Applications of Locally Compact Spaces in Polyhedra:
Dimension and Limits
MAYSOON QOUSINI1, HASAN HDIEB2, EMAN ALMUHUR3
1Faculty of Science and Information Technology,
Al-Zaytoonah University of Jordan,
11183, Amman,
JORDAN
2Faculty of Science,
The University of Jordan,
11942, Amman,
JORDAN
3Department of Basic Science and Humanities,
Applied Science Private University,
11931, Amman,
JORDAN
Abstract: - The study of applications of locally compact spaces in polyhedra in relation to their dimensions as
well as homotopy and extension problems developed in the late 1940s and early 1950s under the leadership of
mathematician. Many mathematicians studied application locally compact in polyhedra. A polyhedron can be
obtained by subdivision, as a simplicial metric complex; thus, re-gluings of polyhedra can also be seen as
simple complexes. Thus, the topology of a simplicial metric complex X is the topology quotient of the
reattachment. The objective of this work is to shed light on the applications in polyhedra of locally
compact spaces and to highlight the limits of these applications. A continuous application f of X in P
defines a finite open overlay of X, and a partition of the unit subordinate to this overlay, f is homotopic to an
application f ', obtained by composing the restriction to A, of an application of X in the KR polyhedron, and a
simplistic application of a sub-polyhedron KR' in P. The problem of extension deserves to be elucidated to
understand how it is possible to get around certain conceptual difficulties.
Key-Words: - polyhedron, compact spaces, locally compact spaces, paracompact spaces, CECH
COHOMOLOGY, homotopy, extension.
Received: April 12, 2023. Revised: November 9, 2023. Accepted: December 4, 2023. Published: February 27, 2024.
1 Introduction
The study of applications of locally compact spaces
in polyhedra in relation to their dimensions as well as
homotopy and extension problems developed in the
late 1940s and early 1950s under the leadership of
mathematician Henry CARDAN, [1], [2]. In a more
general framework, when we consider a polyhedron
P, finite or infinite, but locally finite (thus locally
compact) and supposedly simplicial abstract (we
decompose it if it is not), it is defined by a simplicial
abstract complex K and is identified with a subspace
of the "cube" IK (I designating the bounded closed
interval [0; 1]), [3]. Thus, a point of P is a system (λK)
of real numbers between 0 and 1, all zero except a
finite number, and whose sum is equal to 1; the index
k runs through the set K and the set of k such that λK ≠
0 is a simplex of the simplicial complex K. Our goal
in this work is to study these applications in their
dimension and possibly raise problems of
homotopies and extension.
These continuous applications transform, indeed,
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the unit I ball into a locally compact part
Consequently, a continuous application f of the X
space in P is then defined by the coordinate data λK =
fK(x) of the transform of the point x by f; the fK are
numerical continuous functions, with values in the
segment I, such as:
I. In the (open) UK set of points x where fK > 0,
the fj is zero except for a finite number;
II.The sum Σ fK is equal to one (this sum makes
sense, since at each point all terms are zero except for
a finite number).
If we have a system of continuous functions fK,
with values in I, defined in a locally compact space X,
then the satisfaction of the first condition leads to say
that it is a finite type partition. This partition is finally
a partition of the unit when, in addition to the first
criterion, the second condition is satisfied.
The objective of this work is to shed light on the
applications in polyhedra of locally compact spaces
and to highlight the limits of these applications. The
text is structured as follows: the first section presents
problem statements to highlight locally compact
spaces in polyhedra, the second section deals with
homotopy problems and the last section is devoted to
the limits in the case of extensions.
2 Materials and Methods
2.1 Continuous Applications of a Locally
Compact Space X in Polyhedra
This work could have been treated without leaving
the framework of the Banach spaces, [1], [2], [3], [4].
Moreover, given the results that we are trying to
highlight with regard to polyhedra, we preferred to
place ourselves within the framework of locally
compact spaces. We use without reference the
elementary results of the theory of measurement,
placing ourselves exclusively in the wake of the
measurements on the space of continuous
applications with compact supports on a given
locally compact space.
One of the most important statements is that any
open overlap of X is said to be finite if each set of the
overlap meets only a finite number of them, [5].
Moreover, to each (fk) partition (of finite type) let us
associate the Uk open overlay defined as above: this
open overlay is said to be "associated with the
partition"; it is of finite type.
In [3], Given an open finite (Vk)k∈K overlay, a
partition of the unit (fk)k∈K is said to be subordinate
to the overlay if it has the same set of indices k, and if,
for any k∈K, the set Uk of points x such that fk(x) ≠ 0
is contained in Vk.
From this demonstration, we can say that a
continuous application f of X in P defines a finite
open overlay of X, and a partition of the unit
subordinate to this overlay.
Conversely, let R be an open finite overlay, and
(fk) a partition subordinate to this overlay: then (fk)
defines a continuous application f of X in the
polyhedron KR (topological realization of the "nerve"
of the overlay of R; this nerve is the simplicial
abstract complex KR defined as follows: its vertices
are the sets of the overlay R, and a set of "vertices" is
a "simplex" if the corresponding sets of the overlay
have a non-empty intersection).
Note that from, [6], a polyhedron can be obtained
by subdivision, as a simplicial metric complex; thus
re-gluings of polyhedra can also be seen as simple
complexes. Thus the topology of a simplicial metric
complex X is the topology quotient of the
reattachment.
In order to study the applications of locally
compact spaces, we start from two problems which
are the following:
Problem (2.1):
Given an open overlay of finite type R, are there
partitions subordinate to this overlay? Yes, if the
space is normal (in other words, two disjoint closed
neighbourhoods can be separated by two disjoint
open neighbourhoods). Indeed, if (Uk) is a finite open
overlay of a normal space X, there is an open overlay
(Vk) such as Vk⊂ Uk; or gk a continuous function with
values in the segment [0; 1], such that gk(x) = 1 for all
x ∈ Vk, gk(x)=0 for all x ∉ Uk.Or g(x) equals
summation of over k, which has a meaning
and a numerical continuous function with values
greater than or equal to 1. Just
ask:
( ) ( ) / g(x)
kk
f x g x
to obtain a partition of the
unit subordinate to the overlay (Uk).
From two overlaps (Uk) and (Vj) (not necessarily
having the same set of indices), we say that (Vj) is
finer than (Uk) if any set (Vj) is contained in at least
one set Uk.
Problem (2.2):
X locally compact given, is there a finer finite open
overlap than an arbitrarily given open overlap? Any
compact space is paracompact (trivial) (moreover, in
a compact space, any finite open overlap is finite). A
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paracompact space is normal. The immediate
consequence is that if X is paracompact, there is
always a partition of the unit whose associated
overlay is finer than an arbitrarily given open overlay.
Consequently, there are arbitrarily thin unit
partitions.
Note: it is easy to characterize, among the locally
compact X, those that are paracompact. For this, it is
necessary and sufficient that X is a meeting of
disjoint open subspaces (each of which is therefore
also closed), each of which is an enumerable meeting
of compact subsets. An equivalent condition is: X has
at least one finite overlay of relatively compact open
subspaces, [6].
Any closed sub-space of a paracompact space is
paracompact.
Proposal (2.3): Let R be an open covering of
finished type, KR the polyhedron it defines. Each
partition of the unit whose associated overlap is finer
than R defines (in several ways) a continuous
application of X in KR; all these applications
(whatever partition they correspond to) are
homotopic.
Let R' be the overlay associated with a partition, R'
being finer than R. For each simple application
KR' KR (which associates to a set of R' a set
of R containing it), let us compose the application X
KR' defined by the partition, with KR' KR. By
definition, we thus obtain all the applications of X in
KR defined by the partition. These applications
(whatever the partition whose associated overlap R'
is finer than R) each define a partition of the unit
subordinate to R. To show that they are all homotopic,
it is enough to use the affine structure of the space of
the partitions subordinate to R.
In the next part of this section according to [6], we
will discuss The Approximation of an application f of
a closed part A of X, in a polyhedron P
We assume X paracompact. The application f of A in
P defines an open, finite overlap of A. The joining of
the sets of this overlap is complementary (in X) to a
closed set B, therefore paracompact; if dim (X – A) ≤
n, then the dimension of B is less than or equal to n.
There is an open overlap R, of the finished type of X,
whose trace on A is a finer overlap than that defined
by f. Let us associate with R, a partition of the unit (ϕk)
in X space. The restrictions on A of (ϕk) thus define
an overlap of A that is finer than that defined by f.
The (ϕk)'s define an application ϕ of X in KR, which
applies A in a sub-polyhedron KR' of KR. There is an
application f ' of A in P, consisting of A in KR'
(restriction of ϕ), and a simple application KR' in P. In
A, the applications f and f ' are homotopic (according
to proposal (2.3)).
In summary: f is homotopic to an application f ',
obtained by composing the restriction to A, of an
application of X in the KR polyhedron, and a
simplistic application of a sub-polyhedron KR' in P.
Therefore, to extend f' it is sufficient to know how to
extend the KR' application in P into an application of
KR in P (problem studied previously). If f' is
extendable, then f will also be extendable, by virtue
of the following lemma:
Lemma (2.4):
Let X be a paracompact, and A a closed contained in
X. Any continuous application of A in a polyhedron
P, which. In A, is homotopic to an application
extendable to X, is itself extendable to X.
To demonstrate this assertion, it is sufficient to prove
that any application of A in a polyhedron P is
extendable to a neighbourhood of A (where A is a
closed part of a paracompact space X).
However, it is true that if A is compact then the
image of A is contained in a finished polyhedron. If
A is not compact, then it is sufficient to demonstrate
when X is a countless meeting of compacts, X1 ⊂ X2
⊂ X3 ⊂ … ⊂ Xj ⊂ …. The restriction from f to A ∩
X1 extends to a compact neighbourhood V1 of A ∩ X1;
hence f1 on A ∪ (V1 ∩ X1) = A1. The restriction from
f1 to A ∩ X2 extends to a compact neighbourhood V2
of ... , etc.
It can also be proved that if f and g (applications of x
in the polyhedron P) coincide on a closed part A ⊂ X,
they are homotopic (in X) to two applications f' and
g' which coincide on a neighbourhood of A.
3 Size of A Locally Compact Space
It is important to specify that in all that follows, only
locally compact spaces X with the following property
are considered:
X has arbitrarily fine open overlaps of finite
dimensions. Recall that from, [3], [7], an overlap is
said to be of dimension greater than or equal to n, if
its "nerve" is of dimension greater than or equal to n,
in other words, if each point of the space belongs at
most to n + 1 sets of the overlap, any closed
sub-space of such space X enjoys the same property).
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Definition (3.1):
It is said that dim X ≤ n if there are open overlaps of
dimension greater than or equal to n, arbitrarily fine.
In fact, dim X = n if dim X ≤ n and dim X ≠ n – 1. If
dim X ≤ n and if A is a closed subspace of X, then
dim A ≤ n: it is immediate.
Theorem (3.2):
If dim X ≤ n, any continuous application, in the
sphere Sn, of a closed part A of X is extendable to X.
This is a direct consequence of the extension theorem
of an application in Sn of a sub-polyhedron of a
polyhedron of dimension less than or equal to n.
Notice that as a reciprocity of this theorem:
Suppose only that any compact sub-space X’ ⊂ X
enjoys the property Πn(X’): any continuous
application in Sn of a closed part of X' is extendable
into a continuous application of X' in Sn. Then dim X
≤ n.
Demonstration of this reciprocity:
we first observe that the property Πn(X) leads to
Πn+1(X).
Indeed, suppose that X verifies Πn; that is A a closed
part of X, and f a continuous application of A in Sn+1.
Consider Sn as the equator of Sn+1 and either B =
f-1(Sn) ⊂ A. There is a continuous application g of X
in Sn, which coincides with f on B; on A, f and g
(considered as applications in Sn+1) are homotopic,
because f(x) and g(x) are never diametrically
opposed; since g(x) is extendable to X, f is also
(theorem 1).
Let R be an overlap of X, of finite dimension,
arbitrarily fine, and formed of relatively compact
openings. Let us choose an application f of X in KR,
in the class defined by R. For any integer k, let Xk be
the reciprocal image of the k-skeleton Pk of KR, and fk
the restriction from f to Xk. We will show that fn
extends into a continuous application gn of x in Pn, so
that g(x) belongs to the smallest closed simplex
containing x; then the reciprocal image, by gn, of the
canonical overlap of Pn will be an arbitrarily fine
open overlap of dimension lower or equal to n. To
prove the existence of gn, we define, by downward
recurrence on k ≥ n, an application gk of X in Pk in the
following way: gk = fk for k large enough (in fact: for
k at least equal to the dimension of KR); gk coincides
gk+1 on Xk, and is deduced from gk+1 using the
property Πk for the compacts contained in X.
Corollary (3.3):
For dim X ≤ n, it is necessary and sufficient that dim
Y ≤ n for any compact sub-space Y⊂X.
As a remark, we can add that this could be used as a
definition for the dimension of a locally compact
space X, without any restrictive hypothesis on X).
Before moving on to theorem (5.4), it is essential to
first define the notion of R-application. It is thus a
continuous application f of X in a space Y, such that
the reciprocal image of each point of Y is "small of
order R" (R designating an open overlap of X).
Theorem (3.4):
For a space X to be of dimension less than or equal to
n, it is necessary and sufficient that for any open
overlap R of X, there is an R-application of X in the
polyhedron of dimension less than or equal to n.
The condition is obviously necessary. To show that it
is sufficient, it is sufficient to demonstrate when X is
compact (according to the previous corollary). Then
the reciprocal image of any "fairly small" set of the
polyhedron P (which can be assumed to be finite) is
still small of order R.
Let us take a subdivision P' of P, fine enough so that
the overlap of X defined by the application of X in P'
is finer than R. Since this overlap is of dimension less
than or equal to n, X has many arbitrarily fine open
overlaps of dimension less than or equal to n.
Corollary (3.5):
The dimension of the product of two spaces is at most
equal to the sum of the dimensions of these spaces.
On the other hand, it can be smaller than the sum).
We will notice that the dimension of a quotient space
may be greater than the dimension of the space itself
(Peano curve).
4 CECH COHOMOLOGY
Let R' be a finished type X overlay, thinner than a
finished type R overlay. Then, all the simplistic
applications of KR' in KR (which to a set of R',
associate a set of R containing it) are homotopic
(proposal 1), so in, [4], [5], they define the same
homomorphism of the cohomology groups: H*(KR)
in H*(KR') (any coefficients, fixed once and for all).
Transitivity (obvious) of these homomorphisms.
There is therefore an inductive limit (direct limit) of
the H*(KR) groups, with canonical homomorphisms
of the H*(KR) within this inductive limit. This
inductive limit is the Cech cohomology group H(X),
X being locally compact and paracompact.
NB: this definition is only consistent with Cech's
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definition if X is compact for X not compact, this is
the correct generalisation of Cech's definition). It can
be shown that the cohomology groups thus obtained
are those given by the axiomatics of the beams.
Proposal (4.1.1):
If dim X ≤ n, the cohomology groups Hp(X) are zero
for p > n, whatever the group of coefficients.
This is obvious from the definition of these groups,
and from the definition of the dimension.
Or A a closed part of X. For each overlap R of X, or
RA the overlap of A induced by R; then KRA identifies
itself with a sub-polyhedron of KR. We can consider
the inductive limit of H*(KR mod KRA); this is, by
definition, the relative cohomology group H (X mod
A).
Proposal (4.1.2):
We have an exact sequence of canonical
homomorphisms
… Hn(X mod A) Hn(X)
Hn(A)
Hn+1(X mod A) … (for any system
of coefficients).
Indeed, we have such an exact sequence for the
cohomology of the KR polyhedron and its
sub-polyhedron KRA. However, the inductive limit of
a family of exact sequences is an exact sequence.
Remark: if dim (X – A) ≤ n, then Hp(X mod A) = 0
for p > n.
Let us from, [6], recall without demonstration the
well known fact: by compact X, the group H (X mod
A) depends only on the space X - A, and is identified
with the cohomology group (of Cech) of "second
species", or "with compact supports" of the locally
compact space X - A.
4.1 Effect of Continuous Application
Let f be a continuous application of X in X', which
transforms a closed part A ⊂ X into a closed part A’
⊂ X’. The reciprocal image of any finite open
overlap of X' is a finite open overlap of X.
We deduce, after passing the inductive limit, a
homomorphism of the cohomology groups: Hn(X’
mod A’) in Hn(X mod A).
4.2 Transitivity of these Homeomorphisms
Let us now have two continuous applications f and g
of X in Y, whose restrictions to A are identical. We
deduce a homomorphism (f, g) * of H(Y) in H (X
mod A) (group of arbitrary coefficients). This
homomorphism is defined by crossing the inductive
limit. The homomorphism (f, g) * relating to Cech
cohomology groups has properties similar to those
indicated for singular cohomology.
4.3 Some important Definitions
Definition (4.4.1), [4], Let Y be an aspherical
polyhedron in dimension less than n. We have
already defined the fundamental class of Y as an
element of Hn (Y, Hn(Y)) where for Y the
cohomology of Cech is identified with the singular
cohomology.
We then deduce that the characteristic class of an
application f of X (paracompact) in Y is an element
γ(f) of Hn(X, Hn(Y)), image of the fundamental class
by the homomorphism Hn(Y , Hn(Y)) defined by f.
Definition (4.4.2), [5], The deviation of a pair (f, g)
of applications of X in Y, which coincide on a closed
part A of X: it is an element γ (f, g) of Hn (X mod A,
Hn(Y)), image of the fundamental class by the
homomorphism (f, g) * of Hn (Y, Hn(Y)) in Hn (X
mod A, Hn(Y)).
Definition (4.4.3):
The obstruction of an application f of A in Y (A:
closed part of X paracompact): it is an element β (f)
of Hn+1(X mod A, Hn(Y)), image of the characteristic
class of f (element of Hn (A, Hn(Y)) by the canonical
homomorphism Hn(A , Hn(Y)) in Hn+1(X mod A ,
Hn(Y)).
4.4 Extension and Homotopy Theorems
Throughout the paragraph, X denotes a locally
compact space satisfying the conditions of paragraph
2; A denotes a closed part of X, and Y denotes an
aspherical polyhedron of dimensions less than n (e.g.
a sphere of dimension n).
The idea is to transpose to these cases theorems
(4.2) and (4.4), in which X is a polyhedron and A a
sub-polyhedron (it is true that then Y is not
necessarily a polyhedron). Thus, we define a new
theorem which is in reality only theorem (5.2) stated
above.
Theorem (4.4.1):
We assume dim X ≤ n+1. Then, for an application f
of A in Y to be extendable to X, it is necessary and
sufficient that the obstruction β (f) ∈ Hn+1(X mod A,
Hn(Y)) is null. This is true, strictly speaking, only for
any n ≥ 2; if n = 1, it is furthermore assumed that the
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space Y is i-simple for any integer i ≤ n+1, and then
the conclusion remains.
Theorem (4.4.2):
It is assumed that dim X ≤ n. Then, for two
applications f and g of X in Y, which coincide on A,
to be homotopic modulo A, it is necessary and
sufficient that the deviation
γ (f, g) ∈ Hn (X mod A, Hn(Y)) is zero. Moreover,
given an application f of X in Y, and an arbitrary
element γ of Hn (X mod A, Hn(Y)), there is an
application g of X in Y, equal to f on A, and such that
the deviation γ (f, g) is precisely γ. In particular, (A is
assumed to be empty): the classes of applications of
X in Y are in one-to-one correspondence with the
elements of the (Cech) cohomology group Hn (X,
Hn(Y)).
The conclusions of this theorem are only valid, in
reality, if n ≥ 2. For n = 1, they are valid provided that
it is also assumed that the space Y is i-simple for all i
≤ n (respectively i< n).
It is possible to apply the previous theorems
especially when Y is a sphere of dimension n. But for
n = 1, the nullity of the homotopy groups Πi(S1) for i
≥ 2 makes it possible to give the following extensions
of theorems 1 and 2.
Theorem (4.4.3):
Or a closed part of X. For two applications f and g of
X in S1 to extend to X, it is necessary and sufficient
that the obstruction β (f) ∈ H2(X mod A, Z) is zero.
Theorem (4.4.4):
Or A a closed part of X. For two applications f and g
of X in S1, which coincide on A, to be homotopic
modulo A, it is necessary and sufficient that the
deviation γ (f, g) ∈ H1(X mod A, Z) is zero. The
classes of applications of X in S1 correspond
biunivocally to the elements of the cohomology
group (of Cech) of the X space, with integer
coefficients.
4.5 Application of the Extension Theorem:
Cohomological Characterisation of the
Dimension
Lemma (4.5.1):
If dim(X) ≤ n+1, and if Hn+1(X mod A, Z) = 0 for any
closed part A of X, then any application of A in Sn is
extended into an application of X in Sn.
This is a direct consequence of theorem (5.5.1).
According to the reciprocal of theorem (4.2), we see
that, in the hypotheses of the lemma, the dimension
of X is less than or equal to n. Reciprocally, it is clear
that if dim(X) ≤ n, then Hn+1(X mod A, Z) = 0 for any
closed part A of X. Consequently, we can state a new
theorem.
Theorem (4.5.2):
If X is of finite dimension, the dimension of X is the
largest of the integers n such that there exists a closed
part A of X satisfying Hn (X mod A, Z) ≠ 0. (The
integer coefficients, for cohomology, play a
privileged role for the characterisation of the
dimension; it can be seen that this is due to the fact
that the homology group Hn(Sn) is isomorphic to Z).
Remark (4.5.3):
If X is compact (a case to which we can return,
since the dimension of a non-compact space is the
upper limit of the dimensions of the contained
compacts), the cohomology group Hn (X mod A, Z),
which intervenes in the characterisation of the
dimension is none other than the cohomology group
with compact supports of the open subspace X - A.
As an example: the space IRn is of dimension
(topological) equal to n, because the cohomology
with compact supports of an open ball is not null
for dimension n.
A polyhedron of (simplistic) dimension n is
of(topological) dimension n.
For a closed part of IRn to be of dimension n, it is
necessary and sufficient that it has at least one
interior point. (This is sufficient according to
theorem (5.6.2); it is necessary since if A has no
interior point in any triangulation of IRn, each
n-simplex contains a point which does not belong to
A, which allows (by central projection in each
n-simplex) to find an ε-application of A in a
polyhedron of dimension n - 1.
5 Conclusion
This work has highlighted the importance of locally
compact space applications in the case of polyhedra,
but also their limitations. A continuous application f
of X in P defines a finite open overlay of X, and a
partition of the unit subordinate to this overlay, f is
homotopic to an application f ', obtained by
composing the restriction to A, of an application of X
in the KR polyhedron, and a simplistic application of
a sub-polyhedron KR' in P. Indeed, problems may
remain and are linked to homotopic applications as
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well as to the cohomological character of the
dimension chosen to study. In future work we hope to
highlight some of the problems related to homotopic
applications and to study some applications of locally
compact spaces in further cases other than polyhedra.
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Society, vol. 41, no. 3, pp. 375-481, 1973.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
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Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.14
Maysoon Qousini, Hasan Hdieb, Eman Almuhur
E-ISSN: 2224-2880
124
Volume 23, 2024
