov [11] and Dranishnikov [2] introduced asymptotic
and coarse dimensions of proper metric spaces via quite different ways.
We define coarse and asymptotic dimension of all metric spaces in a uni-
fied manner and we investigate relationships between them generalizing
results of Dranishnikov [2] and Dranishnikov-Keesling-Uspienskij [5].
Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
References
Introduction
Preliminaries
Multiplicity and higher Lebesque numbers
The coarse category
Coarse dimensions
The large scale dimension
Slowly oscillating functions
Coarse dimension and Higson corona
Coarse dimension and absolute extensors
Open problems
1
3
6
9
11
14
17
19
22
23
24
1. Introduction
There are three concepts of dimension associated with variants of the
coarse category of proper metric spaces. The original one, the asymptotic
dimension of Gromov [11], and dimensions asdim∗(X) and dimc(X) intro-
duced by Dranishnikov [2]. All three dimensions are defined in seemingly
different ways:
1. The asymptotic dimension of Gromov (see [11] or Definitions 1-2 in [2]
on p.1103) is the smallest integer n such that for every M > 0 there is a
Date: September 15, 2005.
1991 Mathematics Subject Classification.
18B30, 54D35, 54D40, 20H15.
Key words and phrases. Asymptotic dimension, coarse dimension, coarse category,
Lebesque number.
1
Primary: 54F45, 54C55, Secondary: 54E35,

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2N. BRODSKIY AND J. DYDAK
uniformly bounded family U of Lebesque number at least M and multiplicity
(or order) at most n + 1.
2. The asymptotic dimension asdim∗(X) of Dranishnikov (see Definition
3 in [2] on p.1104) is the smallest integer n such that for every proper function
f : X → R+there is a contracting map φ : X → K to an n-dimensional
asymptotic polyhedron such that for each M > 0 there is a compact subset C
of X with the property that φ−1(B(φ(x),M)) ⊂ B(x,f(x)) for all x ∈ X\C.
3. The coarse dimension dimc(X) of Dranishnikov (see Definition 4 in [2]
on p.1105) is the smallest integer n such that Rn+1is an absolute extensor
of X in the category of proper asymptotically Lipschitz functions. That
dimension coincides with the dimension of the Higson corona ν(X) of X
(see Theorem 6.6 in [2] on p.1111).
One of the main motivations behind the research in asymptotic dimension
is the result of Yu (see [16] and [17]) that the Novikov Conjecture holds for
groups of finite asymptotic dimension.
In this paper we work in the coarse category of all metric spaces and we de-
vise a unified way of defining five dimensions: coarse dimension dimcoa
major coarse dimension dimCOA
RSE(X), asymptotic dimension asdim(X), mi-
nor asymptotic dimension ad(X), and large scale dimension dimlarge
In case of proper metric spaces, three of them coincide with the above
dimensions. Namely, dimCOA
asdim(X) coincides with Gromov’s asymptotic dimension. The fourth one,
the minor asymptotic dimension, is a variant of Gromov’s dimension. The
large scale dimension is always equal to the coarse dimension and the reason
we are introducing it is to simplify proofs of the relations between the three
basic dimensions which we do in a much simpler way than as described in
Dranishnikov’s paper [2]. The main relations between dimensions are as
follows:
rse(X),
scale(X).
RSE(X) = dim∗(X), dimcoa
rse(X) = dimc(X), and
(1) There are two strands of inequalities: asdim(X) ≥ dimCOA
dimcoa
(2) In each strand (for unbounded spaces X), finiteness of a larger
dimension implies its equality with all smaller dimensions in the
strand.
RSE(X) ≥
rse(X) and asdim(X) ≥ ad(X) ≥ dimcoa
rse(X),
We do not know of any unbounded space X such that a larger dimension
in a strand is infinite and a smaller dimension is finite.
Our fundamental concept is that of a coarse family and we follow the well-
established route of defining the covering dimension by refining covers with
covers of a prescribed multiplicity. In classical dimension theory one deals
with two cases: finite covers and infinite covers. There, for paracompact
spaces, the two concepts coincide. In the case of coarse covers we get two
concepts of coarse dimension whose equality remains unresolved.
A finite family U of subsets of X is coarse if and only if there is a slowly
oscillating partition of unity f on X \ B for some bounded subset B of X
whose carriers Carr(f) refine U. That explains why, in the case of a proper

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COARSE DIMENSIONS AND PARTITIONS OF UNITY3
metric space X, its coarse dimension equals the covering dimension of the
Higson corona of X.
Our basic strategy is to associate natural functions with objects and de-
clare those objects to be coarse, asymptotic, or large scale if the function
is coarsely proper. A function f is coarsely proper if f(En) → ∞ whenever
En→ ∞. Elements Enrelated to objects could be points in a metric space,
bounded subsets in a metric space, or covers of a metric space (in which
case divergence to infinity is measured by the size of the Lebesque number).
In [2] (p.1089) coarsely proper functions were defined as those f : X → Y
such that f−1(A) is bounded whenever A is bounded in Y . Notice that our
definition generalizes the one from [2].
The authors are grateful to Jose Higes for helpful comments.
2. Preliminaries
Given a subset A ?= ∅ of a metric space X the most basic function is the
distance function dA: X → R+: dA(x) = dist(x,A).
Definition 2.1. Given a subset A of a metric space (X,dX) the ball B(A,M)
is defined to be the set {x ∈ X | dist(x,A) < M} if M > 0, it is defined to
be the set {x ∈ X | dist(x,X \ A) > −M} if M < 0, and it is simply A if
M = 0.
The distance function leads to the first concept of divergence to infinity:
xn→ ∞ if dX(xn,x0) → ∞ for some (and hence for all) x0∈ X. However,
dist(x,A) is a function of two arguments and we can use the second one
to define divergence to infinity for bounded subsets of X. Here is a more
general concept.
Definition 2.2. A family U of bounded subsets of X is called coarsely
proper if the function U → dU(x0) is coarsely proper for some (and hence
for all) x0∈ X. Here U is considered as a subspace of all bounded subsets
of X with the Hausdorff metric.
Notice that a sequence {An} of bounded subsets of X containing points
xn∈ Anso that xn→ ∞ is coarsely proper if and only if every bounded
subset of X intersects at most finitely many elements of the sequence. In
that case we write An → ∞ and that form of divergence to infinity is of
most interest to us.
Lemma 2.3. If U is a coarsely proper cover of X, then every selection
function φ : X → U (that means x ∈ φ(x)) is coarsely proper.
Proof. Suppose xn → ∞ and xn ∈ Un ∈ U.
Hausdorff metric. Pick x0∈ X. Since dUn(x0) → ∞, every bounded subset
of X intersects at most finitely many elements of the sequence {Un} and
any selection function φ is coarsely proper.
Clearly, Un → ∞ in the
?

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4N. BRODSKIY AND J. DYDAK
Definition 2.4. Given a family U in X, the local Lebesque number LU(x) ∈
R+∪ ∞ is defined as the supremum of dist(x,X \ U), U ∈ U. If U = X for
some U ∈ U it is defined to be infinity.
Notice that either LU≡ ∞ at all points or it is a natural Lipschitz function
associated with U. More precisely |LU(x) − LU(y)| ≤ dX(x,y).
Definition 2.5. The Lebesque number L(U,A) is inf{LU(x) | x ∈ A}.
Definition 2.6. A family of subsets U of a metric space X is called coarse
if LUis coarsely proper (as a function from X to R ∪ ∞).
An alternative way to define coarse families is to require L(U,A) → ∞ as
A → ∞. Yet another way is to state that L(U,X\B(x0,t)) → ∞ as t → ∞.
Proposition 2.7.
X is coarse if and only if X \ A is bounded.
2. A family U = {X1,X2} consisting of two subsets of X is coarse if
and only if dXrestricted to (X \ X1) × (X \ X2) is coarsely proper.
3. A family U = {X1,X2,... ,Xn} consisting of finitely many subsets
1. A family U = {A} consisting of one subset A of
of X is coarse if and only if the function dU(x) :=
n ?
i=1
dist(x,X \Xi)
is coarsely proper.
Proof. 1. If X\A is bounded, then LU(x) ≥ dist(x,X\A) and LUis coarsely
proper. If X \ A is unbounded, then LU(x) = 0 at all x ∈ X \ A and LUis
not coarsely proper.
2. Suppose U = {X1,X2} is coarse and xn → ∞, yn → ∞, for some
xn∈ X\X1, yn∈ X\X2. Notice LU(xn) ≤ dX(xn,yn), so dX(xn,yn) → ∞.
If U = {X1,X2} is not coarse, then there is a sequence zn → ∞ with
LU(zn) bounded by M. We can produce xn ∈ X \ X1 and yn∈ X \ X2
so that dX(zn,xn) < M + 1 and dX(zn,yn) < M + 1 for all n.
dX(xn,yn) < 2M + 2, a contradiction.
3. Notice dU(x) ≥ LU(x) and m · LU(x) ≥ dU(x).
Now,
?
Definition 2.8. Given a function f : X → Y of metric spaces, its Lebesque
number transfer Lf: R+→ R+∪ ∞ is the supremum of all functions α :
R+→ R+∪ ∞ such that L(U,Y ) ≥ t implies L(f−1(U),X) ≥ α(t) for all
families U of subsets of Y .
Definition 2.9. A function f : X → Y of metric spaces is coarse if the
Lebesque number transfer Lfis coarsely proper.
An alternative definition of coarse functions is to require the function
U → L(f−1(U),X) to be coarsely proper on the set of covers of Y .
Let us show that our definition of coarse functions coincides with that of
Roe [14].
Proposition 2.10. A function f : X → Y is coarse if and only if for every
R > 0 there is M > 0 such that dX(x,y) ≤ R implies dY(f(x),f(y)) ≤ M
for all x,y ∈ X.

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COARSE DIMENSIONS AND PARTITIONS OF UNITY5
Proof. Notice that if M > 0 and N > 0 are numbers such that dX(x,y) < M
implies dY(f(x),f(y)) < N, then Lf(N) ≥ M. Therefore f being coarse in
the sense of Roe implies Lfbeing coarsely proper.
Conversely, if Lf(N) ≥ M, then consider the cover U = {B(z,N)}z∈Y
whose Lebesque number is clearly at least N. If dX(x,y) < M, then there
is z so that x,y ∈ f−1(B(z,N)). Hence dY(f(x),f(y)) < 2 · N and f is
coarse.
?
Dranishnikov [2] (p.1088) defined asymptotically Lipschitz functions f :
X → Y as those for which there are constants M > 0 and A such that
dY(f(x),f(y)) ≤ M · dX(x,y) + A for all x,y ∈ X.
concept to the Lebesque number transfer.
Let us relate this
Proposition 2.11. A function f : X → Y is asymptotically Lipschitz if
and only if there is a linear function t → m · t + b so that m > 0 and
Lf(t) ≥ m · t + b for all t.
Proof. Suppose there are constants M > 0 and A such that dY(f(x),f(y)) ≤
M · dX(x,y) + A for all x,y ∈ X. Given a cover U of Y with L(U,Y ) ≥ t
and given x ∈ X, the ball B(x,(t−A−δ)/M) is mapped by f into the ball
B(f(x),t − δ) which is contained in an element of U for all δ > 0. That
shows the Lebesque number of f−1(U) to be at least (t−A)/M). Conversely,
if Lf(t) ≥ m · t + b for all t and m > 0, then we claim dY(f(x),f(y)) <
2·dX(x,y)/m+2(1−b)/m. Indeed, put dX(x,y) = s and consider the cover
U = {B(z,(s + 1 − b)/m)}z∈Y whose Lebesque number is clearly at least
(s + 1 − b)/m. There is z so that x,y ∈ f−1(B(z,(s + 1 − b)/m)). Hence
dY(f(x),f(y)) < 2 · (s + 1 − b)/m and f is asymptotically Lipschitz.
?
Proposition 2.12. Given a function f : X → Y of metric spaces the
following conditions are equivalent:
1. f sends bounded subsets of X to bounded subsets of Y and f−1(U)
is coarse for every coarse family U in Y .
2. f is coarse and coarsely proper.
Proof. 1 =⇒ 2. Given a bounded subset A of Y the family {Y \A} is coarse
(see 2.7). Since {f−1(Y \A)} is coarse and f−1(Y \A) = X\f−1(A), f−1(A)
must be bounded and f is coarsely proper.
If f is not coarse, we find sequences xn,yn∈ X so that dY(f(xn),f(yn)) >
n for each n but dX(xn,yn) < M for all n. Since f sends bounded subsets of
X to bounded subsets of Y , we may assume xn→ ∞, hence yn→ ∞. Put
A = {xn} and B = {yn}. Using 2.7 we see that U = {Y \f(A),Y \f(B)} is
a coarse family in Y . Since f−1(U) is coarse, the family V = {X \A,X \B},
to which U is a shrinking, is coarse as well. That however contradicts 2.7.
2 =⇒ 1. Obviously, coarse functions f : X → Y send bounded subsets
of X to bounded subsets of Y . Put V = f−1(U) for some coarse family U
in Y . To find points x ∈ X such that LV(x) > t we find s > 0 so that
Lf(s) > t and we find u > 0 such that LU(y) > s for y ∈ Y \ B(y0,u). Put

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6N. BRODSKIY AND J. DYDAK
W = U∪{B(y0,u+s)}. Note L(W,Y ) > s. Since L(f−1(W),X) > t, points
x lying outside of the bounded set f−1(B(y0,u + s)) satisfy LV(x) > t.
?
In the end of this section let us demonstrate the usefulness of the concept
of a coarse family by rewording notions from [6].
In section 5.2 of [6] the concept of asymptotic neighborhood W of a subset
A of X is introduced by requiring lim
r→∞dist(A \ B(x0,r),X \ W) = ∞ for
some (and hence for all) x0∈ X.
Proposition 2.13. W is an asymptotic neighborhood of A if and only if
the pair {X \ A,W} is coarse.
Proof. According to part 2 of 2.7 the pair {X \ A,W} is coarse if and only
if dXrestricted to A × (X \ W) is coarsely proper. That can be easily seen
as equivalent to lim
r→∞dist(A\B(x0,r),X \W) = ∞ for some (and hence for
all) x0∈ X.
?
In section 5.2 of [6] (see also [3]) the concept of asymptotically disjoint
subsets A and B of X is introduced by requiring lim
r→∞dist(A \ B(x0,r),B \
B(x0,r)) = ∞ for some (and hence for all) x0∈ X.
Proposition 2.14. A and B are asymptotically disjoint if and only if the
pair {X \ A,X \ B} is coarse.
Proof. Apply part 2 of 2.7.
?
Also notice that the concept of an asymptotic separator of [6] (see section
5.2) can be introduced without referring to the Higson corona.
Definition 2.15. A subset C of X is an asymptotic separator between
asymptotically disjoint subsets A and B if there are asymptotic neighbor-
hoods WAof A and WBof B such that C = X\(WA∪WB) and WA∩WB= ∅.
3. Multiplicity and higher Lebesque numbers
Definition 3.1. Given a family U of subsets of X we define the multiplicity
function mU: X → Z+∪ ∞ by setting mU(x) to be equal to the number of
elements of U containing x. The global multiplicity m(U,A) is the supremum
of mU(x), x ∈ A.
By a coarse refinement V of a coarse family U we mean a coarse family
such that every element V of V is contained in an element U of U. V is
called a shrinking of U if they are indexed by the same set S and Vs⊂ Us
for all s ∈ S. If V is a coarse refinement of U indexed by a different set T,
then one can create a shrinking V′of U as follows: find a function φ : T → S
satisfying Vt⊂ Uφ(t)for all t ∈ T. Define V′
that V′has multiplicity at most that of V and is a coarse shrinking of U.
Given a family φ = {φs : X → R+}s∈S of functions its carrier family
Carr(φ) is the family {φ−1
s(0,∞)}s∈S. The multiplicity m(φ) of φ is defined
sas?{Vt| s = φ(t)}. Notice

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COARSE DIMENSIONS AND PARTITIONS OF UNITY7
as the multiplicity of its carrier family and its Lebesque number L(φ) is
defined as the Lebesque number of its carrier family.
Lemma 3.2. If U = {Us}s∈S is family in X such that LU(x0) = ∞ for
some x0∈ X, then it has a coarse refinement V of multiplicity at most 2.
Proof. Put Vn= {x ∈ X | (n − 1)2≤ d(x,x0) < (n + 1)2} for n ≥ 1.
?
Lemma 3.3. If U = {Us}s∈Sis a family in X of multiplicity at most n+1,
n+1
?
i=1
each x ∈ X and each Viconsists of disjoint sets.
then it can be refined by V =
Visuch that LV(x) ≥ LU(x)/(2n + 2) for
Proof. Define fs(x) = dist(x,X \ Vs). For each finite set T of S define
WT = {x ∈ X | min{ft(x) | t ∈ T} > sup{fs(x) | s ∈ S \ T}}. Notice
WT= ∅ if T contains at least n+2 elements. Also, notice that WT∩WF= ∅
if both T and F are different but contain the same number of elements. Let
us estimate the Lebesque number of W = {WT}T⊂S. Given x ∈ X arrange
all non-zero values fs(x) from the largest to the smallest. Add 0 at the end
and look at gaps between those values. The largest number is at least LU(x),
there are at most n+1 gaps, so one of them is at least LU(x)/(n+1). That
implies the ball B(x,LU(x)/(2n +2)) is contained in one WT (T consists of
all t to the left of the gap). Define Vias {WT}, all T containing exactly i
elements.
?
Lemma 3.4. If U = {Us}s∈Sis a coarse family in X, then it has a coarse
refinement V that is coarsely proper. Moreover, if U is of finite multiplicity,
then we may require V to be of finite multiplicity as well.
Proof. Let V = {Vs,m}(s,m)∈S×N, where Vs,m= {x ∈ Us| 2m< d(x,x0) ≤
2m+2}. Notice V is coarse of multiplicity at most 2·m(U). Also, it consists of
bounded sets so that for any sequence xk→ ∞ the conditions xk∈ Vs(k),m(k)
imply Vs(k),m(k)→ ∞.
?
Proposition 3.5. If U = {Us}s∈S is a coarse family in X, then it has a
coarse shrinking V = {Vs}s∈S such that for any M > 0 there is a bounded
subset AMof X with the property that B(x,M)∩Vs?= ∅ implies B(x,M) ⊂
Usprovided x ∈ X \ AM.
Proof. Pick x0∈ X and define f(x) = min(d(x,x0)/2,LU(x)/2). Notice f
is a coarsely proper function of Lipschitz constant 1/2. For each x ∈ X pick
s(x) ∈ S so that B(x,f(x)) ⊂ Us(x). Define Vsas the union of those balls
B(x,f(x)/2) so that s = s(x). It suffices to observe that B(x,M) ∩ Vs?= ∅
and M < f(x)/3 implies B(x,M) ⊂ Us. Indeed, B(y,f(y)) ⊂ Usfor some
y ∈ B(x,M).Since f(x) − f(y) ≤ d(x,y)/2 < M/2, one has f(y) >
f(x) − M/2 > 3M − M/2 > 2M and B(x,M) ⊂ B(y,f(y)) ⊂ Us.
?
Lemma 3.6. If U is a coarse family in X that is coarsely proper, then there
is a coarsely proper function f : U → R+such that the family {B(U,−f(U))}U∈U
is coarse.

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8N. BRODSKIY AND J. DYDAK
Proof. Define f(U) = inf{LU(x)/4 | x ∈ U}. Notice f is a coarsely proper
function. Pick s(x) ∈ S so that B(x,LU(x)/2) ⊂ Us(x). f(Us(x)) ≤ LU(x)/4
which implies B(x,LU(x)/4) ⊂ B(Us(x),−f(Us(x))). Thus {B(U,−f(U))}U∈U
is coarse.
?
In the large scale geometry one should think of bounded subsets of X as
points. Here is a generalization of the Lebesque number.
Definition 3.7. Let n ≥ 0. Suppose U is a family in X and A is a bounded
subset of X. The n-th Lebesque number Ln(U,A) is the supremum of t ∈
[0,∞] such that U|A has a refinement of multiplicity at most n + 1 and
Lebesque number at least t.
Notice such supremum exists as the cover of A consisting of points is of
Lebesque number 0 and multiplicity 1.
Observe that Ln(U,A), n ≥ 0, form an increasing sequence of numbers
bounded by L(U,A). If U|Ais of finite order, then they eventually stabilize
and are equal to L(U|A,A).
Let us point out that Sperner’s Lemma can be used to estimate higher
Lebesque numbers as follows: Consider a 2-simplex ∆ with vertices labeled
0, 1, and 2. Let U be the cover of ∆ by stars Ui, i = 0,1,2, of its vertices.
Consider a subdivision L of ∆ with mesh M (in this case it coincides with
the longest edge in the subdivision). Let X = A be the set of vertices
of L. Suppose V = {V0,V1,V2} is a shrinking of U|A. Obviously, there
is a shrinking of multiplicity 1. However, if we request V to be of large
Lebesque number, we run into problems. Namely, L1(U,A) ≤ M. Indeed, if
L(V) > M, we assign to each vertex v of L number i such that v ∈ Vi. We
are in the situation of the classical Sperner’s Lemma: vertices on the edges
of ∆ must be labeled with a number of one of the vertices of that edge.
Therefore one has a simplex in L whose vertices were assigned all three
numbers 0,1,2. Since L(V) > M, the three vertices belong to V0∩ V1∩ V2
and multiplicity of V is 3. Thus L1(V,A) ≤ M.
We will use the observation above in the case of M-scale connected spaces.
Definition 3.8. Suppose M is a positive number. A metric space X is
called M-scale connected if for every two points x,y ∈ X there is a chain of
points x = x1,x2,... ,xk= y such that dX(xi,xi+1) < M for all i < k.
Here is an application of Sperner’s Lemma for 1-simplices.
Lemma 3.9. Let M be a positive number and X be an M-scale connected
metric space. If L0(U,X) > M for some cover U of X, then U contains X
as an element.
Proof. Suppose V is a refinement of U of multiplicity at most 1 and Lebesque
number bigger than M. If X is not an element of V, then there are disjoint
non-empty elements V1,V2∈ V. Pick a chain of points x = x1,x2,... ,xk=
y such that dX(xi,xi+1) < M for all i < k and x ∈ V1, y ∈ V2. There is
an index j < k such that xj∈ V1and xj+1 / ∈ V1. The ball B(xj+1,M) is

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COARSE DIMENSIONS AND PARTITIONS OF UNITY9
contained in an element W of V and intersects V1. Therefore W = V1, a
contradiction.
?
4. The coarse category
Let us introduce the coarse category in a way that explains why two coarse
functions are considered equivalent if their distance is bounded.
Definition 4.1. Given a metric space (X,dX) and its two subsets X1and
X2the notation X1≤ X2means there is a positive number R such that X1
is contained in the ball B(X2,R) = {x ∈ X | dist(x,X2) < R}.
Proposition 4.2. A function f : X → Y of metric spaces is coarse if
and only if it preserves the relation ≤ of sets. Thus, X1 ≤ X2 implies
f(X1) ≤ f(X2).
Proof. Suppose f : X → Y preserves the relation ≤ of sets but not in the
sense of Roe. Therefore, for some M > 0 there is a sequence of points xn,yn
so that dX(xn,yn) < M for each n but dY(f(xn),f(yn)) → ∞ as n → ∞.
If f(A) is bounded for some subsequence A of xn, then f(B) is bounded for
the corresponding subsequence of yn(in view of f(B) ≤ f(A)) contradicting
dY(f(xn),f(yn)) → ∞ as n → ∞. Thus f(xn) → ∞ and f(yn) → ∞ as n →
∞. By induction define a subsequence anof {xn}n≥1and the corresponding
subsequence bnof {yn}n≥1with the property that dY(f(ak),f(bi)) > k and
dY(f(bk),f(ai)) > k for all k ≥ i. Since A = {an}n≥1≤ B = {bn}n≥1one
has f(A) ≤ f(B), a contradiction.
Suppose f : X → Y is coarse in the sense of Roe and X1 ≤ X2 in
X.Pick R > 0 so that X1 ⊂ B(X2,R) and choose M > 0 satisfying
dY(f(x),f(y)) < M if dX(x,y) < R for all x,y ∈ X.
pick y ∈ X2 so that dX(x,y) < R since dY(f(x),f(y)) < M one gets
f(X1) ⊂ B(f(X2),M). Thus f(X1) ≤ f(X2).
Given x ∈ X1
?
Notice that X1≤ X2for every bounded subset X1of X provided X2?= ∅.
Also, X1≤ X2implies X1is bounded provided X2is bounded. Therefore
f(A) is bounded for every bounded subset A of X and every coarse function
f : X → Y .
Given a function f : X → Y of metric spaces one can identify it with its
graph Γ(f) ⊂ X × Y . Therefore it makes sense to ponder the meaning of
Γ(f) ≤ Γ(g) for f,g : X → Y .
Proposition 4.3. Suppose f,g : X → Y are functions of metric spaces.
1. If g is coarse, then Γ(f) ≤ Γ(g) implies that the distance dist(f,g)
between f and g is finite. In particular, f is coarse.
2. If dist(f,g) is finite, then Γ(f) ≤ Γ(g).
Proof. 1. Suppose the distance dist(f,g) is not finite, so there are points
xn ∈ X with dY(f(xn),g(xn) > n for all n ≥ 1. Let R > 0 be a num-
ber such that B(Γ(g),R) contains Γ(f). For each n pick yn∈ X satisfying
dX(xn,yn)+dY(f(xn),g(yn)) < R. There is M > 0 so that dY(g(xn),g(yn)) <

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10N. BRODSKIY AND J. DYDAK
M for all n ≥ 1 as g is coarse. Now, dY(f(xn),g(xn) ≤ dY(f(xn),g(yn)) +
dY(g(yn),g(xn)) < R + M for all n ≥ 1, a contradiction.
2. Notice Γ(f) ⊂ B(Γ(g),dist(f,g)).
?
Definition 4.4. Given a function f : X → Y of metric spaces we define
the forward distance transfer function df: R+→ R+∪∞ as the infimum of
all functions α : R+→ R+∪ ∞ with the property that dX(x,y) ≤ t implies
α(t) ≥ dY(f(x),f(y)) for all x,y ∈ X.
The reverse distance transfer function df: R+→ R+∪∞ as the infimum
of all functions α : R+→ R+∪∞ with the property that dY(f(x),f(y))) ≤ t
implies dX(x,y) ≤ α(t) for all x,y ∈ X.
Notice that f is coarse if and only if dfmaps R+to R+, i.e. the values of
dfare finite. Also, f is asymptotically Lipschitz if and only if dfis bounded
by a linear function.
Proposition 4.5. If f,g : X → Y are two coarsely proper coarse functions,
then the following conditions are equivalent:
1. dist(f,g) is finite.
2. For every coarse family U = {Us}s∈S in Y the family {f−1(Us) ∩
g−1(Us)}s∈Sis coarse.
Proof. 1 =⇒ 2. Let dist(f,g) < M. Consider V = {B(Us,−M)}s∈S. It
is a coarse family, so f−1(V) is coarse by 2.12. Notice f−1(B(Us,−M)) ⊂
f−1(Us) ∩ g−1(Us) for all s ∈ S which is sufficient to establish coarseness of
{f−1(Us) ∩ g−1(Us)}s∈S.
2 =⇒ 1. If dist(f,g) is not finite, there is a sequence xn → ∞ such
that dY(f(xn),g(xn)) > n for all n. Put A = {xn}n≥1. By 2.7, the family
U = {Y \f(A),Y \g(A)} is coarse. However, {f−1(Us)∩g−1(Us)}s∈Sis not
coarse as it refines {X \ A} which is not coarse.
?
Our category is that of metric spaces and equivalence classes of coarse
functions. f ∼ g if dY(f(x),g(x)) is a bounded function of x.
Generalizing the concept of A ≤ B for subsets of a given metric space X,
we say Y coarsely dominates X (notation: X ≤coa
functions f : X → Y and g : Y → X such that g ◦ f is at a finite distance
from idX.
rseY ) if there are coarse
Proposition 4.6. Suppose f : X → Y and g : Y → X are coarse functions.
If g ◦ f is at a finite distance from idX, then both f : X → f(X) and
g : f(X) → X are coarsely proper and f ◦g is at finite distance from idf(X).
Proof. Suppose xn → ∞.
bounded as g would send it to a bounded subset of X. Thus f(xn) → ∞. If
f(xn) → ∞, then none of subsequences of {xn} is bounded. Therefore none
of the subsequences of {g(f(xn))} is bounded and g : f(X) → X is coarsely
proper. If dX(g(f(x)),x) < M for all x ∈ X, then dY(f(g(f(x))),f(x)) ≤
df(M) and f ◦ g is at finite distance from idf(X).
None of the subsequences of {f(xn)} can be
?

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COARSE DIMENSIONS AND PARTITIONS OF UNITY11
Proposition 4.7. A surjective coarse function f : X → Y of metric spaces
is a coarse isomorphism if and only if the reverse distance transfer function
dfis finite.
Proof. If there is a coarse function g : Y → X such that g ◦ f is at finite
distance M to idX, then df(a) ≤ dg(a) + 2M is finite.
Assume dfis finite and pick a right inverse g : Y → X. Notice dX(g(x),g(y)) ≤
df(dY(x,y)), so g is coarse.
?
5. Coarse dimensions
Definition 5.1. The coarse dimension dimcoa
coarse dimension dimCOA
RSE(X)) is the smallest integer n such that any fi-
nite coarse family in X (respectively, any coarse family in X) has a coarse
refinement with multiplicity at most n + 1.
rse(X) (respectively, the major
Remark 5.2. Using Proposition 4.4 on p.1104 in [2] (notice that the words
‘uniformly bounded’ are erroneously inserted there) one can show that, for
proper metric spaces X, the major coarse dimension of X coincides with the
asymptotic dimension of Dranishnikov. In view of 8.2, our coarse dimension
and Dranishnikov coarse dimension are identical.
Given a coarse family U = {Us}s∈S in a subset A of X one can extend
it to a coarse family U′= {Us∪ (X \ A)}s∈S in X. Notice that V ∩ A is
a coarse refinement of U for any coarse refinement V of U′. Therefore the
following holds.
Corollary 5.3. If A is a subset of a metric space X, then dimcoa
dimcoa
RSE(X).
Proposition 5.4. If Y coarsely dominates X, then dimcoa
and dimCOA
RSE(Y ).
rse(A) ≤
rse(X) and dimCOA
RSE(A) ≤ dimCOA
rse(X) ≤ dimcoa
rse(Y )
RSE(X) ≤ dimCOA
Proof. The proof is almost the same for both dimensions. Suppose U is a
coarse family in X and f : X → Y , g : Y → X are coarse functions such
that there is M > 0 with dX(x,g(f(x))) < M for all x ∈ X. Replacing Y
by f(X) we may assume f is onto and both f and g are coarsely proper (see
4.6). The idea of the proof is to refine g−1(U) by V and then refine f−1(V)
to obtain a desired refinement W of U of multiplicity at most n + 1, where
n is the dimension of Y . Consider U′= {B(Us,−M)}s∈S. It is a coarse
family in X, so {g−1(B(Us,−M))}s∈Sis coarse and it has a coarse shrinking
V = {Vs}s∈S of multiplicity at most n + 1. Suppose x ∈ f−1(Vs) \ Us.
Since dX(x,g(f(x))) < M, g(f(x)) / ∈ B(Us,−M). However, f(x) ∈ Vs⊂
g−1(B(Us,−M)), a contradiction.
?
Definition 5.5. The minor asymptotic dimension ad(X) (respectively, the
asymptotic dimension asdim(X)) is the smallest integer n such that the
function U → Ln(U,X) is coarsely proper on the space of finite covers
(respectively, arbitrary covers) U of X.

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12N. BRODSKIY AND J. DYDAK
Let us show that our definition of asymptotic dimension is equivalent to
that of Gromov.
Proposition 5.6. asdim(X) ≤ n if and only if for each M > 0 there is
a uniformly bounded family U in X of Lebesque number at least M and
multiplicity at most n + 1.
Proof. If asdim(X) ≤ n as in 5.5 and M > 0, then there is N > 0 such that
every cover V of X satisfying L(V,X) ≥ N has a refinement U of multiplicity
at most n + 1 and Lebesque number at least M. Pick V to be the cover of
X by balls of radius N. The resulting U is uniformly bounded.
Suppose for each M > 0 there is a uniformly bounded family UMof
multiplicity at most n + 1 and Lebesque number at least M. Let α(M)
be the supremum of diameters of elements of UM. Given any family V of
Lebesque number at least α(M)+1, UMis a refinement of of V which proves
that the function V → Ln(V,X) is coarsely proper on the space of all covers
V of X.
?
Quite often it is useful to have even stronger conditions imposed on covers
appearing in 5.6.
Proposition 5.7 (Gromov). If Gromov asymptotic dimension asdim(X)
does not exceed n, then for any M,N > 0 there exist uniformly bounded
families Ui, 1 ≤ i ≤ n + 1, such that each Uiis N-disjoint and U =
n+1
?
i=1
Ui
is of Lebesque number at least M.
Proof. Consider a uniformly bounded family V = {Vs}s∈sof multiplicity at
most n+1 and Lebesque number at least 2(n+1)·(M+N). Lemma 3.3 says it
n+1
?
i=1
each x ∈ X and each Viconsists of disjoint sets. Define Uias {B(W,−N)},
W ∈ Vi.
can be refined by V′=
Visuch that LV(x) ≥ LU(x)/(2n+2) ≥ M+N for
?
Let us characterize spaces of asymptotic dimension 0.
Proposition 5.8. asdim(X) > 0 if and only if there exist a number M > 0
and a coarsely proper sequence {(xn,yn)}∞
that dist(xn,yn) → ∞ and the points xnand yncan be M-scale connected
in X \ B(x0,n).
n=1of pairs of points in X such
Proof. If asdim(X) = 0, then for any M > 0 there exists an M-disjoint
cover of X by uniformly bounded sets. Therefore, the distance between two
points x and y which can be M-scale connected in X is uniformly bounded.
Suppose asdim(X) > 0. Let n be a positive integer and x0be the base
point in X. There is L > 0 such that X does not have a uniformly bounded
cover of Lebesque number bigger than L and multiplicity 1.
equivalence relation on X \ B(x0,n) by saying x ∼ y iff x and y can be 2L-
scale connected in X \ B(x0,n). The cover of X by the equivalence classes
Define an

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COARSE DIMENSIONS AND PARTITIONS OF UNITY13
has Lebesque number at least 2L, therefore these classes are not uniformly
bounded by the choice of L. Thus, there exist points xnand ynwhich can
be 2L-scale connected in X \ B(x0,n) such that dist(xn,yn) is arbitrarily
large.
?
Proposition 5.9. If Y coarsely dominates X, then asdim(X) ≤ asdim(Y )
and ad(X) ≤ ad(Y ).
Proof. The proof is almost the same for both dimensions. Suppose U is a
coarse family in X and f : X → Y , g : Y → X are coarse functions such that
there is M > 0 with dX(x,g(f(x))) < M for all x ∈ X. By replacing Y with
f(X) we may assume f is onto and both f and g are coarsely proper (see
4.6). The idea of the proof is to refine g−1(U) by V and then refine f−1(V)
to obtain a desired refinement W of U of multiplicity at most n + 1, where
n is the dimension of X. Take a coarsely proper function α : R+→ R+
with the property that any finite cover (respectively, arbitrary cover) U of
Y satisfying L(U,Y ) ≥ α(t) has a refinement V of multiplicity at most n+1
so that L(V,Y ) ≥ t.
Given t > 0 pick β(t) so that Lg(β(t)) > α(t) (see 2.9). Assume L(U) >
M+β(t). Consider U′= {B(Us,−M)}s∈S. L(U′) > β(t), so {g−1(B(Us,−M))}s∈S
is of Lebesque number at least α(t) and it has a shrinking V = {Vs}s∈S
of multiplicity at most n + 1 and L(V) ≥ t. Suppose x ∈ f−1(Vs) \ Us.
Since dX(x,g(f(x))) < M, g(f(x)) / ∈ B(Us,−M). However, f(x) ∈ Vs⊂
g−1(B(Us,−M)), a contradiction.
?
Theorem 5.10. The major coarse dimension of X does not exceed the
asymptotic dimension of X.
Proof. Suppose asdim(X) = n < ∞ and U = {Us}s∈Sis a coarse family in
X. By Lemma 3.4 we may assume U is coarsely proper. By induction on k
find a sequence of numbers M0= 1,M1,M2,..., and covers Vk= {Vt}t∈T(k),
k ≥ 1, of multiplicity at most n + 1 and satisfying the following conditions:
a. L(Vk,X) ≥ Mk−1for k ≥ 1.
b. The diameter of each element of Vkis smaller than Mk.
c. The family {B(x,Mk−1) | d(x,x0) ≥ Mk} refines U for each k ≥ 1.
d. Mk+1> 2Mkfor all k ≥ 1.
Find functions j(k) : T(k) → T(k + 1) so that Vt ⊂ Vj(k)(t). Denote
{x : Mk≤ d(x,x0) < Mk+1} by Ak. Given t ∈ T(k) so that Vtis contained
in some element of U define α(t) ∈ S by looking at the sequence Vt ⊂
Vj(k)(t)⊂ ..., picking the latest element contained in some Usand setting
α(t) = s (it is possible each element of the sequence is contained in some Us
in which case all of them are contained in some Usand that s is picked as
α(t)). Define Wsas follows: it is the union of non-empty sets of the form
Vt∩ Akso that Vt∈ Vk−1and α(t) = s. Notice that m(W) ≤ n + 1 as in
the annulus Akthe family W is obtained from Vk−1by assembling some of
its elements together.

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14N. BRODSKIY AND J. DYDAK
We plan to show W is coarse by proving that if Mk≤ d(x,x0) < Mk+1,
then B(x,Mk−3) is contained in some Ws. Indeed, there is t ∈ T(k − 2) so
that B(x,Mk−3) ⊂ Vt. Put r = j(k − 2)(t) and u = j(k − 1)(r). Points
of B(x,Mk−3) can belong to only two of the following three annuli: Ak−1,
Ak, and Ak+1. If z ∈ B(x,Mk−3) ∩ Ak+1, then Vu ⊂ B(z,Mk) ⊂ Us for
some s ∈ S. We might as well put s = α(t) = α(u) = α(r). In this
case B(x,Mk−3) ⊂ Ws. If B(x,Mk−3) misses the last annulus, then only
α(r) is definitely defined (α(u) may not exist) and α(t) = α(r).
B(x,Mk−3) ⊂ Ws, where s = α(r).
Now,
?
Remark 5.11. 5.10 generalizes Proposition 4.5 on p.1105 of [2].
6. The large scale dimension
In this section we prove that any dimension of X (asymptotic, major
coarse, or minor asymptotic), if finite, equals the coarse dimension of X.
That corresponds to results of Dranishnikov [2] that asdim(X) or asdim∗(X),
if finite, are equal to the dimension of the Higson corona of any proper met-
ric space X. Our proofs are direct and become simpler by introducing a new
dimension, the large scale dimension of X. That dimension turns out to be
identical with the coarse dimension.
Definition 6.1. The large scale dimension dimlarge
integer n such that A → Ln(U,A) is a coarsely proper function on the set
of bounded subsets of X for all finite coarse families U in X.
scale(X) of X is the smallest
Notice dimlarge
Obviously, dimlarge
scale(X) = −1 if X is bounded.
scale(X) ≥ dimlarge
scale(A) for any subset A of X.
Proposition 6.2. ad(X) ≥ dimlarge
scale(X) and dimcoa
rse(X) ≥ dimlarge
scale(X).
Proof. The inequality dimcoa
given n = dimcoa
elements one has a coarse refinement V of U such that the multiplicity m(V)
is at most n + 1. In that case
Ln(U,A) ≥ L(V,A) ≥ inf
rse(X) ≥ dimlarge
scale(X) is almost obvious. Indeed,
rse(X) and given a coarse family U in X consisting of m
a∈ALV(a)
and is a coarsely proper function of A.
Suppose ad(X) = n and U is a coarse cover of X consisting of m ele-
ments. Given t > 0 find a bounded subset U of X such that U|(X\U)has a
refinement V of multiplicity at most n + 1 and Lebesque number at least t.
For any bounded subset A of X \ U, Ln(U,A) ≥ L(V,A) ≥ t which proves
dimlarge
scale(X) ≤ n.
As shown in [5], the asymptotic dimension of Rnis at most n (see p.793).
For the convenience of the reader let us reword the argument from [5] as
follows: Given M > 0 consider the triangulation on the unit n-cube Inob-
tained by starring at the center of each face. It is invariant under symmetries
?

Page 15

COARSE DIMENSIONS AND PARTITIONS OF UNITY15
of Inand the cover of Inby stars of vertices has a positive Lebesque number
k and is of multiplicity at most n +1. Rescale Inby the factor of M/k and
extend its triangulation over the whole Rnby reflections. The cover of Rn
by stars of vertices has Lebesque number at least M and is of multiplicity
at most n + 1.
Let us show how to use the large scale dimension to estimate asymptotic
dimension from below.
Proposition 6.3. dimlarge
scale(Rn) ≥ n.
Proof. Since dim(In) = n, there is a finite open cover U of Inwith no open
refinement of multiplicity at most n. Let In
k times with the corresponding cover Uk. We request In
obtained by adding the corresponding elements of Ukis a finite coarse family
∞ ?
k=1
k⊂ Rnbe a copy of Inenlarged
k→ ∞ so that V
on A =
In
k. Notice Ln−1(V,In
k) = 0 for all k. Thus dimlarge
scale(Rn) ≥ n.
?
Proposition 6.4. If dimlarge
0.
scale(X) = 0, then asdim(X) = 0 and dimCOA
RSE(X) =
Proof. It suffices to show asdim(X) = 0 (see 5.10). Suppose asdim(X) >
0. By 5.8 there exist a number M > 0 and a coarsely proper sequence
{(xn,yn)}∞
n=1of pairs of points in X such that dist(xn,yn) → ∞ and the
points xn and yn can be M-scale connected in X \ B(x0,n) by a chain
Pn. Consider a coarse family U consisting of two sets: X \
∞ ?
n=1{xn} and
X \
∞ ?
n=1{yn}.
Since C → L0(U,C) is a coarsely proper function, there is a chain Pnsuch
that L0(U,Pn) > M. This contradicts 3.9 since Pnis M-scale connected and
the cover U is non-trivial on Pn.
?
Definition 6.5. Given a point-finite family U = {Us}s∈Sin X (that means
each point of X belongs to at most finitely many elements of U) by the
canonical partition of unity of U we mean the family of functions {fs/f}s∈S,
where fs(x) = dist(x,X \ Us) and f(x) =
?
s∈S
fs(x). If T is a subset of S,
then XTis defined to be {x ∈ X |?
s∈T
fs(x)/f(x) = 1} and by ∂XTwe mean
the set of all x ∈ XT such that fs(x) = 0 for some s ∈ T.
Notice that f(x) > 0 for all x ∈ X such that LU(x) > 0 and f is a
Lipschitz function if U is of finite multiplicity.
Lemma 6.6. If the large scale dimension of X is at most n, then any
coarse family U in X of finite multiplicity m has a coarse refinement V of
multiplicity at most n + 1.
Proof. Suppose U exists with no coarse refinement of multiplicity at most
n+1. Using 3.4 we reduce the general case to that of U = {Us}s∈Sconsisting