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Mathematical Modeling of Complex Information Processing Systems 123
CHAPTER 4
METHODS OF MATHEMATICAL MODELING
SHAPE FIBRATIONS WITH 0­DIMENSIONAL FIBERS
A. Bykov 1 and M. Texis 1
It is shown that a shape fibration (between metric compacta) with 0­dimensional fibers can be
represented as an inverse limit of covering projections of absolute neighborhood retracts (ANRs).
This representation is used to establish a lifting theorem for these fibrations in terms of shape groups.
1. Introduction. Covering projections play an important role not only in algebraic topology but in
differential geometry, complex analysis, the theory of Lie groups, etc. The central result concerning covering
projections is the solution of the following lifting problem:
Let p : (E; e) ! (B; b) be a covering projection and f : (Y; y) ! (B; b) be a map. Under what conditions
does there exist a lift ¯
f : (Y; y) ! (E; e), i.e., a map ¯
f satisfying p ffi ¯
f = f ?
If Y is connected and locally path­connected, then a necessary and sufficient condition for the lift is the
following lifting criterion:
f# (ú 1 (Y; y)) ae p# (ú 1 (E; e))
This is the classical lifting theorem. This theorem remains true if we take, as a map p, a Hurewicz fibration
with a unique path lifting instead of a covering projection (see [10], Chap. 2, Sec. 4, Th. 5, p. 76).
Obviously, the lifting criterion is obtained by applying the fundamental group functor ú 1 to the equality
p ffi ¯
f = f . Therefore, this criterion is always a necessary condition for a lift. However, it fails to be sufficient
when the space Y is not ``locally nice''.
That is why the following problem arises: to modify the lifting criterion in order to make it sufficient for
a wider class of spaces Y (perhaps, at a sacrifice in reducing the class of maps p). In [4], R. Fox formulated a
lifting criterion in terms of first shape groups for the class of all connected metrizable Y , considering a rather
special class of covering projections p named overlays. Afterwards, in [9] P. Mrozik considerably generalized
Fox's lifting theorem, using inverse systems of fundamental groups (pro­groups) and improving the definition of
an overlay.
In this paper we deal, in fact, with a special class of Hurewicz fibrations with a unique path lifting, namely,
with the class of shape fibrations p having 0­dimensional fibers. Thus, the fibers of p may not be necessarily
discrete, whereas the fibers of overlays, clearly, are always discrete. Theorems 4.2 and 4.3 are lifting theorems
for such kind of fibrations. The corresponding lifting criterion is given in terms of shape and strong shape
groups. Although the spaces Y in these theorems still are rather ``nice'' (pointed one­movable compacta), they
need not be locally path connected.
2. Some basic concepts and definitions. In this section, we give all necessary definitions and results
for convenience of the reader, although some knowledge of shape theory corresponding to [7] is assumed.
Throughout the paper, we deal only with metric topological spaces and continuous maps. The corresponding
category is denoted by M. The notion of shape fibration was first introduced in [8] for metric compacta and
was generalized afterwards in [6] for arbitrary spaces. The definition of shape fibration uses the approximate
homotopy lifting property (AHLP) applied to ANR­resolutions of maps. We shall need only resolutions for
compact metric spaces and for maps of compact metric spaces.
Let E = fE i ; q j
i g be an inverse sequence of ANR­spaces (not necessarily compact) and let E be a com­
pactum. A morphism q = fq i g : E ! E (in pro­M) is an ANR­resolution of E if and only if (E; q) = lim /\Gamma
E and
the following condition holds: for each i and any neighborhood U of q i (E) in E i there exists j ? i such that
g j
i (E j ) ` U .
1 Benem'erita Universidad Aut'onoma de Puebla, Av. San Claudio y Rio Verde, Ciudad Universitaria, Colonia
San Manuel, CP 72570, Puebla, M'exico, e­mail: abykov@fcfm.buap.mx; mtexis@fcfm.buap.mx

124 Mathematical Modeling of Complex Information Processing Systems
Let p = fp i g : E ! B (where p i : E i ! B i ) be a level map of the inverse sequences E = fE i ; q j
i g and
B = fB i ; r j
i g. It can be regarded as an inverse sequence p = fp i ; (q j
i ; r j
i )) in MapM, where MapM is the
category of maps of M. Let p : E ! B be a map of compact spaces. A morphism (q; r) : p ! p is an
ANR­resolution of p if and only if q and r are ANR­resolutions of E and B, respectively.
In the case of maps of metric compacta (more generally, in the case of proper maps of metric spaces),
shape fibrations can be described using the homotopy lifting property (HLP), which is simpler than the AHLP.
Namely, a map p : E ! B of metric compacta is a shape fibration if it admits an ANR­resolution (q; r) : p ! p
such that for any space X, for each map h : X ! E i+1 , and for each homotopy H : X \Theta I ! B i+1 with
p i+1 ffi h = H 0 there exists a homotopy ~
H : X \Theta I ! E i satisfying ~
H 0 = q i+1
i ffi h and p i ~
H = r i+1
i ffi H.
The lifting property can even more be simplified using the regularity of the HLP and the fact that each
map p i is a composition of an SDR­map E i ! coCyl(p i ) and a Hurewicz fibration coCyl(p i ) ! B i , where the
SDR­map is a map that embeds one space into another as a strong deformation retract.
Theorem 2.1. A map p : E ! B of a compact metric space is a shape fibration if and only if p admits an
ANR­resolution p ! p, p = fp i g, such that each p i is a Hurewicz fibration.
For a proof, see [2], Th. 5, p. 203.
A closed subset A of a space X is a shape strong deformation retract of X if and only if there exists an
absolute retract space M and a closed embedding ff : X ,! M such that the following condition holds: for each
pair of neighborhoods (U; V ) of (ff(X); ff(A)) in M there is the homotopy H : X \Theta I ! M rel: A with H 0 = ff,
ImH ` U and H 1 ` V .
A map s : A ,! X is a SSDR­map if and only if s embeds A as a shape strong deformation retract of X.
A closed subset A of a space X is a strong infinite deformation retract of X if and only if there exists a
strong infinite deformation of X onto A, i.e., a map D : X \Theta [0; 1) ! X, such that:
(i) D(a; t) = a if a 2 A and t 2 [0; 1);
(ii) for any neighborhood U of A in X there exists – 2 [0; 1) such that D(X \Theta [–; 1)) ae U .
Note that if A is a strong infinite deformation retract of X, then A ,! X is an SSDR­map.
A space Y is a fibrant space if and only if it has the extension property with respect to SSDR­maps in the
following sense: for every SSDR­map s : A ,! X and every map A ! Y there exists a map ¯
f : X ! Y such
that ¯
f ffi s = f .
Every ANR is a fibrant space. Inverse limits of inverse sequences consisting of fibrants and Hurewicz
fibrations as bonding maps are fibrant. The compact metric topological groups are fibrant. The retracts of
fibrant spaces are obviously fibrant.
A fibrant extension of X consists of a fibrant space ~
X and an SSDR­map s : X ,! ~
X. Throughout the
paper, dealing with fibrant extensions ~
X of X, we always suppose that X ` ~
X .
Every compact metric space has a fibrant extension (see [3]).
Each of the ANR­resolutions q : X ! X (more generally, the resolutions consisting of fibrant spaces) of a
compact space X can be used to construct its fibrant extension. Namely, the cotelescope ~
X = coTel(X) of X
(see [5]) with the natural embedding s : X ,! ~
X serves as a fibrant extension for X. Moreover, s(X) is a strong
infinite deformation retract of ~
X.
The cotelescope construction can be realized in the category MapM. In this category, the Hurewicz
fibrations of fibrant spaces can be regarded as ``fibrant'' objects (for more details, see [2]). In a natural way, the
concepts of strong deformation and strong infinite deformation are defined in MapM.
For example, a strong infinite deformation of ~ p onto p in the category MapM can be represented by a
pair (D; H) of strong infinite deformations
D : ~
E \Theta [0; 1) ! ~
E; H : ~
B \Theta [0; 1) ! ~
B
onto E and B, respectively, such that
~ p ffi D = H ffi (~p \Theta id [0;1) )
As a consequence of Theorem 2.1 and the cotelescope construction, we get the following
Theorem 2.2 ([2], Th. 9, p. 207). If p : E ! B is a shape fibration of metric compacta E and B, then
there exists a Hurewicz fibration ~
p : ~
E ! ~
B of fibrant extensions ~
E and ~
B of E and B, respectively, such that
the following diagram commutes:

Mathematical Modeling of Complex Information Processing Systems 125
­
­
? ?
~
E
~
p
~
B
i
j
p
E
B
Here the inclusions i : E ,! ~
E and j : B ,! ~
B are SSDR­maps. Moreover, there exists a strong infinite
deformation of ~ p onto p in the category MapM.
Let q : X ! X , X = fX i ; q j
i g, be an ANR­resolution of a compact metric space X. Fixing a point x 2 X,
we can pass on to the ``pointed'' ANR­resolution q : (X; x) ! (X ; x), where x = fx i g and x i = q i (x) for each i.
Let G = fG i ; f j
i g be an inverse sequence of groups and homomorphisms. G is said to be Mittag­Leffler if
and only if for each i there exists j ? i such that, for all k ? j, f j
i (G j ) = f k
i (G k ).
Proposition 2.3 (see [7]). If G = fG i ; f j
i g is Mittag­Leffler, then for each i there exists j ? i such that,
for all k ? j, f i (lim /\Gamma
G) = f k
i (G k ), where f i : (lim /\Gamma
G) \Gamma! G i are the natural projections.
Let q : (X; x) ! (X ; x) be an ANR­resolution of a pointed compactum (X; x).
(X; x) is said to be pointed one­movable if and only if fú 1 (X i ; x i ); q j
i# g is Mittag­Leffler, where ú 1 (X i ; x i )
are fundamental groups and q j
i# are homomorphisms of fundamental groups induced by the maps q j
i .
This definition does not depend on the choice of the ANR­resolution q : (X; x) ! (X ; x). Moreover, if X
is connected (i.e., a continuum), then it does not depend on the choice of x 2 X.
Let q : (X; x) ! (X; x) be an ANR­resolution of a pointed compact space (X; x). Then, the group
Ÿ
ú 1 (X; x) = lim /\Gamma
fú 1 (X i ; x i ); q j
i #
g is called the first shape group of (X; x) (see [7]).
Let s : X ,! ~
X be a fibrant extension of a compactum X. The group ¯ ú 1 (X; x) = ú 1 ( ~
X ; s(x)) is called the
first strong shape group of (X; x).
The definitions of shape and strong shape groups do not depend on the choice of an ANR­resolution and
of a fibrant extension of (X; x), respectively. Moreover, for any map f : (X; x) ! (Y; y) of pointed compact
spaces, the homomorphisms Ÿ
f : Ÿ ú 1 (X; x) ! Ÿ
ú 1 (Y; y) and ¯
f : ¯ ú 1 (X; x) ! ¯
ú 1 (Y; y) are naturally defined in such a
way that Ÿ
ú 1 and ¯
ú 1 can be considered as functors from the category of compact metric spaces and continuous
maps to the category of groups and homomorphisms.
3. 0­dimensional shape fibrations as inverse limits of covering projections. Recall that a topo­
logical space X is 0­dimensional if and only if it has a base consisting of sets that are both open and closed.
For example, all discrete sets and inverse limits of discrete sets are 0­dimensional. The condition of 0­
dimensionality formally is the strongest condition of disconnectedness. There are many other conditions of that
kind. Here we mention only two of them.
A topological space X is totally disconnected if and only if it has only single point components. A topological
space X is totally separated if and only if for every pair of points x; y 2 X, x 6= y, there exist closed subsets
A; B ae X such that x 2 A, y 2 B, A [ B = X, and A `` B = ;.
Every 0­dimensional T 0 ­space is totally separated. Every totally separated space is totally disconnected.
Clearly, in our case of compact metric spaces all these three notions coincide.
A shape fibration p : E ! B is 0­dimensional if and only if all its fibers are 0­dimensional (in other words,
dim (p \Gamma1 (b)) = 0 for each b 2 B).
The natural projection p : D ! S 1 of the dyadic solenoid D to the circle S 1 is a 0­dimensional shape
fibration. Its fibers are homeomorphic to the Cantor set.
Lemma 3.1. Let p : ~
X ! X be a surjective Hurewicz fibration of ANRs. There exists a commutative
diagram
­
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma\Psi
@
@
@
@ @R
X
X ?
~
X
p p ?
fl

126 Mathematical Modeling of Complex Information Processing Systems
such that
(a) X ? is an ANR,
(b) p ? is a covering projection,
(c) fl is a Hurewicz fibration with path­connected fibers.
Since components of an ANR are open and also are ANRs, without loss of generality we assume that X
and ~
X are connected. The map p ? is constructed as a covering projection corresponding to p# (ú 1 ( ~
X; ~ x)) for
some ~
x 2 ~
X , whereas the map fl is a lift of p. Note that fl can be considered as a factor map with respect to
the partition of ~
X by the following equivalence relation: ~ x 1 ¸ ~ x 2 if and only if p(~x 1 ) = p(~x 2 ) and, moreover,
~
x 1 ; ~ x 2 can be joined by a path in p \Gamma1 (p(~x 1 )).
Theorem 3.2. A map p : E \Gamma! B between compact metric spaces is a 0­dimensional shape fibration if and
only if there exists an ANR­resolution p = fp i g : E \Lambda \Gamma! B of p, where p i : E i \Gamma! B i is a covering projection
for each i.
Proof. If p admits an ANR­resolution consisting of covering projections, then, obviously, it is a 0­dimensional
shape fibration, since the covering projections are Hurewicz fibrations with 0­dimensional fibers.
Let us assume now that p is a 0­dimensional shape fibration. By Theorem 2.1, there exists an ANR­
resolution p = fp i g : E \Gamma! B of p : E \Gamma! B, E = fE i ; q j
i g, B = fB i ; r j
i g, where p i : E i \Gamma! B i is fibration for
each i.
Using Lemma 3.1, we represent every fibration as a composition of a factor map fl i : E i \Gamma! E \Lambda
i and a
covering projection p \Lambda
i : E \Lambda
i \Gamma! B i of ANR­spaces.
For each i, we define f i+1
i : E \Lambda
i+1 \Gamma! E \Lambda
i by f i+1
i (e \Lambda
i+1 ) = fl i ffi q i+1
i (e i+1 ) where e \Lambda
i+1 2 E \Lambda
i+1 and e i+1 2
fl \Gamma1
i+1 (e \Lambda
i+1 ). The map f i+1
i is well defined. Indeed, if e i+1 ; e 0
i+1 2 fl \Gamma1
i+1 (e \Lambda
i+1 ), then e i+1 , e 0
i+1 are connected by a
path ! : I ! p \Gamma1
i+1 (p i+1 (e i+1 )) and q i+1
i (e i+1 ), q i+1
i (e 0
i+1 ) are connected by q i+1
i ffi ! : I ! p \Gamma1
i (p i ffi q i+1
i (e i+1 )),
but this means that fl i ffi q i+1
i (e i+1 ) = fl i ffi q i+1
i (e 0
i+1 ). Moreover, we have f i+1
i ffi fl i+1 = fl i ffi q i+1
i . In particular,
this implies that f i+1
i is continuous by the definition of factor topology.
Finally, p \Lambda
i ffi f i+1
i = r i+1
i ffi p \Lambda
i+1 . Indeed, if e \Lambda
i+1 2 E \Lambda
i+1 and e i+1 2 fl \Gamma1
i+1 (e \Lambda
i+1 ), then p \Lambda
i ffi f i+1 (e \Lambda
i+1 ) =
p \Lambda
i ffi fl i ffi q i+1
i (e i+1 ) = p i ffi q i+1
i (e i+1 ) = r i+1
i ffi p i+1 (e i+1 ) = r i+1
i ffi p \Lambda
i+1 ffi fl i+1 (e i+1 ) = r i+1
i ffi p \Lambda
i+1 (e \Lambda
i+1 ). Therefore,
we have the following commutative diagram for each i:
oe
? ?
oe
@
@
@
@R
\Phi
\Phi
\Phi
\Phi
\Phiú
oe
@
@
@ @R
\Phi
\Phi
\Phi
\Phi
\Phiú
E i E i+1
E \Lambda
i E \Lambda
i+1 .
B i
q i+1
i
r i+1
i
p i
fl i
p \Lambda
i
fl i+1
p i+1
f i+1
i
B i+1
p \Lambda
i+1
Thus, we obtained the inverse sequence E \Lambda = fE \Lambda
i ; f i+1
i g of ANR­spaces and the level maps p \Lambda = fp \Lambda
i g :
E \Lambda \Gamma! B, fl = ffl i g : E \Gamma! E \Lambda such that p \Lambda ffi fl = p.
Now let p \Lambda = lim /\Gamma
p \Lambda , E \Lambda = lim /\Gamma
E \Lambda , and fl = lim /\Gamma
fl. Then p \Lambda ffi fl = p.
Let us show that fl is a homeomorphism.
First we prove that fl is injective. Let e 1 ; e 2 2 E be such that fl(e 1 ) = fl(e 2 ). Hence, p(e 1 ) = p(e 2 ) and
e 1 ; e 2 2 F = p \Gamma1 (b), where b = p(e 1 ). For each i, the points e 1i = q i (e 1 ) and e 2i = q i (e 2 ) can be joined by a
path in the fiber F i = p \Gamma1
i (r i (b)), since fl i (e 1i ) = fl j (q i (e i )) = f i (fl(e 1 )) = f i (fl(e 2 )) = fl i (q i (e 2 )) = fl i (e 2i ).
Note also that fF i ; q j
i j F j
g is an ANR­resolution of F .
Suppose e 1 6= e 2 . Since F is 0­dimensional, there exist its closed subsets C 1 , C 2 for which C 1 `` C 2 = ;,
C 1 [C 2 = F , and e 1 2 C 1 , e 2 2 C 2 . Clear, that q i (C 1 ) `` q i (C 2 ) = ; for some i, because C 1 and C 2 are compact.
For the closed sets q i (C 1 ) and q i (C 2 ), we choose open (in E i ) sets U 1 ' q i (C 1 ) and U 2 ' q i (C 2 ) such that
U 1 `` U 2 = ;. Then, the set U = U 1 [U 2 is an open neighborhood of q i (F ) in E i and there exists j ? i for which
q j
i (F j ) ae U . The points e 1j = q j (e 1 ) and e 2j = q j (e 2 ) can be joined by a path in F j . Hence, q j
i (e 1j ) and q j
i (e 2j )
can be joined by a path in U = U 1 [ U 2 . However, this is in contradiction with U 1 `` U 2 = ;.
Now we show that fl is surjective. Let e \Lambda 2 E \Lambda . We are going to prove the following assertion: for each i,
fl \Gamma1
i (f i (e \Lambda )) `` q i (E) 6= ;.

Mathematical Modeling of Complex Information Processing Systems 127
Assume that for some i this is not true. Then, we can find an open neighborhood U of q i (E) in E i and
some j ? i such that fl \Gamma1
i (f i (e \Lambda )) `` U = ; and q j
i (E j ) ae U . On the other hand, since fl j is surjective, there
exists e j 2 E j for which fl j (e j ) = f j (e \Lambda ); therefore, fl i ffi q j
i (e j ) = f j
i ffi fl j (e j ) = f j
i ffi f j (e \Lambda ) = f i (e \Lambda ). Hence,
q j
i (e j ) 2 fl \Gamma1
i (f i (e \Lambda )) and q j
i (e j ) 2 q j
i (E j ) ae U . However, this is in contradiction with fl \Gamma1
i (f i (e \Lambda )) `` U = ;.
The above­proved assertion implies that e \Lambda 2 fl(E), where fl(E) is the closure of fl(E) in E \Lambda . Indeed, if U is a
neighborhood of e \Lambda in E \Lambda , then for some i and some neighborhood U i of f i (e \Lambda ) in E \Lambda
i we have f \Gamma1
i (U i ) ae U . If now
e i 2 fl \Gamma1
i (f i (e \Lambda )) `` q i (E), then for some e 2 E the equalities q i (e) = e i and f i (fl(e)) = fl i (q i (e)) = fl i (e i ) = f i (e \Lambda )
hold. Hence, fl(e) 2 f \Gamma1
i (f i (e \Lambda )) ae U .
Since E is compact, fl(E) = fl(E) and we conclude that e \Lambda 2 fl(E). Thus, fl : E ! E \Lambda is a bijection. Hence,
it is a homeomorphism, because E is compact. Therefore, we can identify E and E \Lambda , p and p \Lambda .
In order to complete the proof, we should check only that p \Lambda is an ANR­resolution of p \Lambda = p.
Let U be a neighborhood of q \Lambda
i (E \Lambda ) in E \Lambda
i . Then, V = fl \Gamma1
i (U ) is a neighborhood of q i (E) in E i . Since E
is an ANR­resolution, there exists j ? i such that q j
i (E j ) ` V . Recall that fl i is surjective. Hence,
q \Lambdaj
i (E \Lambda
j ) = q \Lambdaj
i ffi fl j (E j ) = fl i ffi q j
i (E j ) ` fl i (V ) = U
Thus, p \Lambda is as claimed.
Corollary 3.3. Let p : E ! B be a 0­dimensional shape fibration between compact metric spaces E and B.
Then, p is a Hurewicz fibration with a unique path­lifting property.
The following statement is a ``0­dimensional'' version of Theorem 2.2.
Corollary 3.4. If p : E ! B is a 0­dimensional shape fibration of metric compacta E and B, then there
exists a Hurewicz fibration with the unique path lifting ~ p : ~
E ! ~
B of fibrant extensions ~
E and ~
B of E and B,
respectively, such that the following diagram commutes:
­
­
? ?
~
E
~
p
~
B
i
j
p
E
B
Here the inclusions i : E ,! ~
E and j : B ,! ~
B are SSDR­maps. Moreover, there exists a strong infinite
deformation of ~ p onto p in the category MapM.
Clearly, the proof of the corollary is a repetition of the proof of Theorem 2.2 (for the proof, see [2]) with
the following modification: for each morphism ff = (q i+1
i ; r i+1
i ) : p i+1 ! p i in the category MapM , where p i
and p i+1 are covering projections of ANRs, the cocylinder coCyl(ff) is also a covering projection of ANRs and,
hence, ~ p = coTel(p), being an inverse limit of the cocylinders, is a Hurewicz fibration with the unique path
lifting.
Note also that under the conditions of the corollary we have (~p) \Gamma1 (j(B)) = i(E), because no point of
(~p) \Gamma1 (j(B)) moves when we carry out the strong infinite deformation.
4. The lifting problem for 0­dimensional shape fibrations.
Lemma 4.1. Let p : (E; e) \Gamma! (B; b) be a covering projection of ANR­spaces E and B. If Y is a pointed
one­movable continuum, then a map f : (Y; y) ! (B; b) has a (unique) lift ¯
f : (Y; y) ! (E; e), i.e., a map for
which the diagram
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma \Gamma`
?
­
(E; e)
(Y; y) (B; b)
¯
f
f
p

128 Mathematical Modeling of Complex Information Processing Systems
commutes, if and only if f satisfies the lifting criterion: Ÿ
f (Ÿú 1 (Y; y)) ae p# (ú 1 (E; e)).
Proof. It is clear that the lifting criterion is a necessary condition for a lift. We have to prove that it is
sufficient.
We may assume that Y is a closed subset of the Hilbert cube Q. There exists a decreasing system fY i g i2N
of connected ANR­neighborhoods Y in Q such that `` 1
i=1 Y i = Y . In particular, the spaces Y i are locally
path­connected.
Let q j
i : (Y j ; y j ) \Gamma! (Y i ; y i ) and q i : (Y; y) \Gamma! (Y i ; y i ) be inclusions. Then fY i ; q j
i g is an ANR­resolution
of Y .
Since B is an ANR, there is an extension h : U ! B of the map f : Y ! B, where U is some neighborhood
of Y in Q. Find m 2 N such that Ym ae U . So, for each i ? m we define a map f i : (Y i ; y i ) \Gamma! (B; b) by
f i = hj Y i
. Thus, for each i ? m, we have f i ffi q i = f and q i+1
i ffi f i+1 = f i , i.e., the following diagram commutes:
oe oe
¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸¸ :
i i i i i i i i i i i i i i i i i i i i i i i i i1
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma
\Gamma \Gamma` 6
(Ym ; ym) (Ym+2 ; ym+2 )
q m+1
m q m+2
m+1
(Ym+1 ; ym+1 ) . . . (Y,y).
(B,b)
f
fm+2
fm+1
fm
Let i ? m. Since Y is pointed one­movable, by Proposition 2.3 there exists j ? i such that for each k ? j we
have q i# (Ÿú 1 (Y; y)) = q k
i# (ú 1 (Y k ; y k )).
Hence, for each k ? j we obtain
Ÿ
f(Ÿú 1 (Y; y)) = f i# ffi q i# (Ÿú 1 (Y; y)) = f i# ffi q k
i# (ú 1 (Y k ; y k )) = f k# (ú 1 (Y k ; y k ))
Thus, f k# (ú 1 (Y k ; y k )) ae p# (ú 1 (E; e)). By the classical lifting theorem, there exists a unique map ¯
f k :
(Y k ; y k ) \Gamma! (E; e) such that p ffi ¯
f k = f k .
For each l ? k, we get p ffi ( ¯
f k ffi q l
k ) = (p ffi ¯
f k ) ffi q l
k = f k ffi q l
k = f l = p ffi ¯
f l and hence, by the uniqueness of a
lift, we have ¯
f k ffi q l
k = ¯
f l .
Let ¯
f = lim /\Gamma
f ¯
f k g. Then, p ffi ¯
f = p ffi ( ¯
f k ffi q k ) = (p ffi ¯
f k ) ffi q k = f k ffi q k = f . Thus, ¯
f is the required lift.
In the proofs of each of the next two lifting theorems, we show only that the corresponding lifting criteria
are sufficient. Obviously, they are necessary for a lift, too, since Ÿ
ú 1 and ¯
ú 1 are functors.
Theorem 4.2. Let p : (E; e) \Gamma! (B; b) be a 0­dimensional shape fibration between compact metric spaces.
Let Y be a pointed one­movable continuum. A map f : (Y; y) \Gamma! (B; b) has a (unique) lift ¯
f : (Y; y) \Gamma! (E; e)
if and only if Ÿ
f (Ÿú 1 (Y; y)) ae Ÿ
p(Ÿú 1 (E; e)).
Proof. Since p is a 0­dimensional shape fibration, by Theorem 3.2 there exists an ANR­resolution p = fp i g :
E ! B of p, E = fE i ; q j
i g, B = fB i ; r j
i g, where for each i the map p i : E i ! B i is a covering projection and
p = lim /\Gamma
fpg. In other words, the following diagram commutes:
oe
? ?
oe
?
(E1 ; e1 ) (E2 ; e2 )
(B1 ; b1)
q 2
1
r 2
1
p1
(B2 ; b2)
p2
(E; e)
(B; b).
. . . p

Mathematical Modeling of Complex Information Processing Systems 129
For each i we define f i : (Y; y) \Gamma! (B i ; b i ) by f i = r i ffi f , where r i : (B; b) ! (B i ; b i ) is a natural projection.
Since Ÿ
f ( Ÿ
ú 1 (Y; y)) ae Ÿ
p( Ÿ
ú 1 (E; e)) and f i = r i ffi f , we get Ÿ
f i (Ÿú 1 (Y; y)) = Ÿ
r i ffi Ÿ
f (Ÿú 1 (Y; y)) ae Ÿ
r i ffi Ÿ
p(Ÿú 1 (E; e)) =
p i# ffi Ÿ
q i (Ÿú 1 (E; e)) ae p i# (ú 1 (E i ; e i )) and, therefore, Ÿ
f i (Ÿú 1 (Y; y)) ae p i# (ú 1 (E i ; e i )).
According to Lemma 4.1, there exists a unique map ¯
f i : (Y; y) \Gamma! (E i ; e i ) such that p i ffi ¯
f i = f i . Let j ? i.
We have p i ffi (q j
i ffi ¯
f j ) = (p i ffi q j
i ) ffi ¯
f j = (r j
i ffi p j ) ffi ¯
f j = r j
i ffi (p j ffi ¯
f j ) = r j
i ffi f j = f i = p i ffi ¯
f i . On the other hand,
since q j
i ffi f j (y) = q j
i (e i ) = e i = ¯
f i (y) and p i ffi (q j
i ffi f j ) = p i ffi ¯
f i , we get q j
i ffi ¯
f j = ¯
f j by the uniqueness of a
lift. Thus, the family of maps f ¯
f i g : Y \Gamma! E induces a map ¯
f : Y \Gamma! E such that q i ffi ¯
f = ¯
f i for each i. The
map ¯
f is the desired lift. Indeed, r i ffi (p ffi ¯
f ) = p i ffi (q i ffi ¯
f ) = p i ffi ¯
f i = f i and r i ffi (p ffi ¯
f ) = f i = r i ffi f for each i.
Hence, p ffi ¯
f = f .
Theorem 4.3. Let p : (E; e) \Gamma! (B; b) be a 0­dimensional shape fibration between compact metric spaces.
Let Y be a pointed one­movable continuum. A map f : (Y; y) \Gamma! (B; b) has a (unique) lift ~
f : (Y; y) \Gamma! (E; e)
if and only if ¯
f (¯ú 1 (Y; y)) ae ¯
p(¯ú 1 (E; e)).
Proof. By Corollary 3.4, we can consider p : (E; e) ! (B; b) as a restriction of the fibration with the unique
path lifting ~
p : ( ~
E; ~ e) ! ( ~
B; ~ b), where ~
E ' E and ~
B ' B are fibrant extensions. Let ~
Y ' Y be a fibrant
extension of Y . Since ~
B is a fibrant space, there exists an extension F : ~
Y ! ~
B of the map j ffi f , where
j : B ,! ~
B is an inclusion (in fact, it is an SSDR­map). By the definition of strong shape groups, we have
¯
f(¯ú 1 (Y; y)) = F# (ú 1 ( ~
Y ; y)) and ¯
p(¯ú 1 (E; e)) = ~
p# (ú 1 ( ~
E; e)) (in ¯ ú 1 (B; b) = ú 1 ( ~
B; b)).
Hence, F# (ú 1 ( ~
Y ; y)) ae ~
p# (ú 1 ( ~
E; e)).
Since Y is the pointed one­movable continuum, ~
Y is a connected locally path­connected space [1]. Therefore,
we can apply the classical lifting theorem in order to find a unique lift ¯
F : ( ~
Y ; y) ! ( ~
E; e) of F , i.e., a map ¯
F such
that ~ p ffi ¯
F = F . Note that x 2 Y implies ¯
F (x) 2 E, since ~ p \Gamma1 (B) = E. Indeed, ~
p( ¯
F (x)) = F (x) = f(x) 2 B.
Clearly, a map ~
f : (Y; y) ! (E; e) defined by ~
f (x) = ¯
F (x) is the required lift of f .
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