> To the blessed memory of my teacher Yurii Mikhailovich Smirnov

# Shape Theory

Pavel S. Gevorgyan  Moscow Pedagogical State University, Moscow, Russia
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Shape theory was founded by K. Borsuk 50 years ago. In essence, this is
spectral homotopy theory; it occupies an important place in geometric
topology. The article presents the basic concepts and the most important, in
our opinion, results of shape theory. Unfortunately, many other interesting
problems and results related to this theory could not be covered because of
space limitations. The article contains an extensive bibliography for those
who wants to gain a more detailed and systematic insight into the issues
considered in the survey.

Keywords: homotopy, inverse system, homotopy pro-category, shape category,
shape functor, equivariant shape, $Z$-set, movability, stable space, shape
retract, homotopy pro-group, shape group, shape dimension, cell-like map,
$Q$-manifold.

###### Contents

  1. 1 Introduction
  2. 2 Basic Constructions of Shape Theory
  3. 3 The Shape Classification of Spaces
  4. 4 Complement Theorems in Shape Theory
  5. 5 Movability and Other Shape Invariants
  6. 6 Stable Spaces and Shape Retracts
  7. 7 Whitehead’s and Hurewicz’ Theorems in Shape Theory
  8. 8 The Shape Dimension of Spaces and Maps
  9. 9 Embeddings in Shape Theory
  10. 10 Cell-Like Maps and Shape Theory

##  1\. Introduction

Shape theory was initiated in 1968 by the renowned Polish mathematician K.
Borsuk. In his opening speech [2] at the 1979 International Topology
Conference, P. S. Aleksandrov identified three periods in the development of
geometric topology111According to Aleksandrov, the first of these periods was
marked by outstanding works of Brouwer (1909–1913), Fréchet (1907), and
Hausdorff (1914). During the second period (1925–1943), homology and
cohomology theory (including duality theorems) was developed, dimension theory
was set up, and continuum theory was elaborated. The third period began with
Borsuk’s work on retract theory and continued with the development of shape
theory (see also [3]). and mentioned shape theory as one of the most important
research directions during the third period.

Shape theory is the spectral form of homotopy theory; it uses set-theoretic,
geometric, and combinatorial-algebraic ideas and methods of topology. This
theory is related to Aleksandrov–Čech homology and, therefore, to algebraic
topology. It is also closely related to infinite-dimensional topology, in
particular, to the theory of $Q$-manifolds.

As is known, the fundamental notions of homotopy theory can be applied to
spaces with good local structure (such as manifolds, polyhedra, CW-complexes,
and ANRs). Many theorems of homotopy topology are valid for CW-complexes but
not valid for compact metrizable spaces. An example is Whitehead’s celebrated
theorem, which asserts that a map $f\colon X\to Y$ between CW-complexes is a
homotopy equivalence if and only if it induces isomorphisms
$\pi_{n}(f)\colon\pi_{n}(X)\to\pi_{n}(Y)$, $n=1,2,\dots$, of all homotopy
groups. This theorem is not generally true for compact metrizable spaces.
Indeed, consider the so-called _Warsaw circle_ , or _quasi-circle_ , $W$,
which consists of the graph of the function $y=\sin{\dfrac{2\pi}{x}}$,
$0<x\leqslant 1$, the interval $[-1,1]$ of the $y$-axis, and the arc joining
the points $(0,0)$ and $(1,0)$ (see Fig. 1). It has bad local structure and is
not a CW-complex.

\begin{overpic}[scale={0.6}]{wc} \put(69.0,83.0){$W$} \end{overpic}

Fig. 1

Any continuous map of the $n$-sphere $S^{n}$ to the Warsaw circle $W$ is
homotopic to a constant map, and hence all homotopy groups of the Warsaw
circle are trivial. However, $W$ does not have the homotopy type of a point.
Thus, the constant map $f\colon W\to\\{*\\}$ to a one-point space induces
isomorphisms of the homotopy groups for all $n$ but is not a homotopy
equivalence.

Homotopy theory is ideally suited for being applied to spaces with good local
structure but not to arbitrary spaces. The Warsaw circle $W$ and the usual
circle $S^{1}$ are not homotopy equivalent, although they look similar and
share certain important properties. For example, each of them separates the
plane into two parts.

The reason why the Warsaw circle and the circle $S^{1}$ are not homotopy
equivalent is that there are not enough continuous maps from $S^{1}$ to $W$.
The situation changes when continuous maps from $S^{1}$ to arbitrary
neighborhoods of the Warsaw circle $W$ are considered. Such maps are already
sufficiently many. This observation has enabled Borsuk to propose a remarkable
idea of addressing this “drawback” of homotopy theory for compact metrizable
spaces and, thereby, open up a new direction of geometric topology, shape
theory [74]. Borsuk first reported his results at the 1967 Symposium on
Infinite-Dimensional Topology, which was held in Baton Rouge, Louisiana, on
March 27–April 1, 1967, and then at the International Symposium on Topology
and Its Applications in Herceg Novi on August 25–31, 1968 (see [75]).

After the appearance of Borsuk’s pioneering papers [73]–[84], shape theory
began to rapidly develop across the world. Warsaw, where Borsuk and his
students worked, had become the center of scientific activities and research
on shape theory. In the United States the first studies on shape theory were
performed by R. Fox, T. A. Chapman, J. Segal, and J. Keesling, and in Japan,
by K. Morita and Y. Kodama. In Moscow research in shape theory was headed by
Yu. M. Smirnov. In Zagreb a group of topologists working on this theory was
formed; it was led by S. Mardešić. In Frankfurt shape theory was developed by
F. Bauer and his students, and in Great Britain, by T. Porter; many other
topologists worked on it all over the world.

Borsuk’s idea underlying shape theory is based on the fact that any compact
metrizable space embeds in the Hilbert cube $Q$ (as well as in any other
absolute retract) and consists in considering, instead of continuous maps
between compact metrizable spaces $X$ and $Y$, so-called fundamental sequences
$(f_{n}):Q\to Q$ coordinated in a certain way with neighborhoods of $X$ and
$Y$ (see Sec. 2).

In 1972 Fox [168] extended shape theory to metrizable spaces by using Borsuk’s
method. To this end, he embedded an arbitrary metric space $X$ in an absolute
neighborhood retract $M$ as a closed subspace and constructed shape morphisms
by using a system of neighborhoods of the space $X$ in $M$. Importantly, all
these neighborhoods are absolute neighborhood retracts for metrizable spaces.

Borsuk’s geometric method does not apply in the general case, because there
are too few absolute neighborhood retracts in the category of all topological
spaces.

In 1970 Mardešić and Segal [287], [288] constructed shape theory for arbitrary
compact Hausdorff spaces by means of inverse systems. They essentially used
the well-known theorem that _any compact Hausdorff set $X$ is the limit of an
inverse system ${\bf X}=\\{X_{\alpha},p_{\alpha\alpha^{\prime}},A\\}$ of
compact ANR spaces $X_{\alpha}$_.

In 1975 Morita [308] extended shape theory to arbitrary topological spaces by
the method of inverse ANR-systems. He had succeeded thanks to Čech’s functor
$\check{C}$; to each topological space $X$ this functor assigns an inverse
system $\check{C}(X)$ of the nerves $X_{\alpha}$ of normal locally finite open
covers $\alpha$ with the natural projections generated by the refinement of
covers. Morita introduced the notion of the _associated inverse ANR-system_
and proved that, for any topological space $X$, the inverse ANR-system
$\check{C}(X)$ is associated with $X$. Moreover, all inverse ANR-systems
associated with a given space $X$ are isomorphic to each other and, in
particular, to the system $\check{C}(X)$.

The spectral method for constructing shape theory has turned out to be
universal. Thus, in [74], given a compact metrizable space $X$, Borsuk
essentially used an associated countable ANR-system of its neighborhoods which
arises under an embedding of $X$ in an absolute retract $M$. Moreover, since
the ANR-system obtained from the system of all neighborhoods of a closed
subset $X$ of an absolute neighborhood retract $N$ in the class of metrizable
spaces is associated with $X$, we see that Fox [168] essentially considered
inverse ANR-systems associated with metrizable spaces. Finally, if a compact
Hausdorff space $X$ is the limit of an inverse system ${\bf
X}=\\{X_{\alpha},p_{\alpha\alpha^{\prime}},A\\}$ of compact
$\mathrm{ANR}$-spaces $X_{\alpha}$, i.e.,
$X=\lim\limits_{\longleftarrow}\\{X_{\alpha},p_{\alpha\alpha^{\prime}},A\\}$,
then the ANR-system $\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ in
the homotopy category H-CW is associated with $X$. This has enabled Mardešić
and Segal [287], [288] to construct shape theory for arbitrary compact
Hausdorff spaces.

In the 1973 paper [278] Mardešić gave a short categorical definition of shape
theory. The objects of the shape category Sh-Top are topological spaces, and
morphism $F\colon X\to Y$ are maps which take each homotopy class
$\boldsymbol{\psi}\colon Y\to P$, where $P$ is an ANR, to a homotopy class
$\boldsymbol{\varphi}\colon X\to P$ such that, given any ANR $P^{\prime}$ and
any homotopy classes $\boldsymbol{\psi^{\prime}}\colon Y\to P^{\prime}$ and
$\mathbf{q}\colon P^{\prime}\to P$ satisfying the relation
$\mathbf{q}\boldsymbol{\psi^{\prime}}=\boldsymbol{\psi}$, we have
$\mathbf{q}\boldsymbol{\varphi^{\prime}}=\boldsymbol{\varphi}$.

The shape functor
$S\colon\text{$\mathrm{H}$-$\mathrm{Top}$}\to\text{$\mathrm{Sh}$-$\mathrm{Top}$}$
satisfies the following two conditions:

(i) $S(X)=X$ for any topological space $X$ and

(ii) for any shape morphism $F\colon X\to Q$ to an ANR $Q$, there exists a
unique homotopy class $\mathbf{f}\colon X\to Q$ such that $S(\mathbf{f})=F$.

These conditions have the universality property, and it is natural to call
them the _axioms of shape theory_. The first axiomatics of shape theory for
the class of all compact metrizable spaces was proposed by Holsztyński [211].
In the general case, an axiomatic characterization of shape theory was given
by Mardešić [278].

For compact metrizable spaces, all methods for constructing shape theory are
equivalent to Borsuk’s. However, in the class of metrizable spaces, the shape
theories of Borsuk [80], Fox [168], and Bauer [62] differ from each other.
Thus, there exist different shape theories (likewise, in algebraic topology,
there are different homology and cohomology theories).

The shape category is weaker than the homotopy category. The former is not a
generalization of the latter; it is rather a natural correction, or extension,
of the homotopy category, because the shape functor
$S\colon\text{$\mathrm{H}$-$\mathrm{Top}$}\to\text{$\mathrm{Sh}$-$\mathrm{Top}$}$
is an isomorphism onto the subcategory H-CW of all spaces with the homotopy
type of an ANR. Thus, for the class of spaces having the homotopy type of a
polyhedron, which includes all CW-complexes and ANRs, shape theory coincides
with homotopy theory. The Warsaw circle and the circle $S^{1}$ are examples of
compact metrizable spaces which have different homotopy types but are shape
equivalent.

In essence, shape theory is spectral homotopy theory, because it is based on
the idea of replacing the singular by the approximative; in this respect, it
is closely related to Aleksandrov–Čech spectral homology. Shape theory is
intended for application to spaces with bad local structure, which
increasingly frequently arise in very diverse areas of mathematics; they also
result from applying many topological constructions, such as bundles, cell-
like maps, fixed point sets, attractors of dynamical systems, spectra of
linear operators, remainders of compactifications, boundaries of groups, etc.

Ideas and problems closely related to shape theory have been known long before
the publication of Borsuk’s first papers [74] and [75], in which the shape
category for compact metrizable spaces was constructed. As early as in 1895
Poincaré laid the foundations of algebraic topology and introduced the
important notions of a chain, a cycle, a boundary, the Betti numbers,
homologous cycles, etc. in his celebrated treatise _Analysis Situs_ [326].
Poincaré also conjectured (but not proved) that the Betti numbers are
topological invariants. The first rigorous proof of this fact was given by
Alexander in 1915 [39], which had put Poincaré’s intuitive ideas on solid
mathematical grounds. In 1922 Veblen [387] proved the topological invariance
of singular homology. The application of singular homology is based on the
consideration of a family of continuous maps from a polyhedron $P$ to a
topological space $X$. Homotopy theory is based on the same idea, because each
homotopy group $\pi_{n}(X,*)$ is defined by using continuous maps from the
$n$-sphere $S^{n}$ to a pointed space $(X,*)$. However, this idea does not
work for spaces with bad local structure. This is why shape theory is based on
the dual idea of considering families of continuous maps from a topological
space $X$ to a polyhedron $P$. This approach was first applied by Aleksandrov
to define inverse systems of spaces [43], [44] and introduce the notion of the
nerve of a cover of a topological space [45]. Thus, the introduction of Čech
homology in various forms by Aleksandrov [43], [44], Vietoris [391], and Čech
[97] can be justly regarded as the rudiment of shape theory.

Shape theory is related to many fields of topology, and its ideas and results
turn out to be important and useful for these fields. Because of space
limitations, we could not cover all interesting problems and remarkable
results of shape theory. Nevertheless, we have tried to touch upon the most
important points in the development of shape theory during the past five
decades.

The literature on shape theory is fairly extensive (see the bibliography at
the end of this paper); we mention Borsuk’s monograph [10], the books [289] by
Mardešić and Segal and [148] by Dydak and Segal, and survey papers [24] by
Smirnov, [85] by Borsuk and Dydak, and [271] and [283] by Mardešić.

##  2\. Basic Constructions of Shape Theory

###  2.1. The Shape Category of Compact Metrizable Spaces (Borsuk’s Method)

The main idea of Borsuk’s construction of the shape category of compact
metrizable spaces is the replacement of the class of continuous maps by a
larger class of morphisms and the introduction of the notion of a homotopy
between such morphisms.

Let $X$ and $Y$ be compact metrizable spaces contained in ARs $M$ and $N$,
respectively.

######  Definition 1.

A sequence of maps $f_{n}\colon M\to N$, $n\in\mathbb{N}$, is called a
_fundamental sequence from $X$ to $Y$_ and denoted by $(f_{n})_{MN}\colon X\to
Y$ if, for any neighborhood $V$ of the compact set $Y$ in $N$, there exists a
neighborhood $U$ of the compact set $X$ in $M$ and a positive integer
$n_{V}\in\mathbb{N}$ such that, first, $f_{n}(U)\subset V$ for all $n\geqslant
n_{V}$ and, secondly,

$f_{n}|U\simeq f_{m}|U$

in $V$ for all $n,m\geqslant n_{V}$.

The fundamental sequence $(1_{n})_{MM}\colon X\to X$, where $1_{n}=1_{M}\colon
M\to M$ is the identity map for all $n\in\mathbb{N}$, is called the _identity
fundamental sequence_. The composition of fundamental sequences
$(f_{n})_{MN}\colon X\to Y$ and $(g_{n})_{NP}\colon Y\to Z$, where $Z$ is a
compact metrizable space contained in an AR $P$, is defined by

$(g_{n})(f_{n})=(g_{n}f_{n}).$

Clearly, $(g_{n}f_{n})_{MP}\colon X\to Z$ is a fundamental sequence, and the
composition operation is associative.

The notion of homotopy naturally extends to fundamental sequences.

######  Definition 2.

Two fundamental sequences $(f_{n})_{MN}\colon X\to Y$ and
$(f^{\prime}_{n})_{MN}\colon X\to Y$ are said to be _homotopic_ if, for any
neighborhood $V$ of the compact set $Y$ in $N$, there exists a neighborhood
$U$ of the compact set $X$ in $M$ and a positive integer $n_{V}\in\mathbb{N}$
such that

$f_{n}|U\simeq f^{\prime}_{n}|U$

in $V$ for all $n\geqslant n_{V}$. In this case, we write
$(f_{n})\simeq(f^{\prime}_{n})$.

This relation is reflexive, symmetric, and transitive. The equivalence class
of a fundamental sequence $(f_{n})_{MN}\colon X\to Y$ is called a _fundamental
class_ and denoted by $[(f_{n})]$. It is easy to see that if
$(f_{n})\simeq(f^{\prime}_{n})$ and $(g_{n})\simeq(g^{\prime}_{n})$, then
$g_{n}f_{n}\simeq g^{\prime}_{n}f^{\prime}_{n}$. Therefore, we can define the
composition of fundamental classes $[(f_{n})]$ and $[(g_{n})]$ as

$[(g_{n})][(f_{n})]=[(g_{n})(f_{n})].$

Obviously, this composition operation is associative as well, and

$[(f_{n})][(1_{n})]=[(1_{n})][(f_{n})]=[(f_{n})].$

Thus, we obtain the so-called _fundamental category_ , whose objects are all
pairs $(X,M)$, where $M$ is an AR and $X$ is a compact metrizable subspace of
$M$, and morphisms are classes of fundamental sequences. Isomorphic objects in
this category are said to be _fundamentally equivalent_. In other words, pairs
$(X,M)$ and $(Y,N)$ are fundamentally equivalent if there exist fundamental
sequences $(f_{n})_{MN}\colon X\to Y$ and $(g_{n})_{NM}\colon Y\to X$ such
that $[(g_{n})][(f_{n})]=[(1_{n})_{MM}]$ and
$[(f_{n})][(g_{n})]=[(1_{n})_{NN}]$.

We say that a fundamental sequence $(f_{n})_{MN}\colon X\to Y$ is _generated
by a map $f\colon X\to Y$_ if $f_{n}|X=f$ for all $n\in\mathbb{N}$. Clearly,
any map $f\colon X\to Y$ generates a fundamental sequence $(f_{n})_{MN}\colon
X\to Y$. Indeed, since $N$ is an AR, it follows that there exists an extension
$\tilde{f}\colon M\to N$ of the map $f\colon X\to Y$. Setting
$f_{n}=\tilde{f}$ for each $n=1,2,\dots$, we obtain a fundamental sequence
generated by $f$.

The following theorem shows that the fundamental class of a fundamental
sequence $(f_{n})_{MN}\colon X\to Y$ generated by a map $f\colon X\to Y$
depends only on the homotopy class of this map.

######  Theorem 1.

Suppose that fundamental sequences $(f_{n})_{MN}\colon X\to Y$ and
$(f^{\prime}_{n})_{MN}\colon X\to Y$ are generated by maps $f\colon X\to Y$
and $f^{\prime}\colon X\to Y$, respectively, and $f\simeq f^{\prime}$. Then
$(f_{n})\simeq(f^{\prime}_{n})$.

###### Proof.

Consider any neighborhood $V$ of the compact set $Y$ in $N$. By Definition 1
there exists a neighborhood $U_{1}$ of the compact set $X$ in $M$ and a
positive integer $n^{\prime}_{V}\in\mathbb{N}$ such that $f_{n}|U_{1}\simeq
f_{m}|U_{1}$ in $V$ for all $n,m\geqslant n^{\prime}_{V}$. Similarly, there
exists a neighborhood $U_{2}$ of the compact set $X$ in $M$ and a positive
integer $n^{\prime\prime}_{V}\in\mathbb{N}$ such that
$f^{\prime}_{n}|U_{2}\simeq f^{\prime}_{m}|U_{2}$ in $V$ for all $n,m\geqslant
n^{\prime\prime}_{V}$. We set $\widetilde{U}=U_{1}\cap U_{2}$ and
$n_{V}=\max\\{n^{\prime}_{V},n^{\prime\prime}_{V}\\}$. Clearly,

(1) $f_{n}|\widetilde{U}\simeq f_{m}|\widetilde{U}\quad\text{and}\quad
f^{\prime}_{n}|\widetilde{U}\simeq f^{\prime}_{m}|\widetilde{U}$

in $V$ for all $n,m\geqslant n_{V}$.

Now consider the maps $f_{n_{V}},f^{\prime}_{n_{V}}\colon\widetilde{U}\to V$.
By the assumption of the theorem, we have $f_{n_{V}}|X=f$,
$f^{\prime}_{n_{V}}|X=f^{\prime}$, and $f\simeq f^{\prime}$. Let $F\colon
X\times I\to Y$ be a homotopy between $f$ and $f^{\prime}$. Since $V$ is an
ANR, it follows that there exists a neighborhood $U$ of $X$ in $M$ such that
$U\subset\widetilde{U}$ and a homotopy $\widetilde{F}\colon U\times I\to V$
between $f_{n_{V}}|U$ and $f^{\prime}_{n_{V}}|U$ such that
$\widetilde{F}|X\times I=F$ (see [289], Theorem 8, p. 40). Thus,

(2) $f_{n_{V}}|U\simeq f^{\prime}_{n_{V}}|U.$

It remains to note that

$f_{n}|U\simeq f^{\prime}_{n}|U$

in $V$ for all $n\geqslant n_{V}$; this follows from (1) and (2). ∎

######  Corollary 1.

Let $(f_{n})_{MN}\colon X\to Y$ and $(f^{\prime}_{n})_{MN}\colon X\to Y$ be
fundamental sequences generated by a map $f\colon X\to Y$. Then
$(f_{n})\simeq(f^{\prime}_{n})$.

This corollary implies, in particular, that all fundamental sequences
$(i_{n})_{MM}\colon X\to X$ generated by the map $1_{X}\colon X\to X$ are
homotopic to the identity fundamental sequence $(1_{n})_{MM}\colon X\to X$.

The following important theorem shows that the fundamental equivalence
relation is absolute in the sense that it does not depend on the choice of the
ARs in which metrizable compact spaces are embedded.

######  Theorem 2.

Let $X\subset M\cap M^{\prime}$. Then the pairs $(X,M)$ and $(X,M^{\prime})$
are fundamentally equivalent.

###### Proof.

Consider fundamental sequences $(i_{n})_{MM^{\prime}}\colon X\to X$ and
$(i_{n})_{M^{\prime}M}\colon X\to X$ generated by the identity map
$1_{X}\colon X\to X$. Obviously, the compositions
$(i_{n})_{M^{\prime}M}(i_{n})_{MM^{\prime}}$ and
$(i_{n})_{MM^{\prime}}(i_{n})_{M^{\prime}M}$ are generated by the identity map
$1_{X}\colon X\to X$ as well; hence, according to Corollary 1, they are
homotopic to the identity fundamental sequences $(1_{n})_{MM}$ and
$(1_{n})_{M^{\prime}M^{\prime}}$, respectively. Thus,
$[(i_{n})_{M^{\prime}M}][(i_{n})_{MM^{\prime}}]=[(1_{n})_{MM}]$ and
$[(i_{n})_{MM^{\prime}}][(i_{n})_{M^{\prime}M}]=[(1_{n})_{M^{\prime}M^{\prime}}]$,
i.e., $(X,M)$ and $(X,M^{\prime})$ are fundamentally equivalent. ∎

######  Remark 1.

Since each metrizable compact space $X$ is homeomorphic to a compact subset of
the Hilbert cube $Q$, in constructing the fundamental category, is suffices to
consider pairs of the form $(X,Q)$.

The category whose objects are compact metrizable spaces and morphisms are
classes of fundamental sequences between them is called the _shape category_
of compact metrizable spaces. We denote it by Sh(CM). Since the fundamental
equivalence relation is an equivalence relation, it follows that the class of
all compact metrizable spaces splits into pairwise disjoint classes of spaces,
which are called _shapes_. The shape containing a space $X$ is called the
_shape of the space_ $X$ and denoted by sh($X$). We say that two compact
spaces $X$ and $Y$ _are shape equivalent_ , or _have the same shape_ , and
write $\operatorname{sh}(X)=\operatorname{sh}(Y)$, if they are fundamentally
equivalent.

###  2.2. The Spectral Method for Constructing Shape Theory (the
Mardešić–Morita Method [278], [308])

To define the shape category of arbitrary topological spaces, we need the pro-
homotopy category pro-N-TOP, which was introduced by Grothendieck in [196].
The objects of this category are inverse systems
$\mathbf{X}=\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ in the
homotopy category H-Top of topological spaces. The morphisms of the category
pro-N-TOP are defined in two steps. First, a morphism
$(\mathbf{f}_{\beta},\varphi)\colon\mathbf{X}\to\mathbf{Y}=\\{Y_{\beta},\mathbf{q}_{\beta\beta^{\prime}},B\\}$
between inverse systems is defined as a pair of maps $\varphi\colon B\to A$
and $\mathbf{f}_{\beta}\colon X_{\varphi(\beta)}\to Y_{\beta}$ satisfying the
following condition:

(*) for any $\beta,\beta^{\prime}\in B$, $\beta^{\prime}\geqslant\beta$, there
exists an index $\alpha\geqslant\varphi(\beta),\varphi(\beta^{\prime})$ such
that
$\mathbf{f}_{\beta}\mathbf{p}_{\varphi(\beta)\alpha}=\mathbf{q}_{\beta\beta^{\prime}}\mathbf{f}_{\beta^{\prime}}\mathbf{p}_{\varphi(\beta^{\prime})\alpha}$,
i.e., the following diagram is commutative:

$\textstyle{{X_{\alpha}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{p}_{\varphi(\beta)\alpha}}$$\scriptstyle{\mathbf{p}_{\varphi(\beta^{\prime})\alpha}}$$\textstyle{X_{\varphi(\beta)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{f}_{\beta}}$$\textstyle{X_{\varphi(\beta^{\prime})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{f}_{\beta^{\prime}}}$$\textstyle{Y_{\beta}}$$\textstyle{Y_{\beta^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{q}_{\beta\beta^{\prime}}}$

Then, the notion of equivalent morphisms is introduced. Morphisms
$(\mathbf{f}_{\beta},\varphi)$ and $(\mathbf{g}_{\beta},\psi)$ from an inverse
system $\mathbf{X}=\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ to
an inverse system
$\mathbf{Y}=\\{Y_{\beta},\mathbf{q}_{\beta\beta^{\prime}},B\\}$ are said to be
_equivalent_ if the following condition holds:

(**) for any $\beta\in B$, there exists an index
$\alpha\geqslant\varphi(\beta),\psi(\beta)$ such that
$\mathbf{f}_{\beta}\mathbf{p}_{\varphi(\beta)\alpha}=\mathbf{g}_{\beta}\mathbf{p}_{\psi(\beta)\alpha}$,
i.e., the following diagram is commutative:

$\textstyle{{X_{\alpha}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{p}_{\varphi(\beta)\alpha}}$$\scriptstyle{\mathbf{p}_{\psi(\beta)\alpha}}$$\textstyle{X_{\varphi(\beta)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{f}_{\beta}}$$\textstyle{X_{\psi(\beta)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathbf{g}_{\beta}}$$\textstyle{Y_{\beta}}$

This relation is indeed an equivalence relation. Thus, the set of all
morphisms from an inverse system
$\mathbf{X}=\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ to an
inverse system $\mathbf{Y}=\\{Y_{\beta},\mathbf{q}_{\beta\beta^{\prime}},B\\}$
splits into equivalence classes $[(\mathbf{f}_{\beta},\varphi)]$; it is these
classes which are morphisms $\mathbf{f}\colon\mathbf{X}\to\mathbf{Y}$ of the
pro-homotopy category pro-N-TOP. The composition of two morphisms
$\mathbf{f}\colon\mathbf{X}\to\mathbf{Y}$ and
$\mathbf{g}\colon\mathbf{Y}\to\mathbf{Z}=\\{Z_{\gamma},\mathbf{r}_{\gamma\gamma^{\prime}},\Gamma\\}$
is defined by using their representatives
$\mathbf{f}=[(\mathbf{f}_{\beta},\varphi)]$ and
$\mathbf{g}=[(\mathbf{g}_{\gamma},\psi)]$ as
$\mathbf{g}\circ\mathbf{f}=[(\mathbf{g}_{\gamma}\circ\mathbf{f}_{\psi(\gamma)},\varphi\circ\psi)]$.

######  Remark 2.

The morphisms between inverse systems
$\mathbf{X}=\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ and
$\mathbf{Y}=\\{Y_{\beta},\mathbf{q}_{\beta\beta^{\prime}},B\\}$ in the pro-
homotopy category pro-N-TOP are described by Grothendieck’s formula [196]

$\operatorname{Mor}(\mathbf{X},\mathbf{Y})=\lim_{\underset{\beta}{\longleftarrow}}\lim_{\underset{\alpha}{\longrightarrow}}[X_{\alpha},Y_{\beta}],$

where the $[X_{\alpha},Y_{\beta}]$ are homotopy classes of maps.

In constructing shape theory, of special interest is the pro-homotopy category
pro-H-CW of inverse systems in the complete subcategory H-CW of the category
H-Top. The following well-known result gives an important characterization of
spaces in the category H-CW (see [289], Chap. I, Sec. 4.1, Theorem 1).

######  Lemma 1.

For a topological space $X$, the following conditions are equivalent:

(a) $X$ has the homotopy type of a CW-complex;

(b) $X$ has the homotopy type of a simplicial complex with the metric
topology;

(c) $X$ has the homotopy type of an ANR (the class of metrizable spaces).

Taking into account this lemma, we can say that H-CW is the homotopy category
of spaces having the homotopy type of an ANR. In what follows, we shall adhere
to this approach, because in shape theory only properties of ANRs are used; we
refer to inverse systems in the category pro-H-CW as _ANR-systems_.

An important role in the construction of shape theory for the class of all
topological spaces is played by the notion of an associated ANR-system, which
was introduced by Morita in [308].

######  Definition 3 (Morita [308]).

An inverse ANR-system
$\mathbf{X}=\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ is said to
be _associated with a topological space $X$_ if there exist homotopy classes
$\mathbf{p}_{\alpha}\colon X\to X_{\alpha}$, $\alpha\in A$, satisfying the
following conditions:

(i)
$\mathbf{p}_{\alpha}=\mathbf{p}_{\alpha\alpha^{\prime}}\mathbf{p}_{\alpha^{\prime}}$
for all $\alpha\leqslant\alpha^{\prime}$;

(ii) for any ANR-space $P$ and any homotopy class $\mathbf{f}\colon X\to P$,
there exists an index $\alpha\in A$ and a homotopy class
$\mathbf{h}_{\alpha}\colon X_{\alpha}\to P$ such that
$\mathbf{h}_{\alpha}\mathbf{p}_{\alpha}=\mathbf{f}$;

(iii) if
$\boldsymbol{\varphi}\mathbf{p}_{\alpha}=\boldsymbol{\psi}\mathbf{p}_{\alpha}$
for two homotopy classes $\boldsymbol{\varphi},\boldsymbol{\psi}\colon
X_{\alpha}\to P$, then there exists an index $\alpha^{\prime}\geqslant\alpha$
such that
$\boldsymbol{\varphi}\mathbf{p}_{\alpha\alpha^{\prime}}=\boldsymbol{\psi}\mathbf{p}_{\alpha\alpha^{\prime}}$.

######  Theorem 3 (Morita [308]).

For any topological space $X$, there exists an associated inverse ANR-system.

###### Proof.

Let $\mathcal{U}_{\alpha},\alpha\in A$, be all normal locally finite open
covers of the space $X$. If a cover $\mathcal{U}_{\alpha^{\prime}}$ is a
refinement of $\mathcal{U}_{\alpha}$, then we write $\alpha^{\prime}>\alpha$.
Let $X_{\alpha}$ be the nerve of $\mathcal{U}_{\alpha}$; this is a simplicial
complex with the weak topology. For each $\alpha\in A$, consider a continuous
map $p_{\alpha}\colon X\to X_{\alpha}$ satisfying the condition

$p_{\alpha}^{-1}(St(u,X_{\alpha}))\subset U,$

where $u$ is the vertex of $X_{\alpha}$ corresponding to the element $U$ of
the cover $\mathcal{U}_{\alpha}$. Such a map is said to be _canonical_. All
canonical maps are homotopic to each other. If $\alpha^{\prime}>\alpha$, then
there exists a simplicial map $p_{\alpha\alpha^{\prime}}\colon
X_{\alpha^{\prime}}\to X_{\alpha}$ such that $p_{\alpha\alpha^{\prime}}(u)=v$
implies $U\subset V$, where $u$ and $v$ are the vertices of the simplicial
complexes $X_{\alpha}$ and $X_{\alpha^{\prime}}$ corresponding to the elements
$U$ and $V$ of the covers $\mathcal{U}_{\alpha}$ and
$\mathcal{U}_{\alpha^{\prime}}$, respectively. The map
$p_{\alpha\alpha^{\prime}}$ is called a _canonical projection_. Any two
canonical projections are homotopic to each other. Moreover,

$\mathbf{p}_{\alpha\alpha^{\prime}}\mathbf{p}_{\alpha^{\prime}}=\mathbf{p}_{\alpha}$

for all $\alpha,\alpha^{\prime}\in A$, $\alpha>\alpha^{\prime}$.

The inverse ANR-system $\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$
thus obtained is associated with the space $X$. ∎

Inverse ANR-systems associated with different spaces can be constructed by
different methods. For example, any compact Hausdorff space $X$ is the limit
of an inverse system $\\{X_{\alpha},p_{\alpha\alpha^{\prime}},A\\}$ consisting
of compact ANRs $X_{\alpha}$:
$X=\lim\limits_{\longleftarrow}\\{X_{\alpha},p_{\alpha\alpha^{\prime}},A\\}$.
It turns out that the inverse ANR-system
$\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ is associated with
$X$. However, importantly, all inverse ANR-systems associated with the same
space $X$ are equivalent in the category pro-H-CW.

Thus, each topological space $X$ is assigned an inverse ANR-system $S(X)$ in
the category pro-H-CW. Moreover, this assignment can be uniquely extended in a
natural way to morphisms of the category H-Top, i.e., each homotopy class
$\mathbf{f}\colon X\to Y$ can be assigned a morphism $S(\mathbf{f})\colon
S(X)\to S(Y)$ of the category pro-H-CW. Thereby, we obtain a functor S: H-Top
$\to$ pro-H-CW, which is called the _shape functor_.

The shape functor $S$ makes it possible to naturally define the _shape
category_ Sh-Top for the class of all topological spaces. The objects of the
category Sh-Top are those of the category H-Top (i.e., these are topological
spaces), and the morphisms are those of the category pro-H-CW:
$S(X,Y)=\text{MOR}(S(X),S(Y))$. The shape classification of spaces is weaker
than the homotopy classification, but on the class H-CW, both classifications
coincide. In the class of all compact metrizable spaces, Mardešić–Morita shape
theory coincides with Borsuk’s one.

Shape theory can be constructed without employing associated inverse systems
as follows (see Mardešić [278]). To each topological space $X$ we assign a
category $W^{X}$ whose objects are homotopy classes $\mathbf{f}\colon X\to P$,
$P\in\text{H-CW}$, and morphisms
$\mathbf{u}\colon\mathbf{f}\to\mathbf{f^{\prime}}$, where
$\mathbf{f^{\prime}}\colon X\to P^{\prime}$, $P^{\prime}\in\text{H-CW}$, are
homotopy classes $\mathbf{u}\colon P\to P^{\prime}$ such that
$\mathbf{u}\mathbf{f}=\mathbf{f^{\prime}}$. Note that the identity morphism
$1_{\mathbf{f}}\colon\mathbf{f}\to\mathbf{f}$ is the homotopy class
$\mathbf{1}_{P}\colon P\to P$, and the compositions of morphisms in the
category $W^{X}$ are the compositions of the corresponding homotopy classes in
the category H-CW. Now the shape morphisms $F\colon X\to Y$ are defined as the
covariant functors $F\colon W^{Y}\to W^{X}$ satisfying the following two
conditions: (i) if $\mathbf{g}\in W^{Y}$, $\mathbf{g}\colon Y\to P$, then
$F(\mathbf{g})=\mathbf{f}\in W^{X}$, where $\mathbf{f}\colon X\to P$; (ii) if
$\mathbf{u}\colon\mathbf{g}\to\mathbf{g^{\prime}}$ is the morphism determined
by a homotopy class $\mathbf{u}\colon P\to P^{\prime}$, then the morphism
$F(\mathbf{u})\colon F(\mathbf{g})\to F(\mathbf{g^{\prime}})$ is determined by
the same homotopy class $\mathbf{u}\colon P\to P^{\prime}$.

By using this definition of the shape category, is easy to prove the following
criterion for a continuous map to be a shape equivalence.

######  Theorem 4.

A continuous map $f\colon X\to Y$ is a shape equivalence if and only if, for
any ANR $P$, the induced map $f^{*}\colon[Y,P]\to[X,P]$ is one-to-one.

Here $[Y,P]$ denotes the set of homotopy classes $\mathbf{g}\colon Y\to P$ and
$f^{*}$ is the map assigning the homotopy class
$f^{*}(\mathbf{g})=\mathbf{g}\mathbf{f}\in[X,P]$ to each $\mathbf{g}\colon
Y\to P$.

###  2.3. Equivariant Shape Theory

There are many ways to construct shape category for $G$-spaces (continuous
transformation groups with a fixed action of a group $G$). Smirnov [25], [26]
constructed shape theory for the categories of metrizable $G$-spaces and
compact $G$-spaces in the case of a compact acting group $G$ by the Borsuk–Fox
method [10], [168]. Equivariant shape theory for arbitrary $G$-spaces in the
case of a finite group $G$ was constructed independently by Pop [330] and
Matumoto [293] and in the case of a compact group $G$, by Antonyan and
Mardešić [50] and Gevorgyan [186]. Čerin [99] constructed the shape category
for arbitrary $G$-spaces in the case of any acting group $G$ by using
$G$-homotopy classes of families of set-valued maps.

In [186] equivariant shape theory was constructed by a method based on the
application of all invariant continuous pseudometrics on a given $G$-space.
Let $X$ be any $G$-space, and let $\mu$ be an invariant continuous
pseudometric on $X$. Consider the equivalence relation on $X$ defined by
setting $x\sim x^{\prime}$ if and only if $\mu(x,x^{\prime})=0$. We denote the
corresponding quotient space $X|_{\sim}$ by $X_{\mu}$ and the equivalence
class of an element $x\in X$ by $[x]_{\mu}$. The quotient space $X_{\mu}$ is a
$G$-space with action $g[x]_{\mu}=[gx]_{\mu}$, and
$\rho([x]_{\mu},[x^{\prime}]_{\mu})=\mu(x,x^{\prime})$ is an invariant metric
on $X_{\mu}$. Note also that the quotient map $p_{\mu}\colon X\to X_{\mu}$
defined by $p_{\mu}(x)=[x]_{\mu}$ is continuous and equivariant. Thus, the
following assertion is valid.

######  Proposition 1.

The quotient space $X_{\mu}$ is an invariant metrizable $G$-space, and the
quotient map $p_{\mu}\colon X\to X_{\mu}$ is an equivariant continuous map.

The following theorem makes it possible to construct equivariant shape theory
(see Gevorgyan’s paper [186]) for the class of all $G$-spaces in the case a
compact group $G$ by the Mardešić–Morita method of inverse systems.

######  Theorem 5.

For any $G$-space $X$, there exists an associated inverse system
$\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ in the equivariant
homotopy category H-G-ANR.

###### Proof.

Consider the family $\mathcal{P}$ of all invariant pseudometrics on the
$G$-space $X$. We introduce a natural order on this family by setting
$\mu\leqslant\mu^{\prime}$ if
$\mu(x,x^{\prime})\leqslant\mu^{\prime}(x,x^{\prime})$ for $x,x^{\prime}\in
X$.

Given any pseudometrics $\mu,\mu^{\prime}\in\mathcal{P}$,
$\mu\leqslant\mu^{\prime}$, we define a map $p_{\mu\mu^{\prime}}\colon
X_{\mu^{\prime}}\to X_{\mu}$ by
$p_{\mu\mu^{\prime}}([x]_{\mu^{\prime}})=[x]_{\mu}$, where
$[x]_{\mu^{\prime}}\in X_{\mu^{\prime}}$ and $[x]_{\mu}\in X_{\mu}$. This map
is well-defined, because it does not depend on the choice of a representative
in the class $[x]_{\mu^{\prime}}$. Indeed, let
$[x]_{\mu^{\prime}}=[x^{\prime}]_{\mu^{\prime}}$, that is,
$\mu^{\prime}(x,x^{\prime})=0$. Then
$\mu(x,x^{\prime})\leqslant\mu^{\prime}(x,x^{\prime})=0$, whence
$\mu(x,x^{\prime})=0$. This means that $[x]_{\mu}=[x^{\prime}]_{\mu}$, that
is,
$p_{\mu\mu^{\prime}}([x]_{\mu^{\prime}})=p_{\mu\mu^{\prime}}([x^{\prime}]_{\mu^{\prime}})$.
It is easy to verify that the map $p_{\mu\mu^{\prime}}\colon
X_{\mu^{\prime}}\to X_{\mu}$ satisfies the conditions
$p_{\mu\mu^{\prime}}p_{\mu^{\prime}}=p_{\mu}$ and
$p_{\mu\mu^{\prime}}p_{\mu^{\prime}\mu^{\prime\prime}}=p_{\mu\mu^{\prime\prime}}$
for any $\mu,\mu^{\prime},\mu^{\prime\prime}\in\mathcal{P}$,
$\mu\leqslant\mu^{\prime}\leqslant\mu^{\prime\prime}$.

Thus, $\\{X_{\mu},p_{\mu\mu^{\prime}}\\}$ is an inverse system of invariant
metrizable $G$-spaces. The $G$-space $X_{\mu}$ is isometrically and
equivariantly embedded in some normed $G$-space $M(X_{\mu})$ as a closed
subspace, and the equivariant map $p_{\mu\mu^{\prime}}\colon
X{{}_{\mu}^{\prime}}\to X{{}_{\mu}}$ extends to equivariant maps
$\bar{p}_{\mu\mu^{\prime}}\colon M(X{{}_{\mu}^{\prime}})\to M(X{{}_{\mu}})$,
which satisfy the relation
$\bar{p}_{\mu\mu^{\prime}}\bar{p}_{\mu^{\prime}\mu^{\prime\prime}}=\bar{p}_{\mu\mu^{\prime\prime}}$
(see Gevorgyan’s paper [182]). Thus, we obtain the inverse system
$\\{M(X_{\mu}),\bar{p}_{\mu\mu^{\prime}}\\}$ of normed $G$-spaces.

Let $A$ be the set of all pairs $(\mu,U)$, where $\mu\in\mathcal{P}$, and let
$U$ be an open invariant neighborhood of the metrizable $G$-space $X_{\mu}$ in
the normed $G$-space $M(X_{\mu})$. On the set $A$ we define a natural order by
setting $\alpha\leqslant\alpha^{\prime}$ for $\alpha=(\mu,U)$ and
$\alpha^{\prime}=(\mu^{\prime},U^{\prime})$ if and only if
$\mu\leqslant\mu^{\prime}$ and $\bar{p}_{\mu\mu^{\prime}}(U^{\prime})\subset
U$. Now we set $X_{\alpha}=U$ and define an equivariant map
$p_{\alpha\alpha^{\prime}}\colon X_{\alpha^{\prime}}\to X_{\alpha}$ by
$p_{\alpha\alpha^{\prime}}=\bar{p}_{\mu\mu^{\prime}}|_{U^{\prime}}\colon
U^{\prime}\to U$. Note that if $\mu=\mu^{\prime}$, then
$\bar{p}_{\mu\mu^{\prime}}=id\colon M(X{{}_{\mu}})\to M(X{{}_{\mu}})$;
therefore, $p_{\alpha\alpha^{\prime}}=i\colon U^{\prime}\hookrightarrow U$.

Thus, we have constructed the inverse system
$\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ in the category H-G-
ANR. Let us prove that it is equivariantly associated with the $G$-space $X$.

For each $\alpha=(\mu,U)\in A$, we define an equivariant map $p_{\alpha}\colon
X\to X_{\alpha}=U$ by $p_{\alpha}=ip_{\mu}$, where $p_{\mu}\colon X\to
X_{\mu}$ is the quotient map and $i\colon X_{\mu}\to U$ is an embedding. It is
easy to see that $p_{\alpha}=p_{\alpha\alpha^{\prime}}p_{\alpha^{\prime}}$ if
$\alpha\leqslant\alpha^{\prime}$.

Now let $f\colon X\to Q$ be a continuous equivariant map, where $Q$ is any
$G$-ANR. Consider an invariant metric $\rho$ on the space $Q$, whose existence
follows from the compactness of the group $G$. On $X$ we define a continuous
invariant pseudometric $\mu$ by $\mu(x,x^{\prime})=\rho(f(x),f(x^{\prime}))$.
Note that the map $\varphi\colon X_{\mu}\to Q$ given by
$\varphi([x]_{\mu})=f(x)$ is an equimorphism of the $G$-spaces $X_{\mu}$ and
$f(X)$, and $\varphi p_{\mu}=f$. Since $X_{\mu}$ is a closed invariant subset
of the normed $G$-space $M(X_{\mu})$ and $Q$ is a $G$-ANR, it follows that
there exists an equivariant extension $h\colon U\to Q$ of the map
$\varphi\colon X_{\mu}\to Q$, where $U$ is an invariant neighborhood of
$X_{\mu}$ in $M(X_{\mu})$. In other words, $h|_{X_{\mu}}=\varphi$, or
$hi=\varphi$.

Take any index $\alpha=(\mu,U)\in A$. We have $X_{\alpha}=U$,
$p_{\alpha}=ip_{\mu}$, and $hp_{\alpha}=f$. Indeed,
$hp_{\alpha}=hip_{\mu}=\varphi p_{\mu}=f$. Now suppose that $h_{0},h_{1}\colon
X_{\alpha}\to Q$ is an equivariant map to the G-ANR $Q$ such that
$h_{0}p_{\alpha}\simeq_{G}h_{1}p_{\alpha}$. Since $X_{\alpha}=U$ and
$p_{\alpha}=ip_{\mu}$, it follows that the equivariant maps $h_{0}|_{X_{\mu}}$
and $h_{1}|_{X_{\mu}}$ are equivariantly homotopic. Hence there exists an
invariant neighborhood $V\subset U$ of the closed invariant subset $X_{\mu}$
such that $h_{0}|_{V}$ and $h_{1}|_{V}$ are equivariantly homotopic as well.
This means that
$h_{0}p_{\alpha\alpha^{\prime}}\simeq_{G}h_{1}p_{\alpha\alpha^{\prime}}$,
where $\alpha^{\prime}=(\mu,V)$. ∎

##  3\. The Shape Classification of Spaces

The shape category, as any other category, generates a classification of all
of its objects, that is, a shape classification of topological spaces. Of
primary interest are classes of spaces in which the shape classification
coincides with the homotopy classification. One of such classes is the class
of ANRs. However, there are also other classes for which the shape
classification coincides even with the topological one. An example is the
class of all zero-dimensional spaces (see Godlewski’s paper [195] and Mardešić
and Segal’s paper [288]).

######  Theorem 6.

Zero-dimensional spaces $X$ and $Y$ have the same shape if and only if they
are homeomorphic.

######  Corollary 2.

There exist $\aleph_{1}$ countable compact metrizable spaces with pairwise
different shapes.

Indeed, all countable compact metrizable sets are zero-dimensional, and
Mazurkiewicz and Sierpiński [294] proved that there are $\aleph_{1}$ different
topological types of countable compact metrizable spaces.

Corollary 2 leads to the conclusion that the number of different shapes of
compact sets in $\mathbf{R}^{1}$ is uncountable.

Yet another class of spaces in which the shape classification coincides with
the topological one is the class of all $P$-adic solenoids $S_{P}$, where
$P=(p_{1},p_{2},\ldots)$ is a sequence of primes. Recall that a $P$-adic
solenoid is defined as the limit of an inverse sequence
$\\{X_{n},p_{nn+1},N\\}$, where $X_{n}=S^{1}$ and $p_{nn+1}$ is a map of
degree $p_{n}$ for each $n\in\mathbb{N}$.

######  Theorem 7 ([195], [288]).

Solenoids $S_{P}$ and $S_{Q}$ have the same shape if and only if they are
homeomorphic.

Godlewski [195] showed that the shape of any solenoid is completely determined
by its first cohomology group, in much the same way that the shape of any
plane continuum is determined by its first Betti number.

Another interesting case in which shape theory gives nothing new is that of
all compact connected Abelian groups. The shape morphisms between groups are
in one-to-one correspondence with the continuous homomorphisms of these
groups. Moreover, Keesling [233] proved the following theorem.

######  Theorem 8.

Let $X$ and $Y$ be compact connected Abelian groups. Then
$\operatorname{sh}X=\operatorname{sh}Y$ if and only if $X$ and $Y$ are
isomorphic (algebraically and topologically).

It should be mentioned that this result turns out to be very useful for
constructing various counterexamples in shape theory. The class of compact
connected Abelian groups is substantially larger than the class of all
solenoids. Moreover, the following theorem of Keesling is valid [230] (see
also [149]).

######  Theorem 9.

Let $X$ be a $T^{n}$-like continuum, where $T^{n}=S^{1}\times\ldots\times
S^{1}$ is the $n$-torus. Then $X$ has the shape of a compact connected Abelian
topological group.

However, if $\Pi$ is the family of all compact connected Lie groups and $X$ is
a $\Pi$-like continuum, then the shape of $X$ may be different from that of a
compact connected topological group. The corresponding example was constructed
in [230].

In the case of plane continua, shape depends only on the first Betti number of
a continuum, i.e., on the number of domains into which the plane is separated
by this continuum.

######  Theorem 10.

[10] Two plane continua $X$ and $Y$ have the same shape if and only if their
first Betti numbers coincide.

In particular, $\operatorname{sh}X\geqslant\operatorname{sh}Y$ if and only if
the number of domains in $\mathbf{R}^{2}\backslash X$ is greater than or equal
to that of domains in $\mathbf{R}^{2}\backslash Y$. Therefore, for any two
plane continua $X$ and $Y$, we have either
$\operatorname{sh}X=\operatorname{sh}Y$,
$\operatorname{sh}X<\operatorname{sh}Y$, or $\operatorname{sh}X>Y$. However,
for continua in Euclidean 3-space $\mathbf{R}^{3}$, the situation is
different: there exist continua $X,Y\subset\mathbf{R}^{3}$ such that
$\operatorname{sh}X\leqslant\operatorname{sh}Y$ and
$\operatorname{sh}X\geqslant\operatorname{sh}Y$ but
$\operatorname{sh}X\neq\operatorname{sh}Y$.

Theorem 10 implies the existence of countably many different shapes of plane
continua. Representatives of these shapes are a point (the trivial shape), the
wedge of $n$ circles for every $n\in\mathbb{N}$, and the wedge of infinitely
many circles. This gives a complete shape classification of plane continua.

It is easy to show that the family of all different shapes of compact sets in
the plane has cardinality $2^{\aleph_{0}}$. Godlewski [194] proved that in
$\mathbf{R}^{3}$ there exist $2^{\aleph_{0}}$ continua with pairwise different
shapes (see also [288]). In fact, these different shapes can be found among
solenoids.

Complete shape classifications of $\Pi$-similar continua for various classes
$\Pi$ can be found in [288], [354], [355], [356], [201], [202], [204], [230],
[149], and [403].

##  4\. Complement Theorems in Shape Theory

Shape theory is a very useful tool for solving classical problems of infinite-
dimensional and geometric topology. This has become quite clear after deep
results of Chapman [105], [106], Geoghegan and Summerhill [180], Edwards
[160], West [407], and other authors, which have demonstrated the
effectiveness of shape theory methods in geometric topology. On the other
hand, methods of infinite-dimensional topology are very useful for studying
the shapes of metrizable compact sets. As far as we know, in homotopy theory,
these methods were first applied by Borsuk [81]. He proved a theorem about the
homotopy type of a quotient space of Euclidean space $\mathbf{R}^{n}$ by using
Klee’s theorem [240] on extending homeomorphisms of compact sets in a Hilbert
space to the entire Hilbert space. These methods were also applied by
Henderson [209].

A special role in the application of methods of infinite-dimensional topology
to shape theory is played by the so-called complement theorems, which answer
the following question in various situations: When are the complements
$M\backslash X$ and $M\backslash Y$ of compact sets $X$ and $Y$ embedded in an
ambient space $M$ in a special way homeomorphic? One of the first results is
this kind was Borsuk’s theorem [10] that if $X$ and $Y$ are plane continua,
then the complements $\mathbf{R}^{2}\backslash X$ and
$\mathbf{R}^{2}\backslash Y$ are homeomorphic if and only if
$\operatorname{sh}(X)=\operatorname{sh}(Y)$. Remarkable results in this
direction were also obtained by Chapman [105], [106]. In essence, Chapman’s
theorems assert that, under certain constraint on the embedding of two compact
spaces $X$ and $Y$ in the Hilbert cube $Q$, these spaces have the same shape
if and only if their complements are homeomorphic. Therefore, if $X$ and $Y$
are absolute neighborhood retracts, then they have the same homotopy type if
and only if their complements are homeomorphic.

We represent the Hilbert cube $Q$ in the form the product
$\prod\limits_{n=1}^{\infty}I_{n}$, where $I_{n}=[0,1]$. The set
$s=\prod\limits_{n=1}^{\infty}\mathring{I}_{n}$, where
$\mathring{I}_{n}=(0,1)$, is called the _pseudointerior_ of the Hilbert cube
$Q$.

Recall that a compact subset $X$ of the Hilbert cube $Q$ is called a _$Z$
-set_ if the identity map $1_{Q}$ can be approximated arbitrarily closely by
continuous maps from $Q$ to $Q\backslash X$, i.e., for any $\varepsilon>0$,
there exists a continuous map $f\colon Q\to Q\backslash X$ such that
$d(f,1_{Q})<\varepsilon$. This notion was introduced by Anderson [46] and is
one of the fundamental notions of the theory of $Q$-manifolds [104].

Note that any compact metrizable space is embedded in the Hilbert cube as a
$Z$-set. Compact sets lying in the pseudointerior $s$ of the Hilbert cube $Q$
form an important class of $Z$-sets. The following theorem shows that any
$Z$-set in the Hilbert cube $Q$ can be mapped to the pseudointerior $s$ by a
homeomorphism $h\colon Q\to Q$.

######  Theorem 11.

Let $X\in Q$ be a $Z$-set. Then, for any $\varepsilon>0$, there exists a
homeomorphism $h\colon Q\to Q$ such that $h(X)\subset s$ and
$d(h,1_{Q})<\varepsilon$.

This theorem, which was proved by Anderson [46], is exceptionally important,
because it reduces studying $Z$-sets to the theory of compact sets in the
pseudointerior $s$.

Chapman [105] proved that the shape of any $Z$-set (in particular, of any
compact set in $s$) depends only on the topological type of its complement. To
be more precise, the following remarkable _complement theorem_ is valid (see
[105]).

######  Theorem 12.

Let $X$ and $Y$ be any $Z$-sets in the Hilbert cube $Q$. Then they have the
same shape if and only if their complements $Q\backslash X$ and $Q\backslash
Y$ are homeomorphic.

If $X$ and $Y$ are absolute neighborhood retracts, then they have the same
homotopy type if and only if their complements are homeomorphic.

To prove his theorem, Chapman applied deep results on $Q$-manifolds (see [46],
[48], [49], and [415]).

Note that a similar assertion for subsets of the Hilbert space $l_{2}$ is
false, because, according to one of Anderson’s theorems [47], $l_{2}$ is
homeomorphic to $l_{2}\backslash X$ for any compact set $X\subset l_{2}$.

Chapman [106] also proved the first finite-dimensional analogue of Theorem 12
about complements. This theorem has attracted attention of many experts in
geometric topology, who proved new finite-dimensional complement theorems. We
mention papers [180] by Geoghegan and Summerhill, [390] by Venema, [218] by
Ivanšić, Sher, and Venema, and [314] by Mrozik. In all these papers, it was
assumed that compact metrizable spaces $X$ and $Y$ are appropriately embedded
in $\mathbf{R}^{n}$ and satisfy certain dimension conditions, and it was
proved that $X$ and $Y$ have the same shape if and only if their complements
$\mathbf{R}^{n}\backslash X$ and $\mathbf{R}^{n}\backslash Y$ are
homeomorphic.

######  Theorem 13 (Chapman [106]).

Let $X$ and $Y$ be compact metrizable spaces of dimension $\leqslant k$. Then
the following assertions hold:

(a) for any $n\geqslant 2k+2$, there exist embeddings $i\colon
X\to\mathbf{R}^{n}$ and $j\colon Y\to\mathbf{R}^{n}$ such that if
$\operatorname{sh}(X)=\operatorname{sh}(Y)$, then $\mathbf{R}^{n}\backslash
i(X)\cong\mathbf{R}^{n}\backslash j(Y)$;

(b) for any $n\geqslant 3k+3$, there exist embeddings $i\colon
X\to\mathbf{R}^{n}$ and $j\colon Y\to\mathbf{R}^{n}$ such that if
$\mathbf{R}^{n}\backslash i(X)\cong\mathbf{R}^{n}\backslash j(Y)$, then
$\operatorname{sh}(X)=\operatorname{sh}(Y)$.

To state and prove their complement theorem, Geoghegan and Summerhill [180]
introduced the notions of an $\varepsilon$-push and a strong $Z_{k}$-set. Let
$X$ be a closed subset of $\mathbf{R}^{n}$. A homeomorphism
$h\colon\mathbf{R}^{n}\to\mathbf{R}^{n}$ is called an _$\varepsilon$ -push_ of
the pair $(\mathbf{R}^{n},X)$ if there exists an isotopy
$H\colon\mathbf{R}^{n}\times I\to\mathbf{R}^{n}$ such that
$d(x,H_{t}(x)<\varepsilon)$ for all $x\in\mathbf{R}^{n}$ and $t\in I$ (this
isotopy is called an $\varepsilon$-isotopy), $H_{0}=1$, $H_{1}=h$, and
$H_{t}(x)=x$ for all $t\in I$ and all $x$ satisfying condition
$\operatorname{dist}(x,X)\geqslant\varepsilon$.

A closed subset $X$ of $\mathbf{R}^{n}$ is called a _strong $Z_{k}$-set_
$(k\geqslant 0)$ if, for each compact polyhedron $P$ in $\mathbf{R}^{n}$ of
dimension $\dim P\leqslant k+1$ and any $\varepsilon>0$, there exists an
$\varepsilon$-push $h$ of the pair $(\mathbf{R}^{n},X)$ such that $P\cap
h(X)=\emptyset$.

######  Theorem 14 (Geoghegan and Summerhill [180]).

Let $X$ and $Y$ be compact metrizable spaces being strong $Z_{n-k-2}$-sets in
$\mathbf{R}^{n}$ $\left(k\geqslant 0,n\geqslant 2k+2\right)$.

Then the following conditions are equivalent:

(i) $\operatorname{sh}(X)=\operatorname{sh}(Y)$;

(ii) the pairs $(\mathbf{R}^{n}/X,\\{X\\})$ and $(\mathbf{R}^{n}/Y,\\{Y\\})$
are homeomorphic;

(iii) the triples
$(\mathbf{R}^{n}/X,(\mathbf{R}^{n}/X)\backslash\\{X\\},\\{X\\})$ and
$(\mathbf{R}^{n}/Y,(\mathbf{R}^{n}/Y)\backslash\\{Y\\},\\{Y\\})$ have the same
homotopy type;

(iv) $\mathbf{R}^{n}\backslash X\cong\mathbf{R}^{n}\backslash Y$.

In relation to this theorem, it is important that _any metrizable compact set
$X$ of dimension $\dim X\leqslant k$ is embedded in $\mathbf{R}^{n}$
$(n\geqslant 2k+1)$ as a strong $Z_{n-k-2}$-set_.

Note that the property of being a strong $Z_{k}$-set is not homotopy
invariant. However, homotopy-invariant properties often imply the property of
being a strong $Z_{n-k-2}$-set. This has enabled Geoghegan and Summerhill
[180] to prove the following complement theorem, which is more convenient for
applications.

######  Theorem 15.

Let $X$ and $Y$ be nonempty compact sets in $\mathbf{R}^{n}$ such that their
complements $\mathbf{R}^{n}\backslash X$ and $\mathbf{R}^{n}\backslash Y$ are
uniformly locally 1-connected and $\max\\{\dim X,\dim Y\\}\leqslant k$,
$\max\\{2k+2,5\\}\leqslant n$. Then
$\operatorname{sh}(X)=\operatorname{sh}(Y)$ if and only if
$\mathbf{R}^{n}\backslash X\cong\mathbf{R}^{n}\backslash Y$.

Recall that a metric space $X$ is said to be _uniformly locally $k$-connected_
(written as $X\in ULC^{k}$) if, for any $\varepsilon>0$, there exists a
$\delta>0$ such that any map $f\colon S^{k}\to X$ satisfying the condition
$\operatorname{diam}f(S^{k})<\delta$ is homotopic to a constant map on any set
$V\subset X$ of diameter $\operatorname{diam}V<\varepsilon$.

######  Corollary 3.

Let $X$ and $Y$ be closed $k$-dimensional submanifolds of $\mathbf{R}^{n}$,
where $\max\\{2k+2,5\\}\leqslant n$. Then $X$ is homotopy equivalent to $Y$ if
and only if $\mathbf{R}^{n}\backslash X\cong\mathbf{R}^{n}\backslash Y$.

The condition $ULC^{1}$ in Theorem 15 cannot be omitted, because the
complement of Blankinship’s wild arc [72] in $\mathbf{R}^{n}$, $n\geqslant 3$,
is not simply connected. It is also important that the codimension be
sufficiently large. Henderson [209] constructed an example of non–homotopy
equivalent two-dimensional compact polyhedra with homeomorphic complements in
$\mathbf{R}^{3}$.

In [112] the so-called _small loops condition_ (SLC) was introduced. A compact
space $X\subset\mathbf{R}^{n}$ satisfies the SLC if, for any neighborhood $U$
of $X$ there exists a smaller neighborhood $V$ ($X\subset V\subset U$) and a
number $\varepsilon>0$ such that any loop in $V\backslash X$ of diameter
$<\varepsilon$ is homotopic to zero in $U\backslash X$. This notion is a
generalization of McMillan’s _cellularity criterion_ (CS) [297], which
characterizes cellular embeddings in manifolds. The CC is obtained from the
SLC at $\varepsilon=\infty$.

Using results of [112], Hollingsworth and Rushing [210] were able to prove the
following generalization of Theorem 15.

######  Theorem 16.

Let $X,Y\subset\mathbf{R}^{n}$ be compact sets satisfying the SLC, and let
$\max\\{\dim X,\allowbreak\dim Y\\}\leqslant k$, where
$\max\\{2k+2,5\\}\leqslant n$. Then
$\operatorname{sh}(X)=\operatorname{sh}(Y)$ if and only if
$\mathbf{R}^{n}\backslash X\cong\mathbf{R}^{n}\backslash Y$.

This theorem has two interesting corollaries.

######  Corollary 4.

Let $X,Y\subset\mathbf{R}^{n}$ be compact ANRs satisfying the SLC, and let
$\max\\{\dim X,\allowbreak\dim Y\\}\leqslant k$, where
$\max\\{2k+2,5\\}\leqslant n$. Then $X$ and $Y$ have the same homotopy type if
and only if $\mathbf{R}^{n}\backslash X\cong\mathbf{R}^{n}\backslash Y$.

######  Corollary 5.

Suppose that compact spaces $X,Y\subset\mathbf{R}^{n}$ are homeomorphic and
satisfy the SLC. If $\dim X=\dim Y\leqslant k$ and $\max\\{2k+2,5\\}\leqslant
n$, then $\mathbf{R}^{n}\backslash X\cong\mathbf{R}^{n}\backslash Y$.

Venema [390] has succeeded in replacing the dimension $\dim$ in Theorem 16 by
the shape dimension $\operatorname{sd}$. For this purpose, he defined the
_inessential loop condition_ (ILC). A compact subset $X$ of a manifold $M$
satisfies the ILC if, for any neighborhood $U$ of $X$ in $M$, there exists a
smaller neighborhood $V$, $X\subset V\subset U$, such that any loop in
$V\backslash X$ which is inessential (homotopic to zero) in $V$ is also
inessential in $U\backslash X$. Note that the ILC implies the SLC. If
$X\subset M^{n}$ and $\dim X\leqslant n-2$, then these two conditions are
equivalent.

######  Theorem 17 (Venema [390]).

Suppose that compact sets $X,Y\subset\mathbf{R}^{n}$ satisfy the ILC,
$\max\\{\operatorname{sd}X,\operatorname{sd}Y\\}\leqslant k$, and
$\max\\{2k+2,5\\}\leqslant n$. Then
$\operatorname{sh}(X)=\operatorname{sh}(Y)$ if and only if
$\mathbf{R}^{n}\backslash X\cong\mathbf{R}^{n}\backslash Y$.

In the case where $\dim X\leqslant n-3$ and $Y$ is a finite polyhedron of
dimension $\leqslant k$ such that $2k+2\leqslant n$, this theorem was proved
in [112].

The most general complement theorem was proved by Ivanšić, Sher, and Venema
[218].

######  Theorem 18 ([218]).

Suppose that compact sets $X,Y\subset\mathbf{R}^{n}$ are shape $r$-connected
and satisfy the ILC. Suppose also that
$\max\\{\operatorname{sd}X,\operatorname{sd}Y\\}\leqslant k$ and
$n\geqslant\max\\{2k+2-r,5\\}$. If $n\geqslant k+3$, then
$\operatorname{sh}(X)=\operatorname{sh}(Y)$ implies $\mathbf{R}^{n}\backslash
X\cong\mathbf{R}^{n}\backslash Y$. The converse is true if $n\geqslant k+4$.

This theorem generalizes almost all preceding results.

Complement theorems in various categories and in more general situations were
proved by Mrozik [314].

##  5\. Movability and Other Shape Invariants

The notion of movability for compact metrizable spaces was introduced by
Borsuk [78].

######  Definition 4.

A compact metrizable space $X$ embedded in an AR $M$ is said to be _movable_
if, given any neighborhood $U$ of $X$, there exists a neighborhood
$U^{\prime}\subset U$ of $X$ such that, for any neighborhood
$U^{\prime\prime}\subset U$ of $X$, there exists a homotopy $H\colon
U^{\prime}\times I\to U$ satisfying the conditions $H(x,0)=x$ and $H(x,1)\in
U^{\prime\prime}$ for all $x\in U^{\prime}$.

This definition does not depend on the choice of the AR $M$ and the embedding
of $X$ in $M$.

For arbitrary topological spaces, the notion of movability was defined by
Mardešić and Segal [287] in terms of ANR-systems.

######  Definition 5.

An inverse ANR-system
$\mathbf{X}=\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ is said to
be _movable_ if the following condition holds:

(M) given any $\alpha\in A$, there is an $\alpha^{\prime}\in A$,
$\alpha^{\prime}\geqslant\alpha$, such that, for any $\alpha^{\prime\prime}\in
A$, $\alpha^{\prime\prime}\geqslant\alpha$, there exists a homotopy class
$\mathbf{r}^{\alpha^{\prime}\alpha^{\prime\prime}}\colon
X_{\alpha^{\prime}}\to X_{\alpha^{\prime\prime}}$ satisfying the condition
$\mathbf{p}_{\alpha\alpha^{\prime}}=\mathbf{p}_{\alpha\alpha^{\prime\prime}}\circ\mathbf{r}^{\alpha^{\prime}\alpha^{\prime\prime}}$.

An inverse ANR-system
$\mathbf{X}=\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ is
_uniformly movable_ if

(UM) given any $\alpha\in A$, there exists an $\alpha^{\prime}\in A$,
$\alpha^{\prime}\geqslant\alpha$, and a morphism $\mathbf{r}\colon
X_{\alpha^{\prime}}\to\mathbf{X}$ in the category pro-H-CW such that
$\mathbf{p}_{\alpha}\circ\mathbf{r}=\mathbf{p}_{\alpha\alpha^{\prime}}$, where
$\mathbf{p}_{\alpha}\colon\mathbf{X}\to X_{\alpha}$ is the morphism in pro-H-
CW generated by the identity homotopy class $\mathbf{1}_{X_{\alpha}}$.

A topological space $X$ is said to be _(uniformly) movable_ if there exists an
associated (uniformly) movable ANR-system
$\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$.

If a space $X$ is (uniformly) movable, then all ANR-systems associated with it
are (uniformly) movable, i.e., the notion of (uniform) movability depends only
on the space $X$ itself. In the case of compact metrizable spaces, Definitions
4 and 5 are equivalent.

The following theorem shows that the notion of a movable space can be defined
without resorting to associated inverse systems.

######  Theorem 19 (Gevorgyan [14]).

A topological space $X$ is movable if and only if the following condition
holds:

(*) given any homotopy class $\mathbf{f}\colon X\to Q$, where
$Q\in\mathrm{ANR}$, there are homotopy classes $\mathbf{f^{\prime}}\colon X\to
Q^{\prime}$ and $\boldsymbol{\eta}\colon Q^{\prime}\to Q$, where
$Q^{\prime}\in\mathrm{ANR}$ and
$\mathbf{f}=\boldsymbol{\eta}\mathbf{f^{\prime}}$, such that, for any homotopy
classes $\mathbf{f^{\prime\prime}}\colon X\to Q^{\prime\prime}$ and
$\boldsymbol{\eta}^{\prime}\colon Q^{\prime\prime}\to Q$, where
$Q^{\prime\prime}\in\mathrm{ANR}$ and
$\mathbf{f}=\boldsymbol{\eta}^{\prime}\mathbf{f^{\prime\prime}}$, there exists
a homotopy class $\boldsymbol{\eta}^{\prime\prime}\colon Q^{\prime}\to
Q^{\prime\prime}$ satisfying the condition
$\boldsymbol{\eta}=\boldsymbol{\eta}^{\prime}\boldsymbol{\eta}^{\prime\prime}$
(see Diagram 1).

---
$\textstyle{Q^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\boldsymbol{\eta}}$$\scriptstyle{\boldsymbol{\eta}^{\prime\prime}}$$\textstyle{Q}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
11.38109pt\mathbf{f}}$$\scriptstyle{\mathbf{f^{\prime}}}$$\scriptstyle{\mathbf{f^{\prime\prime}}}$$\textstyle{Q^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\boldsymbol{\eta}^{\prime}}$

Diagram 1

Movability is a monotone shape invariant, i.e., _if
$\operatorname{sh}X\leqslant\operatorname{sh}Y$ and $Y$ is movable, then so is
$X$_. Movability is preserved under taking products (Borsuk [78] and Bogatyi
[8]) and suspensions (Borsuk [83]), but it is not preserved under
intersection, union (Cox [117]), and cell maps (Keesling [224]). If each
component of a compact space $X$ is movable, then the whole space $X$ is
movable. However, the converse is not true: there exists a movable compact
space which has a non-movable component (see [10]).

Movable spaces are rather many. All ANRs are movable. Moreover, all _stable
spaces_ (i.e., spaces having the shape of an ANR) are movable. All compact
sets in the plane are movable as well [78]. However, in $\mathbf{R}^{3}$ there
exist non-movable continua; e.g., so are all solenoids [10]. This follows from
the non-movability of the first homology pro-group of any solenoid. The point
is that movability is preserved under functorial passages. Therefore, if a
space $X$ is movable, then so are its homotopy pro-groups
$\text{$\mathrm{pro}$-}\pi_{n}(X,*)$ and homology pro-groups pro-$H_{n}(X)$.

Movability, which can be defined in any pro-category (see [312]), is important
because the passage to the limit in movable inverse systems can be performed
without losing the algebraic information about the system. In the case of a
shape morphism $F\colon(X,*)\to(Y,*)$ of movable compact metrizable spaces,
the induced homomorphisms
$\check{\pi}_{n}(F)\colon\check{\pi}_{n}(X,*)\to\check{\pi}_{n}(Y,*)$ and
$\text{$\mathrm{pro}$-}\pi_{n}(F)\colon\text{$\mathrm{pro}$-}\pi_{n}(X,*)\to\text{$\mathrm{pro}$-}\pi_{n}(Y,*)$
of the shape groups and pro-groups are, or are not, isomorphisms
simultaneously (see Theorem 50). Therefore, in the case of movable spaces,
some theorems remain valid under the replacement of homotopy pro-groups by
shape groups. This is the case with the shape versions of Whitehead’s theorem
(see Theorem 49), Hurewicz’ theorem (see Theorem 57), and so on.

In the category pro-Set movability has a simple characterization: an inverse
system
$\mathbf{X}=\\{X_{\alpha},p_{\alpha\alpha^{\prime}},A\\}\in\text{$\mathrm{pro}$-}\mathrm{Set}$
is movable if and only if the following Mittag-Leffler (ML) condition holds:
_for any $\alpha\in A$, there exists an $\alpha^{\prime}\geqslant\alpha$ such
that
$p_{\alpha\alpha^{\prime}}(X_{\alpha^{\prime}})=p_{\alpha\alpha^{\prime\prime}}(X_{\alpha^{\prime\prime}})$
for any $\alpha^{\prime\prime}\geqslant\alpha^{\prime}$_. All movable pro-
groups satisfy the Mittag-Leffler condition. However, the converse is not true
(see [289]).

The family of all movable compact spaces is _shape bounded_ ; to be more
precise, the following theorem of Spież is valid [366].

######  Theorem 20.

There exists a movable compact space $X_{0}$ such that
$\operatorname{sh}(X)\leqslant\operatorname{sh}(X_{0})$ for any movable
compact space $X$.

Such a compact space $X_{0}$ is called a _majorant_ of the family of all
movable compact spaces. This theorem implies that the family of all compact
sets in the plane is shape bounded, or has a majorant. This is not true for
the family of all compact sets in $\mathbf{R}^{3}$. This follows from a result
of Borsuk and Holsztyński [86], according to which there does not exist a
compact space whose shape is larger than that of any solenoid.

In equivariant shape theory, the counterpart of Theorem 20 is false, i.e.,
_the class of all $G$-movable compact spaces has no majorants_. However, the
following theorem is valid.

######  Theorem 21 (Gevorgyan [12]).

Let $G$ be a second-countable compact group. Then, in any class of weakly
shape comparable $G$-movable compact spaces, there exists a majorant.

The _weak shape comparability_ of $G$-spaces $X$ and $Y$ means that there
exist $G$-shape morphisms both from $X$ in $Y$ and from $Y$ to $X$. The proof
of the last theorem is based on McCord’s construction [296] of a universal
compact space and on the equivariant counterpart of Brown’s theorem [12].

The first results on equivariant movability were obtained by the author in
[12]–[15] and [185], where, in particular, the following theorems were proved.

######  Theorem 22.

Let $G$ be a compact Lie group, and let $X$ be a $G$-movable metrizable
$G$-space. Then $X$ is $H$-movable for any closed subgroup $H$ of $G$.

The $G$-movability of a metrizable $G$-space implies also the movability of
the $H$-fixed point set for any closed subgroup $H$ of $G$. In particular,
_the $G$-movability of a metrizable $G$-space $X$ implies its movability_. The
converse is not true even if the acting group $G$ is the cyclic group
$\mathbb{Z}_{2}$ (see [185, Example 5.1]).

######  Theorem 23.

Let $G$ be a compact group, and let $X$ be a metrizable $G$-space. If $X$ is
$G$-movable, then, for any closed normal subgroup $H$ of $G$, the $H$-orbit
space $X|_{H}$ is $G$-movable as well.

In particular, the $G$-movability of a $G$-space $X$ implies the movability of
the orbit space $X|_{G}$. The converse is not true. However, if a compact Lie
group acts freely on a metrizable space $X$, then $G$-movability is equivalent
to movability. The assumptions of the theorem that the acting group $G$ is Lie
and the action is free are essential.

The movability of topological groups was studied by Keesling [229], [233],
[236] and by Kozlowski and Segal [254]. In particular, Keesling [229] proved
the following theorem.

######  Theorem 24.

A connected compact Abelian group is movable if and only if it is locally
connected.

For non-Abelian groups, this theorem does not hold (see [229]).

The equivariant movability of topological groups was studied by the author in
[183]. We mention the following result.

######  Theorem 25 (Gevorgyan [183]).

A second-countable compact group $G$ is Lie if and only if it is $G$-movable.

This theorem gives, in particular, new examples of movable but equivariantly
non-movable $G$-spaces (see [183]).

_Movable shape morphisms_ were defined and studied by Gevorgyan and Pop [187].
A morphism
$(\mathbf{f},\varphi)\colon\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}\to\\{Y_{\beta},\mathbf{q}_{\beta\beta^{\prime}},B\\}$
in the category pro-H-CW is said to be _movable_ if, given any $\beta\in B$,
there is an $\alpha\in A$, $\alpha\geqslant\varphi(\beta)$, such that, for any
$\beta^{\prime}\geqslant\beta$, there exists a homotopy class
$\mathbf{u}\colon X_{\alpha}\to Y_{\beta^{\prime}}$ satisfying the condition
$\mathbf{f}_{\beta}\mathbf{p}_{\varphi(\beta)\alpha}=\mathbf{q}_{\beta\beta^{\prime}}\mathbf{u}$.
The notion of a movable shape morphism agrees with that of a movable space in
the sense that _a space $X$ movable if and only if so is the identity map
$1_{X}$_. Movable morphisms are important, in particular, because the
movability assumption on a space can sometimes be replaced by the weaker
assumption that a shape morphism $F\colon X\to Y$ is movable; see, e.g.,
Whitehead’s Theorem 53 for movable morphisms $F\colon X\to Y$.

Movable shape morphisms were also introduced and studied (by different methods
and for different purposes) by other authors [158], [100], [416], [417].

In [10] Borsuk introduced the notion of $n$-movability, which is a shape
invariant, too. A metrizable compact space $X$ lying in AR-space $M$ is said
to be _$n$ -movable_ if, given any neighborhood $U$ of $X$ in $M$, there
exists a neighborhood $U^{\prime}\subset U$ of $X$ such that, for any
neighborhood $U^{\prime\prime}\subset U$ of $X$, any metrizable compact space
$K$ of dimension $\dim K\leqslant n$, and any map $f\colon K\to U^{\prime}$,
there exists a map $g\colon K\to U^{\prime\prime}$ homotopic to $f$ in $U$.
For arbitrary spaces, this notion is introduced by using inverse systems.

######  Definition 6.

An inverse ANR-system $\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$
is said to be _$n$ -movable_ if, given any $\alpha\in A$, there is an
$\alpha^{\prime}\in A$, $\alpha^{\prime}\geqslant\alpha$, such that, for any
$\alpha^{\prime\prime}\in A$, $\alpha^{\prime\prime}\geqslant\alpha$, and any
homotopy class $\mathbf{h}\colon P\to X_{\alpha^{\prime}}$, where $P$ an ANR
of dimension $\dim P\leqslant n$, there exists a homotopy class
$\mathbf{r}\colon P\to X_{\alpha^{\prime\prime}}$ satisfying the condition
$\mathbf{p}_{\alpha\alpha^{\prime}}\mathbf{h}=\mathbf{p}_{\alpha\alpha^{\prime\prime}}\mathbf{r}$.

A topological space $X$ is _$n$ -movable_ if there exists an associated
$n$-movable ANR-system
$\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$.

It is easy to see that $(n+1)$-movability always implies $n$-movability. It is
also clear that the movability of a space $X$ implies its $n$-movability for
any $n$. If $\dim X\leqslant n$, then the converse is also true: the
$n$-movability of a space $X$ implies its movability (in the case
$\operatorname{sd}X\leqslant n$, this was proved in [8]). The equivariant
counterpart of this statement has been proved in the case of a finite acting
group $G$: any equivariantly $n$-movable $G$-compact set of dimension $\dim
X\leqslant n$ is equivariantly movable (see Gevorgyan [15]). Therefore, a
finite-dimensional $G$-compact space is equivariantly movable if and only if
it is equivariantly $n$-movable for all $n$. In the general case, the
$n$-movability does not imply movability: the Kahn compact space [221] is not
movable, but it is $n$-movable for all $n$ (see [8], [255]). Any $LC^{n-1}$
paracompact space is $n$-movable (see [253]).

The particularly important case of 1-movability was studied in detail by Dydak
[125], McMillan [298], Krasinkiewicz [257], [259], [258], and other authors.
If a metrizable continuum $(X,*)$ is 1-movable for a point $x\in X$, then it
is also 1-movable for any other point of $X$. Moreover, if a metrizable
continuum $(X,x)$ is 1-movable and
$\operatorname{sh}(X)=\operatorname{sh}(Y)$, then
$\operatorname{sh}(X,x)=\operatorname{sh}(Y,y)$ for any $y\in Y$ (Dydak
[125]). There exists a nontrivial example of a metrizable continuum $X$ such
that it is 1-movable but $(X,*)$ is not movable (Dydak [144]). A continuous
image of a 1-movable metrizable continuum is 1-movable. The following
important criterion for the 1-movability of a metrizable continuum is due to
Krasinkiewicz [258].

######  Theorem 26.

A metrizable continuum $(X,*)$ is 1-movable if and only if its homotopy pro-
group $\text{$\mathrm{pro}$-}\pi_{1}(X,*)$ satisfies the Mittag-Leffler
condition.

######  Theorem 27.

A metrizable continuum $(X,*)$ is 1-movable if and only if it has the shape of
a locally connected continuum.

It follows from these theorems that a solenoid $X$ does not have the shape of
a locally connected continuum, because the pro-group
$\text{$\mathrm{pro}$-}\pi_{1}(X)$ does not satisfy the Mittag-Leffler
condition.

Many results of shape theory are easy to transfer from the case of pointed
spaces to that of nonpointed spaces. However, the reverse transfer of results
is not always easy or even possible (see [176], [289]). For example, the
movability of a pointed space $(X,*)$ readily implies that of $X$, but the
question of whether the converse is true is difficult and has not been
answered so far. A partial answer is given by the following theorem.

######  Theorem 28 (Krasinkiewicz [257]).

Let $(X,*)$ be a 1-movable metrizable continuum. Then the movability of $X$
implies that of $(X,*)$.

The notion of $n$-movability has initiated _$n$ -shape theory_, which was
developed by Chigogidze in [27]. This theory is an important tool in the study
of $k$-dimensional Menger manifolds, which was begun by Bestvina in [67].

There are several more modifications of the notions of movability and
$n$-movability (see papers [4] by Bogatyi, [241] by Kodama, [281] by Mardešić,
and [312] by Moszynska; we also mention papers [61] by Bauer, [134] by Dydak,
[254] by Koslowski and Segal, [368] by Spież, and [401] by Watanabe). The
notion of the _$n$ -movability of shape morphisms_ was introduced and studied
by Gevorgyan and Pop [188].

Yet another important shape invariant is _strong movability_. This notion has
played an important role, primarily in the study of the stability of
topological spaces and absolute neighborhood shape retracts. A space $X$ is
said to be _strongly movable_ if there exists an associated _strongly movable
ANR-system_ $\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$, which
means that, given any $\alpha\in A$, there is an
$\alpha^{\prime}\geqslant\alpha$ such that, for any
$\alpha^{\prime\prime}\geqslant\alpha$, there exists an
$\alpha^{*}\geqslant\alpha^{\prime},\alpha^{\prime\prime}$ and a homotopy
class $\mathbf{r}^{\alpha^{\prime}\alpha^{\prime\prime}}\colon
X_{\alpha^{\prime}}\to X_{\alpha^{\prime\prime}}$ satisfying the conditions
$\mathbf{p}_{\alpha\alpha^{\prime\prime}}\mathbf{r}^{\alpha^{\prime}\alpha^{\prime\prime}}=\mathbf{p}_{\alpha\alpha^{\prime}}$
and
$\mathbf{r}^{\alpha^{\prime}\alpha^{\prime\prime}}\mathbf{p}_{\alpha^{\prime}\alpha^{*}}=\mathbf{p}_{\alpha^{\prime\prime}\alpha^{*}}$.
Obviously, strong movability implies movability. Strong movability is
preserved under shape domination. For a connected space $(X,*)$, strong
movability is equivalent to stability (Dydak [137], Watanabe [401]).

The notion of movability was extended to and studied in more general cases by
Segal [353], Shostak [31], Gevorgyan [14], [184], Gevorgyan and Pop [187],
[188], [190], and Avakyan and Gevorgyan [55].

Avakyan and Gevorgyan [55] (see also the author’s papers [14] and [184])
introduced the notion of a _movable category_ and proved criteria for the
movability and strong movability of a topological space.

######  Definition 7.

Let $K$ and $L$ be any categories, and let $\Phi\colon K\to L$ be any
covariant functor. The category $K$ is said to be _movable with respect to the
category $L$ and the functor $\Phi\colon K\to L$_ if, given any object
$X\in\operatorname{Ob}(K)$, there is an object $Y\in\operatorname{Ob}(K)$ and
a morphism $f\in\operatorname{Mor}_{K}(Y,X)$ such that, for any object
$Z\in\operatorname{Ob}(K)$ and any morphism $g\in\operatorname{Mor}_{K}(Z,X)$,
there exists a morphism $h\in\operatorname{Mor}_{L}(\Phi(Y),\Phi(Z))$
satisfying the condition $\Phi(g)h=\Phi(f)$.

If $K$ is a full subcategory of the category $L$ and $\Phi\colon
K\hookrightarrow L$ is the embedding functor, then the subcategory $K$ is said
to be _movable with respect to the category $L$_.

If $K=L$ and $\Phi=1_{K}$, then the category $K$ is called _strongly movable_.

######  Theorem 29 (Avakyan and Gevorgyan [55]).

A topological space $X$ is movable if and only if the category $W^{X}$ is
movable with respect to the category H-CW and forgetful functor $\Omega\colon
W^{X}\to\text{H-CW}$.

The _forgetful functor_ $\Omega\colon W^{X}\to\text{H-CW}$ takes each object
$f\colon X\to Q$ to the object $Q\in\text{H-CW}$ and each morphism
$\eta\colon(f\colon X\to Q)\to(f^{\prime}\colon X\to Q^{\prime})$, $\eta\circ
f=f^{\prime}$, of the category $W^{X}$ to a morphism $\eta\colon Q\to
Q^{\prime}$ of the category H-CW.

######  Theorem 30 (Avakyan and Gevorgyan [55]).

The topological space $X$ is strongly movable if and only if so is the
category $W^{X}$.

Similar theorems for _uniformly movable_ spaces and categories were proved by
Gevorgyan and Pop [190].

######  Definition 8.

A category $K$ is said to be _uniformly movable_ if, given any object $X\in
K$, there exists an object $M(X)\in K$ and a morphism $m_{X}\colon M(X)\to X$
satisfying the following conditions:

(i) for any object $Y\in K$ and any morphism $p\colon Y\to X$ of the category
$K$, there exists a morphism $u(p)\colon M(X)\to Y$ such that $pu(p)=m(X)$;

(ii) for any objects $Y,Z\in K$ and morphisms $p\colon Y\to X$, $q\colon Z\to
X$, and $r\colon Z\to Y$ of $K$ satisfying the condition $pr=q$, the relation
$ru(q)=u(p)$ holds.

######  Theorem 31 (Gevorgyan and Pop [190]).

A topological space $X$ is uniformly movable if and only if so is the category
$W^{X}$.

##  6\. Stable Spaces and Shape Retracts

A topological space $X$ is said to be _stable_ if it has the shape of an ANR.
Stability in the shape category was first considered by Porter [337] and
systematically studied by Demers [120], Edwards and Geoghegan [153], [154],
[155], Dydak [125], [137], [143], Geoghegan and Lacher [179], Porter [335],
[336], and other authors.

Stability is a shape invariant. The notions of stability and pointed stability
are equivalent, i.e., the following theorem is valid (see Dydak’s paper [143]
and Geoghegan’s paper [174]).

######  Theorem 32.

A pointed space $(X,*)$ is stable if and only if so is $X$.

######  Theorem 33.

If a connected pointed space $(X,*)$ is shape dominated by a pointed ANR-space
$(P,*)$, then $(X,*)$ is stable.

This theorem was independently proved by Demers [120] and by Edwards and
Geoghegan [153], [154], who used different methods.

The following theorem of Dydak [137] shows that, for connected pointed spaces,
the notions of stability and strong movability coincide (see also Watanabe’s
paper [401]).

######  Theorem 34.

A connected space $(X,*)$ is stable if and only if it is strongly movable.

If a pointed space $(X,*)$ is stable, then the pro-groups
$\text{$\mathrm{pro}$-}\pi_{k}(X,*)$ are isomorphic to the shape groups
$\check{\pi}_{k}(X,*)$. Therefore, the stability of $(X,*)$ implies that of
the pro-groups $\text{$\mathrm{pro}$-}\pi_{k}(X,*)$. The converse is true for
finite-dimensional connected spaces. Thus, stability admits the following
algebraic characterization.

######  Theorem 35.

Let $(X,*)$ be a connected space of finite shape dimension
$\operatorname{sd}X$. Then $(X,*)$ is stable if and only if so are all
homotopy pro-groups $\text{$\mathrm{pro}$-}\pi_{k}(X,*)$.

This theorem was first proved by Edwards and Geoghegan [154], [155], who used
fairly sophisticated abstract tools. Subsequently, elementary proofs of this
theorem were found by Dydak [143] and Geoghegan [174].

Let $X$ be a subspace of a topological space $Y$. A shape morphism $R\colon
Y\to X$ is called a _shape retraction_ if $Ri=1_{X}$, where $i\colon X\to Y$
is a shape embedding. The space $X$ is then called a _shape retract_ of $Y$. A
subspace $X\subset Y$ is called a _neighborhood shape retract_ of $Y$ if $X$
is a shape retract of some neighborhood $U$ of $X$ in $Y$.

Absolute (neighborhood) shape retracts in shape theory are defined by analogy
with the corresponding notions for the class of metrizable spaces. We say that
a metrizable space $X$ is an _absolute (neighborhood) shape retract_ in the
class of metrizable spaces and write $X\in\text{ASR}$ (respectively,
$X\in\text{ANSR}$) if, for any closed embedding of $X$ in a metrizable space
$Y$, the space $X$ is a (neighborhood) shape retract of $Y$ (see papers [10]
by Borsuk, [277] and [281] by Mardešić, [352] by Segal, and [31] by Shostak).
It is easy to see that these notions are shape invariants.

The notion of an absolute (neighborhood) shape retract in the class of compact
metrizable spaces coincides with that of a _fundamental absolute
(neighborhood) retract_ , abbreviated as FAR (respectively, FANR), which was
introduced by Borsuk in [10].

One of the first questions arising in shape theory is as follows: When does a
space have the shape of a point? It turns out that this property characterizes
absolute shape retracts.

######  Theorem 36 (Mardešić [277]).

A metrizable space $X$ is an ASR if and only if $\operatorname{sh}(X)=0$.

For metrizable compact spaces, this theorem was proved by Borsuk [73], and in
the class of all $p$-paracompact spaces, by Shostak [31]. For arbitrary
topological spaces, Theorem 36 is not true (Godlewski [193]).

The triviality of the shape of a space $X$ can also be characterized in the
language of associated ANR-systems as follows.

######  Theorem 37.

Let $\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ be an ANR-system
associated with a space $X$. Then $X$ has trivial shape if and only if, for
each $\alpha\in A$, there exists an $\alpha^{\prime}\geqslant\alpha$ such that
the projection $p_{\alpha\alpha^{\prime}}$ is homotopic to a constant map.

In the case of compact Hausdorff spaces, this theorem was proved by Mardešić
[277]. Of interest is also the following theorem (see Dydak’s paper [143]).

######  Theorem 38.

A space $X$ is an absolute shape retract if and only if
$\operatorname{sd}X<\infty$ and $X$ is shape $n$-connected for some $n$.

######  Theorem 39.

A space $X$ is an absolute shape retract if and only if it is movable and
shape $n$-connected for all $n$.

Recall that a space $X$ is said to be _shape $n$-connected_ if
$\text{$\mathrm{pro}$-}\pi_{k}(X,*)=0$ for all $k\leqslant n$. For compact
metrizable spaces, the last theorem was proved by Bogatyi [7] and Borsuk [78].

The notion of an ASR is remarkable in many respects. In particular, on this
notion the definition of a cell-like map is based, which makes it possible to
answer the natural question of under what assumptions a map $f\colon X\to Y$
generates a shape isomorphism (see Theorems 83 and 84).

The class of ANSRs is a natural extension of the class of ANRs, and the global
properties of ANSRs resemble to a certain extent those of ANRs. Clearly, each
compact space having the shape of an ANR is an ANSR. Borsuk [79] asked the
converse question: Is it true that any compact metrizable ANSR has the shape
of a compact metrizable ANR? Edwards and Geoghegan [153] gave a negative
answer to this question. They constructed an example of a two-dimensional
metrizable continuum $X$ which is an ANSR but does not have the shape of a
finite complex. This continuum $X$ cannot have the shape of a compact ANR,
because, as is well known (West [406]), any compact metrizable ANR has the
homotopy type (and hence the shape) of a finite complex. Nevertheless, any
plane compact ANSR has the shape of a compact ANR. Indeed, a compact set $X$
in the plane is an ANSR if and only if its Betti numbers $p_{0}(X)$ and
$p_{1}(X)$ are finite. Therefore, the shape of a plane compact ANSR $X$ equals
the shape of a finite union of pairwise disjoint planar graphs, and such
unions are compact ANRs. Since one-dimensional ANSRs have the shape of a
compact set in the plane (see [383]), we can say that Borsuk’s problem has a
positive solution for one-dimensional compact ANSRs. As shown by the example
of Edwards and Geoghegan mentioned above, for two-dimensional compact ANSRs,
the solution is negative.

The ANSRs are characterized by the following theorem due to Mardešić and Segal
[289].

######  Theorem 40.

A metrizable space $X$ is an ANSR if and only if it is shape dominated by an
ANR $P$.

Importantly, in the case of compact metrizable spaces, the shape dominating
ANR $P$ can be chosen compact in this theorem.

######  Theorem 41.

A compact metrizable space $X$ is an ANSR if and only if it is shape dominated
by a compact ANR $P$.

Theorem 40 is also valid in the pointed case.

######  Theorem 42.

A metrizable space $(X,*)$ is an ANSR if and only if it is shape dominated by
an ANR $(P,*)$.

Theorems 32, 33, and 42 imply the following important property of pointed ANSR
(see [289]).

######  Theorem 43.

A connected metrizable space $(X,*)$ is an ANSR if and only if $(X,*)$ is
stable, i.e., has the shape of an ANR $(P,*)$.

For compact metrizable spaces, this theorem was proved by Edwards and
Geoghegan [154], [155]. In Theorem 43 an ANR $(P,*)$ cannot be replaced by a
compact ANR even in the case of a metrizable continuum. Indeed, the above-
mentioned two-dimensional metrizable continuum constructed by Edwards and
Geoghegan [153] is an ANSR but does not have the shape of a compact ANR. The
question of when a compact metrizable space $(X,*)$ has the shape of a compact
ANR has been answered by using tools of algebraic $K$-theory. To be more
precise, a metrizable continuum $(X,*)$ has the shape of a finite complex
precisely when the so-called Wall obstruction [393] $\sigma(X,*)$, which takes
values in the reduced Grothendieck group
$\widetilde{K}^{0}(\check{\pi}_{1}(X,*))$, is trivial (see [153]).

Theorems 35 and 43 imply the following algebraic characterization of ANSRs.

######  Theorem 44.

Let $(X,*)$ be a connected space of finite shape dimension
$\operatorname{sd}X$. Then $(X,*)$ is an ANSR if and only if all homotopy pro-
groups $\text{$\mathrm{pro}$-}\pi_{k}(X,*)$ are stable.

The following important result was obtained by Hastings and Heller [206],
[207].

######  Theorem 45.

Let $X$ be a connected ANSR. Then $(X,*)$ is a pointed ANSR.

The proof of this theorem has become possible after the splitting problem for
homotopy idempotents was solved. Recall that a map $f\colon X\to X$ is called
a _homotopy idempotent_ if $f^{2}\simeq f$. A map $f\colon X\to X$ is said to
_split_ in the category H-CW if there exists a CW-complex $P$ and maps
$u\colon P\to X$ and $v\colon X\to P$ such that $v\circ u\simeq 1_{P}$ and
$u\circ v\simeq f$.

######  Theorem 46.

Any homotopy idempotent $f\colon P\to P$ of a finite-dimensional CW-complex
$P$ splits.

In the case of infinite-dimensional CW-complexes, this statement is false.
Dydak [125] and Minc, and also Freyd and Heller [172], independently proved
the existence of a group $G$ and a homomorphism $\phi\colon G\to G$ which
induces a nonsplitting homotopy idempotent $f\colon K(G,1)\to K(G,1)$ of the
infinite-dimensional Eilenberg–MacLane complex $K(G,1)$. Moreover, the group
$G$ is universal in the sense that if $f^{\prime}\colon X\to X$ is a
nonsplitting homotopy idempotent of an infinite-dimensional CW-complex $X$,
then there exists an injection $G\to\pi_{1}(X)$ equivariant with respect to
$f_{*}$ and $f^{\prime}_{*}$.

However, it is well known that any homotopy idempotent $f\colon(X,*)\to(Y,*)$
between pointed connected CW-complexes splits (see papers [87] by Brown, [153]
by Edwards and Geoghegan, and [171] by Freyd).

Note that, in the nonpointed case, the problem of splitting homotopy
idempotents arises in various areas of topology. In homotopy theory this
problem is closely related to Brown’s theorem on the representability of half-
exact functors (see papers [172] by Freyd and Heller and [208] by Heller), and
in the shape theory, to the study of ANSRs.

It follows from Theorems 32 and 45 that Theorem 43 remains valid in the
nonpointed case.

######  Theorem 47.

A connected metrizable space $X$ is an ANSR if and only if $X$ is stable.

##  7\. Whitehead’s and Hurewicz’ Theorems in Shape Theory

It was mentioned in the introduction that Whitehead’s classical theorem is not
valid for arbitrary spaces instead of CW-complexes. Extending Whitehead’s
theorem to more general spaces is one of the objectives of shape theory.

Let $(X,*)$ be a pointed space, and let
$(\mathbf{X},\boldsymbol{*})=((X_{\alpha},*),\mathbf{p}_{\alpha\alpha^{\prime}},A)$
be an inverse ANR-system associated with it. Applying the functors $\pi_{n}$,
$n=0,1,\dots$, to this system, we obtain the _homotopy pro-groups_ of the
pointed space $(X,*)$:

$\text{$\mathrm{pro}$-}\pi_{n}(X,*)=(\pi_{n}(X_{\alpha},*),\mathbf{p}_{\alpha\alpha^{\prime}\\#},A).$

The homotopy pro-groups $\text{$\mathrm{pro}$-}\pi_{n}(X,*)$ are an analogue
of the homotopy groups $\pi_{n}(X,*)$ in shape theory.

Recall that any shape morphism $F\colon(X,*)\to(Y,*)$ is determined by a
morphism
$\mathbf{f}\colon(\mathbf{X},\boldsymbol{*})\to(\mathbf{Y},\boldsymbol{*})$ in
the category pro-H-CW; this suggest the natural definition of a _morphism_
$\text{$\mathrm{pro}$-}\pi_{n}(F)\colon\text{$\mathrm{pro}$-}\pi_{n}(X,*)\to\text{$\mathrm{pro}$-}\pi_{n}(Y,*)$
_of homotopy pro-groups_.

It is easy to show that $\text{$\mathrm{pro}$-}\pi_{n}$ is a covariant functor
from the shape category $\text{$\mathrm{Sh}$-$\mathrm{Top}$}_{*}$ to the
category $\text{$\mathrm{pro}$-}\mathrm{Set}_{*}$ for $n=0$ and to the
category $\text{$\mathrm{pro}$-}\mathrm{Grp}_{*}$ for $n=1,2,\dots$ . In
particular, $\text{$\mathrm{pro}$-}\pi_{n}(X,*)$ is a shape invariant.

The inverse limits of the pro-homotopy groups
$\text{$\mathrm{pro}$-}\pi_{n}(X,*)$ are called the _shape groups_ of the
pointed space $(X,*)$ and denoted by $\check{\pi}_{n}(X,*)$:

$\check{\pi}_{n}(X,*)=\lim_{\longleftarrow}(\pi_{n}(X_{\alpha},*),p_{\alpha\alpha^{\prime}\\#},A).$

If $F\colon(X,*)\to(Y,*)$ is a shape morphism, then the passage to the limit
yields a homomorphism
$\check{\pi}_{n}(F)\colon\check{\pi}_{n}(X,*)\to\check{\pi}_{n}(Y,*)$ of shape
groups. Therefore, $\check{\pi}_{n}$ is a covariant functor from the category
$\text{$\mathrm{Sh}$-$\mathrm{Top}$}_{*}$ to the category $\mathrm{Set}_{*}$
(for $n=0$) or to the category $\mathrm{Grp}_{*}$ (for $n=1,2,\ldots$).

The _homology pro-group_ $\text{$\mathrm{pro}$-}H_{n}(X)$ is defined in a
similar way:

$\text{$\mathrm{pro}$-}H_{n}(X)=(H_{n}(X_{\alpha}),H_{n}(p_{\alpha\alpha^{\prime}}),A).$

Passing to the limit in this system, we obtain the well-known _Čech homology
group_ $\check{H}_{n}(X)$ of the space $X$.

It should be mentioned that, for spaces with good local structure, such as
ANRs, the homotopy (homology) pro-groups can be replaced by shape groups (Čech
homology groups). However, in the general case, a part of information is lost
under the passage to the limit. Therefore, the homotopy or homology pro-groups
contain more information about the space $X$ than the shape or Čech homology
groups.

The following theorem of Morita [309] is the most general version of
Whitehead’s theorem in shape theory.

######  Theorem 48 (shape version of Whitehead’s theorem, Morita [309]).

A shape morphism $F\colon(X,*)\to(Y,*)$ of finite-dimensional (in the sense of
dimension sd) connected topological spaces is a shape equivalence if and only
if the induced homomorphisms
$\text{$\mathrm{pro}$-}\pi_{n}(F)\colon\text{$\mathrm{pro}$-}\pi_{n}(X,*)\to\text{$\mathrm{pro}$-}\pi_{n}(Y,*)$
of homotopy pro-groups are isomorphisms for all $n$.

Unlike Whitehead’s classical theorem, this theorem is valid for any spaces,
and it deals with homotopy pro-groups rather than homotopy groups. However, it
involves a new constraint, namely, the finite dimensionality of the spaces $X$
and $Y$. This constraint cannot be removed. The corresponding example was
constructed in [122] by using the Kahn metric continuum [221], which is
defined as follows. First, a space $X_{0}$ is defined as the CW-complex
obtained by attaching a $(2p+1)$-cell to the sphere $S^{2p}$ by a map of prime
degree $p$. Then, for each $n\geqslant 0$, a space $X_{n+1}$ is constructed by
induction as the $(2p-2)$-fold suspension of $X_{n}$:

$X_{n+1}=\Sigma^{2p-2}(X_{n}).$

Between these spaces maps $f_{n}\colon X_{n}\to X_{n-1}$, $n>1$, are defined,
also by induction, as

$f_{n}=\Sigma^{2p-2}(f_{n-1}),$

beginning with map $f_{1}\colon X_{1}\to X_{0}$ constructed in a certain
special way.

The Kahn compact space $K$ is the limit of the inverse sequence

$X_{0}\overset{f_{1}}{\longleftarrow}X_{1}\overset{f_{2}}{\longleftarrow}X_{2}\longleftarrow\cdots$

Since $\pi_{k}\left(\Sigma^{n(2p-2)},*\right)=0$ for $n(2p-2)\geqslant k$, it
follows that $\text{$\mathrm{pro}$-}\pi_{k}(K,*)=0$ for all $k$. However, $K$
does not have the shape of a point, because the composition

$f_{1}\circ f_{2}\circ\ldots\circ f_{n}\colon X_{n}\to X_{0}$

is essential (i.e., not homotopic to a constant) for any $n$ (see Adams’
papers [36] and [37]). This fact follows also from Toda’s results [379].

The first Whitehead-type theorem in shape theory was proved by Moszynska
[311]. She proved Theorem 48 for metrizable continua $(X,*)$ and $(Y,*)$.
Mardešić [274] proved this theorem in the case where $(X,*)$ and $(Y,*)$ are
compact and $(Y,*)$ is metrizable and in the case where $(X,*)$ and $(Y,*)$
are any spaces and the shape morphism $F\colon(X,*)\to(Y,*)$ is induced by a
continuous map.

For movable spaces, the shape analogue of Whitehead’s theorem (Theorem 48)
takes a simpler form [311], [231], [141], [143].

######  Theorem 49.

Let $(X,*)$ and $(Y,*)$ be movable metrizable continua of finite shape
dimension. Then a shape morphism $F\colon(X,*)\to(Y,*)$ is a shape equivalence
if and only if all induced homomorphisms
$\check{\pi}_{n}(F)\colon\check{\pi}_{n}(X,*)\to\check{\pi}_{n}(Y,*)$ of shape
groups are isomorphisms.

This theorem is a consequence of Theorem 48 and the following result of
independent interest [231], [143].

######  Theorem 50.

Let $F\colon(X,*)\to(Y,*)$ be a shape morphism movable metrizable continua.
If, for some $n$, the induced homomorphism
$\check{\pi}_{n}(F)\colon\check{\pi}_{n}(X,*)\to\check{\pi}_{n}(Y,*)$ is an
isomorphism (epimorphism) of shape groups, then
$\text{$\mathrm{pro}$-}\pi_{n}(F)\colon\text{$\mathrm{pro}$-}\pi_{n}(X,*)\to\text{$\mathrm{pro}$-}\pi_{n}(Y,*)$
is an isomorphism (epimorphism) of pro-groups.

The first “infinite-dimensional” Whitehead-type theorems were proved by
Edwards and Geoghegan in [152] and generalized in [137] and [141]. The most
general results are the following two theorems of Dydak [143].

######  Theorem 51.

Let $(X,*)$ and $(Y,*)$ be connected spaces. Suppose that $(X,*)$ is movable
and $F\colon(X,*)\to(Y,*)$ is a shape domination. If
$\text{$\mathrm{pro}$-}\pi_{n}(F)\colon\text{$\mathrm{pro}$-}\pi_{n}(X,*)\to\text{$\mathrm{pro}$-}\pi_{n}(Y,*)$
is an isomorphism for every $n$, then $F$ is a shape equivalence.

######  Theorem 52.

Let $F\colon(X,*)\to(Y,*)$ be a shape morphism of connected spaces. Suppose
that one of the following two conditions holds:

(i) $\operatorname{sd}X<\infty$ and $(Y,*)$ movable;

(ii) $(X,*)$ is movable and $\operatorname{sd}Y<\infty$.

If
$\text{$\mathrm{pro}$-}\pi_{n}(F)\colon\text{$\mathrm{pro}$-}\pi_{n}(X,*)\to\text{$\mathrm{pro}$-}\pi_{n}(Y,*)$
is an isomorphism for every $n$, then $F$ is a shape equivalence.

In particular, if $(X,*)$ is movable and all pro-groups
$\text{$\mathrm{pro}$-}\pi_{n}(X,*)$ are trivial, then
$\operatorname{sh}(X,*)=0$. In the last theorem the finite-dimensionality of
spaces cannot be replaced by their movability. Draper and Keesling [122]
constructed a map $f\colon(X,*)\to(Y,*)$ of movable metrizable continua which
induces isomorphisms of all homotopy pro-groups but is not a shape
equivalence. A similar example was also constructed by Kozlowski and Segal
[252].

A shape analogue of Whitehead’s theorem for movable morphisms was obtained by
Gevorgyan and Pop [187].

######  Theorem 53.

Let $(X,*)$ and $(Y,*)$ be connected spaces. Suppose that
$\operatorname{sd}X<\infty$ and $F\colon(X,*)\to(Y,*)$ is a movable shape
morphism being a shape domination. If
$\text{$\mathrm{pro}$-}\pi_{n}(F)\colon\text{$\mathrm{pro}$-}\pi_{n}(X,*)\to\text{$\mathrm{pro}$-}\pi_{n}(Y,*)$
is an isomorphism for every $n$, then $F$ is a shape equivalence.

Whitehead’s theorem has also homology versions (see [275], [309], [341]). In
their proofs the key role is played by the following result (see [309],
[341]).

######  Theorem 54.

Let $F\colon(X,*)\to(Y,*)$ be a shape morphism. If

$\text{$\mathrm{pro}$-}\pi_{0}(X,*)=\text{$\mathrm{pro}$-}\pi_{1}(X,*)=\text{$\mathrm{pro}$-}\pi_{0}(Y,*)=\text{$\mathrm{pro}$-}\pi_{1}(Y,*)=0,$

then the following two conditions are equivalent for each $n\geqslant 2$:

(i) $\text{$\mathrm{pro}$-}\pi_{k}(F)$ is an isomorphism for $k<n$ and an
epimorphism for $k=n$;

(ii) $\text{$\mathrm{pro}$-}H_{k}(F)$ is an isomorphism for $k<n$ and an
epimorphism for $k=n$.

The well-known _Hurewicz homomorphism_ $\phi_{n}\colon\pi_{n}(X,*)\to
H_{n}(X)$ naturally generates a morphism
$\text{$\mathrm{pro}$-}\phi_{n}\colon\text{$\mathrm{pro}$-}\pi_{n}(X,*)\to\text{$\mathrm{pro}$-}H_{n}(X)$.
The passage to the limit yields also the _spectral Hurewicz homomorphism_
$\check{\phi}_{n}\colon\check{\pi}_{n}(X,*)\to\check{H}_{n}(X)$.

Hurewicz’ classical theorem asserts that _if $(X,*)$ is an $(n-1)$-connected
space, i.e., $\pi_{k}(X,*)=0$ for all $0\leqslant k\leqslant n-1$, then, for
any $n\geqslant 2$, the following conditions hold: (i) $H_{k}(X)=0$,
$0\leqslant k\leqslant n-1$; (ii) $\phi_{n}\colon\pi_{n}(X,*)\to H_{n}(X)$ is
an isomorphism; (iii) $\phi_{n+1}$ is an epimorphism; (iv)
$\phi_{1}\colon\pi_{1}(X,*)\to H_{1}(X)$ is an epimorphism._

This theorem was transferred to shape theory in various forms by many authors.
The following version of Hurewicz’ theorem for the pro-category can be found
in [289].

######  Theorem 55.

Let $(X,*)$ be a shape $(n-1)$-connected space, i.e.,
$\text{$\mathrm{pro}$-}\pi_{k}(X,*)=0$, $k\leqslant n-1$. Then, for all
$n\geqslant 2$, the following conditions hold:

(i) $\text{$\mathrm{pro}$-}H_{k}(X)=0$, $1\leqslant k\leqslant n-1$;

(ii)
$\text{$\mathrm{pro}$-}\phi_{n}\colon\text{$\mathrm{pro}$-}\pi_{n}(X,*)\to\text{$\mathrm{pro}$-}H_{n}(X)$
is an isomorphism of pro-groups;

(iii)
$\text{$\mathrm{pro}$-}\phi_{n+1}\colon\text{$\mathrm{pro}$-}\pi_{n+1}(X,*)\to\text{$\mathrm{pro}$-}H_{n+1}(X)$
is an epimorphism if pro-groups.

For $n=1$,

(iv)
$\text{$\mathrm{pro}$-}\phi_{1}\colon\text{$\mathrm{pro}$-}\pi_{1}(X,*)\to\text{$\mathrm{pro}$-}H_{1}(X)$
is an epimorphism.

Somewhat different statements of this theorem were presented in [54], [291],
and [341].

Kuperberg [261] proved the following theorem, which is a special case of a
more general result due to Artin and Mazur [54].

######  Theorem 56.

If a compact metrizable space $(X,*)$ is shape $(n-1)$-connected for
$n\geqslant 2$, then the Hurewicz spectral homomorphism
$\check{\phi}_{n}\colon\check{\pi}_{n}(X,*)\to\check{H}_{n}(X)$ is an
isomorphism.

For movable compact metrizable spaces $(X,*)$, the assumption that $(X,*)$ is
shape $(n-1)$-connected can be replaced by the weaker assumption
$\check{\pi}_{k}(X,*)=0$, $k\leqslant n-1$ (see [261]).

######  Theorem 57.

Let $(X,*)$ be a movable compact metrizable space such that
$\check{\pi}_{k}(X,*)=0$ for all $k\leqslant n-1$, $n\geqslant 2$. Then
$\check{\phi}_{n}\colon\check{\pi}_{n}(X,*)\to\check{H}_{n}(X)$ is an
isomorphism.

The movability assumption in this theorem is essential. Indeed, let $X$ be the
2-fold suspension of the 3-adic solenoid. The space $X$ is connected, and
$\check{\pi}_{k}(X,*)=0$ for all $k\leqslant 3$. However,
$\check{\pi}_{4}(X,*)$ is not isomorphic to $\check{H}_{4}(X)$, because
$\check{\pi}_{4}(X,*)\neq 0$, while $\check{H}_{4}(X)=0$ (see [260]).

The metrizability assumption in Theorem 57 is essential as well. Keesling
[235] proved that, for any $n\geqslant 2$, there exists a nonmetrizable
movable continuum $(X,*)$ such that $\check{\pi}_{k}(X,*)=0$ for all
$k\leqslant n-1$ but $\check{\pi}_{n}(X,*)$ is not isomorphic to
$\check{H}_{n}(X)$.

In papers [331] by Porter, [291] by Mardešić and Ungar, and [384] by Ungar
Theorem 56 was proved for pairs $(X,A)$ (see also [309] and [396]). A Hurewicz
theorem with an application of Steenrod homology is due to Kodama and Koyama
[244].

It should be mentioned that the first spectral Hurewicz theorem was proved by
Christie [111] but only in strong shape theory. Subsequently, similar theorems
with an application of Steenrod homology were proved in [22], [244], and
[340].

##  8\. The Shape Dimension of Spaces and Maps

The dimension of topological spaces is not a shape invariant. For example, a
contractible space, which may have arbitrarily large dimension, has the shape
of a point, which is zero-dimensional. This leads to the necessity of
introducing a numerical shape invariant playing the same role in shape theory
as dimension plays in topology. Such an invariant, called _fundamental
dimension_ , was introduced by Borsuk [75], who defined it first for compact
metrizable spaces.

######  Definition 9.

The _fundamental dimension_ of a compact metrizable space $X$, denoted by
$\operatorname{Fd}X$, is defined as the least dimension $\dim Y$ of a compact
metrizable space $Y$ such that
$\operatorname{sh}Y\geqslant\operatorname{sh}X$:

$\operatorname{Fd}X=\min\\{\dim
Y:\operatorname{sh}Y\geqslant\operatorname{sh}X\\}.$

The fundamental dimension of compact metrizable spaces was thoroughly studied
by Novák [316, 317, 318, 319, 321, 322, 323] and Spież [367, 369].

For arbitrary topological spaces, Dydak [142] introduced the notion of
_deformation dimension_ ; its definition in a slightly modified form is as
follows.

######  Definition 10.

A topological space $X$ has _deformation dimension_
$\operatorname{ddim}X\leqslant n$ if any map $f\colon X\to P$ to an ANR $P$
can be homotopically factored through an ANR $P^{\prime}$ of dimension $\dim
P^{\prime}\leqslant n$, i.e., $f\simeq h\circ g$, where $g\colon X\to
P^{\prime}$ and $h\colon P^{\prime}\to P$ are some maps.

Mardešić and Segal [289] defined _shape dimension_ as follows.

######  Definition 11.

A topological space $X$ has _shape dimension_ $\operatorname{sd}X\leqslant n$
if there exists an associated inverse ANR-system
$\mathbf{X}=\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ such that
each $X_{\alpha}$, $\alpha\in A$, is homotopy dominated by an ANR of dimension
$\leqslant n$.

If $\operatorname{sd}X\leqslant n$ and $n$ is the least such number, then we
say that $X$ has _shape dimension $n$_ and write $\operatorname{sd}X=n$.

If $\operatorname{sd}X\leqslant n$ for no integer $n\geqslant 0$, then we say
that $X$ has _infinite shape dimension_ and write $\operatorname{sd}X=\infty$.

The following theorem is valid [142].

######  Theorem 58.

A topological space $X$ has shape dimension $\operatorname{sd}X\leqslant n$ if
and only if there exists an associated inverse ANR-system
$\mathbf{X}=\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ such that
$\dim X_{\alpha}\leqslant n$ for all $\alpha\in A$.

###### Proof.

Let $\operatorname{sd}X\leqslant n$. Consider an associated inverse ANR-system
$\mathbf{X}=\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ satisfying
the condition in Definition 11. This means that, for each space $X_{\alpha}$,
there exists an ANR $Y_{\alpha}$ of dimension $\dim Y_{\alpha}\leqslant n$ and
homotopy classes $\mathbf{f}_{\alpha}\colon X_{\alpha}\to Y_{\alpha}$ and
$\mathbf{g}_{\alpha}\colon Y_{\alpha}\to X_{\alpha}$ such that

(3) $\mathbf{g}_{\alpha}\mathbf{f}_{\alpha}=\mathbf{1}_{X_{\alpha}}.$

For any $\alpha,\alpha^{\prime}\in A$, $\alpha\leqslant\alpha^{\prime}$, we
define a homotopy class $\mathbf{q}_{\alpha\alpha^{\prime}}\colon
Y_{\alpha^{\prime}}\to Y_{\alpha}$ by

(4)
$\mathbf{q}_{\alpha\alpha^{\prime}}=\mathbf{f}_{\alpha}\mathbf{p}_{\alpha\alpha^{\prime}}\mathbf{g}_{\alpha^{\prime}}.$

Note that if $\alpha\leqslant\alpha^{\prime}\leqslant\alpha^{\prime\prime}$,
then

(5)
$\mathbf{q}_{\alpha\alpha^{\prime}}\mathbf{q}_{\alpha^{\prime}\alpha^{\prime\prime}}=\mathbf{q}_{\alpha\alpha^{\prime\prime}}.$

Indeed, according to (3) and (4), we have

$\mathbf{q}_{\alpha\alpha^{\prime}}\mathbf{q}_{\alpha^{\prime}\alpha^{\prime\prime}}=\mathbf{f}_{\alpha}\mathbf{p}_{\alpha\alpha^{\prime}}\mathbf{g}_{\alpha^{\prime}}\mathbf{f}_{\alpha}^{\prime}\mathbf{p}_{\alpha^{\prime}\alpha^{\prime\prime}}\mathbf{g}_{\alpha^{\prime\prime}}=\mathbf{f}_{\alpha}\mathbf{p}_{\alpha\alpha^{\prime}}\mathbf{1}_{X_{\alpha^{\prime}}}\mathbf{p}_{\alpha^{\prime}\alpha^{\prime\prime}}\mathbf{g}_{\alpha^{\prime\prime}}=\mathbf{f}_{\alpha}\mathbf{p}_{\alpha\alpha^{\prime\prime}}\mathbf{g}_{\alpha^{\prime\prime}}=\mathbf{q}_{\alpha\alpha^{\prime\prime}}.$

Thus, we have constructed an inverse ANR
$\mathbf{Y}=\\{Y_{\alpha},\mathbf{q}_{\alpha\alpha^{\prime}},A\\}$, where
$\dim Y_{\alpha}\leqslant n$ for all $\alpha\in A$. Let us prove that this
inverse system is associated with the space $X$ (see Definition 3).

Let $\alpha\in A$ be any index. Consider the homotopy class
$\mathbf{q}_{\alpha}\colon X\to Y_{\alpha}$ defined by

(6) $\mathbf{q}_{\alpha}=\mathbf{f}_{\alpha}\mathbf{p}_{\alpha}.$

This homotopy class satisfies the condition

(7)
$\mathbf{q}_{\alpha\alpha^{\prime}}\mathbf{q}_{\alpha^{\prime}}=\mathbf{q}_{\alpha},$

where $\alpha\leqslant\alpha^{\prime}$. Indeed, by virtue of (3), (4), and (6)
we have

$\mathbf{q}_{\alpha\alpha^{\prime}}\mathbf{q}_{\alpha^{\prime}}=\mathbf{f}_{\alpha}\mathbf{p}_{\alpha\alpha^{\prime}}\mathbf{g}_{\alpha^{\prime}}\mathbf{f}_{\alpha^{\prime}}\mathbf{p}_{\alpha^{\prime}}=\mathbf{f}_{\alpha}\mathbf{p}_{\alpha\alpha^{\prime}}\mathbf{1}_{X_{\alpha^{\prime}}}\mathbf{p}_{\alpha^{\prime}}=\mathbf{f}_{\alpha}\mathbf{p}_{\alpha}=\mathbf{q}_{\alpha}.$

Now, let us verify condition (ii) in Definition 3. Let $P$ be any ANR, and let
$\mathbf{f}\colon X\to P$ be any homotopy class. Then there exists an index
$\alpha\in A$ and a homotopy class $\mathbf{h}_{\alpha}\colon X_{\alpha}\to P$
such that

(8) $\mathbf{h}_{\alpha}\mathbf{p}_{\alpha}=\mathbf{f}.$

Consider $\tilde{\mathbf{h}}_{\alpha}\colon Y_{\alpha}\to P$ defined by

(9) $\tilde{\mathbf{h}}_{\alpha}=\mathbf{h}_{\alpha}\mathbf{g}_{\alpha}.$

We claim that $\tilde{\mathbf{h}}_{\alpha}\mathbf{q}_{\alpha}=\mathbf{f}$.
Indeed, taking into account (3), (6), (8), and (9), we obtain

$\tilde{\mathbf{h}}_{\alpha}\mathbf{q}_{\alpha}=\mathbf{h}_{\alpha}\mathbf{g}_{\alpha}\mathbf{f}_{\alpha}\mathbf{p}_{\alpha}=\mathbf{h}_{\alpha}\mathbf{1}_{X_{\alpha}}\mathbf{p}_{\alpha}=\mathbf{h}_{\alpha}\mathbf{p}_{\alpha}=\mathbf{f}.$

It remains to verify condition (iii) in Definition 3. Suppose that
$\boldsymbol{\varphi},\boldsymbol{\psi}\colon Y_{\alpha}\to P$ are homotopy
classes such that

(10)
$\boldsymbol{\varphi}\mathbf{q}_{\alpha}=\boldsymbol{\psi}\mathbf{q}_{\alpha}.$

By virtue of (6) this relation implies

(11)
$\boldsymbol{\varphi}\mathbf{f}_{\alpha}\mathbf{p}_{\alpha}=\boldsymbol{\psi}\mathbf{f}_{\alpha}\mathbf{p}_{\alpha}.$

Hence there exists an $\alpha^{\prime}\in A$,
$\alpha^{\prime}\geqslant\alpha$, for which

(12)
$\boldsymbol{\varphi}\mathbf{f}_{\alpha}\mathbf{p}_{\alpha\alpha^{\prime}}=\boldsymbol{\psi}\mathbf{f}_{\alpha}\mathbf{p}_{\alpha\alpha^{\prime}}.$

Relations (2) and (12) directly imply

$\boldsymbol{\varphi}\mathbf{q}_{\alpha\alpha^{\prime}}=\boldsymbol{\varphi}\mathbf{f}_{\alpha}\mathbf{p}_{\alpha\alpha^{\prime}}\mathbf{g}_{\alpha^{\prime}}=\boldsymbol{\varphi}\mathbf{f}_{\alpha}\mathbf{p}_{\alpha\alpha^{\prime}}\mathbf{g}_{\alpha^{\prime}}=\boldsymbol{\psi}\mathbf{f}_{\alpha}\mathbf{p}_{\alpha\alpha^{\prime}}\mathbf{g}_{\alpha^{\prime}}=\boldsymbol{\psi}\mathbf{f}_{\alpha}\mathbf{p}_{\alpha\alpha^{\prime}}\mathbf{g}_{\alpha^{\prime}}=\boldsymbol{\psi}\mathbf{q}_{\alpha\alpha^{\prime}}.$

The sufficiency part of the theorem is obvious. ∎

The following theorem [289] shows that, for any topological space, the notions
of deformation dimension and shape dimension are equivalent.

######  Theorem 59.

Let $X$ be topological space. Then $\operatorname{sd}X\leqslant n$ if and only
if $\operatorname{ddim}X\leqslant n$.

###### Proof.

The implication $\operatorname{sd}X\leqslant
n\implies\operatorname{ddim}X\leqslant n$ easily follows from Theorem 58.

Let us prove the implication $\operatorname{ddim}X\leqslant
n\implies\operatorname{sd}X\leqslant n$. Consider an inverse ANR-system
$\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ associated with $X$.
In view of the CW-approximation theorem, we can assume without loss of
generality that all $X_{\alpha}$ are CW-complexes and all
$p_{\alpha\alpha^{\prime}}$ are cell maps.

Since $\operatorname{ddim}X\leqslant n$, it follows that, for each map
$p_{\alpha}\colon X\to X_{\alpha}$, there exists a CW-complex $P_{\alpha}$
with $\dim P_{\alpha}\leqslant n$ and maps $u_{\alpha}\colon X\to P_{\alpha}$
and $v_{\alpha}\colon P_{\alpha}\to X_{\alpha}$ such that $p_{\alpha}\simeq
v_{\alpha}\circ u_{\alpha}$. In view of the CW-approximation theorem, we can
also assume that $v_{\alpha}\colon P_{\alpha}\to X_{\alpha}$ is a cell map,
i.e., a map to the $n$-skeleton $X_{\alpha}^{n}\subset X_{\alpha}$.

Now, consider the inverse system
$\\{X_{\alpha}^{n},\mathbf{q}_{\alpha\alpha^{\prime}},A\\}$, where each
$X_{\alpha}^{n}$ is the $n$-skeleton of the CW-complex $X_{\alpha}$ and
$q_{\alpha\alpha^{\prime}}=p_{\alpha\alpha^{\prime}}|X_{\alpha^{\prime}}^{n}$.
For each $\alpha\in A$, we define a map $q_{\alpha}\colon X\to X_{\alpha}^{n}$
by $q_{\alpha}=v_{\alpha}u_{\alpha}$. It is easy to show that this inverse
ANR-system is associated with the space $X$. To complete the proof, it remains
to apply Theorem 58. ∎

The following theorem is a corollary of Theorem 58.

######  Theorem 60.

For any topological space $X$, $\operatorname{sd}X\leqslant\dim X$.

###### Proof.

Let $\dim X\leqslant n$, and let
$\\{X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A\\}$ be an associated
inverse system consisting of CW-complexes $X_{\alpha}$. In view of the CW-
approximation theorem, we can assume that all $p_{\alpha\alpha^{\prime}}$ are
cell maps, i.e., $p_{\alpha\alpha^{\prime}}(X_{\alpha^{\prime}}^{n})\subset
X_{\alpha}^{n}$, where $X_{\alpha}^{n}$ is the $n$-skeleton of the CW-complex
$X_{\alpha}$. Consider the inverse system
$\\{X_{\alpha}^{n},\mathbf{q}_{\alpha\alpha^{\prime}},A\\}$, where
$q_{\alpha\alpha^{\prime}}=p_{\alpha\alpha^{\prime}}|X_{\alpha^{\prime}}$.
Since $\dim X\leqslant n$, it follows that there exists a map
$q_{\alpha}\colon X\to X_{\alpha}^{n}$ such that $i\circ q_{\alpha}\simeq
p_{\alpha}$, where $i\colon X_{\alpha}^{n}\to X_{\alpha}$ is the inclusion
map. It is easy to prove that this system with limit projections
$q_{\alpha}\colon X\to X_{\alpha}^{n}$, $\alpha\in A$, is associated with the
space $X$. Now, applying Theorem 58, we obtain $\operatorname{sd}X\leqslant
n$. ∎

######  Theorem 61.

If $X$ and $Y$ are any spaces and
$\operatorname{sh}X\leqslant\operatorname{sh}Y$, then
$\operatorname{sd}X\leqslant\operatorname{sd}Y$.

###### Proof.

Suppose that $\operatorname{sh}X\leqslant\operatorname{sh}Y$ and
$\operatorname{sd}Y\leqslant n$. Consider any map $f\colon X\to P$ to an ANR
$P$. By assumption there exist shape morphisms $F\colon X\to Y$ and $G\colon
Y\to X$ such that $G\circ F=1_{X}$. To the map $f\colon X\to P$ the shape
morphism $G$ assigns a map $G(f)\colon Y\to P$ (see Definition 9). Since
$\operatorname{sd}Y\leqslant n$, it follows that there exists an ANR
$P^{\prime}$ with $\dim P^{\prime}\leqslant n$ and maps $u\colon Y\to
P^{\prime}$ and $v\colon P^{\prime}\to P$ such that $v\circ u\simeq G(f)$. But
then $v\circ F(u)\simeq F(G(f))=f$. Therefore, according to Theorem 59, we
have $\operatorname{sd}X\leqslant n$. ∎

######  Corollary 6.

If $\operatorname{sh}X=\operatorname{sh}Y$, then
$\operatorname{sd}X=\operatorname{sd}Y$.

Thus, the dimension $\operatorname{sd}$ is a shape invariant.

It follows from Corollary 6 that the shape dimension of a contractible space
equals $0$. Thus, the shape dimension of the disk equals $0$. On the other
hand, the shape dimension of the circle $S^{1}$ equals $1$. Therefore, the
shape dimension $\operatorname{sd}X$ is not a monotone function of $X$.
However, as Theorem 61 show, it is a monotone function of the shape
$\operatorname{sh}X$.

######  Theorem 62 (Holsztyński, see [322]).

Let $X$ be a compact metrizable space with $\operatorname{sd}X=n$. Then there
exists a compact metrizable space $Y$ with $\dim Y=n$ such that
$\operatorname{sh}X=\operatorname{sh}Y$.

######  Corollary 7.

Let $X$ be a compact metrizable space. Then $\operatorname{sd}X\leqslant n$ if
and only if there exists a compact metrizable space $Y$ such that
$\operatorname{sh}X\leqslant\operatorname{sh}Y$ and $\dim Y\leqslant n$.

It immediately follows from this corollary that
$\operatorname{sd}X=\operatorname{Fd}X$ for any compact metrizable space $X$,
i.e., Borsuk’s notion of fundamental dimension (see Definition 9) coincides
with Mardešić’s shape dimension (see Definition 11).

######  Remark 3.

Theorem 62 shows that the condition
$\operatorname{sh}Y\geqslant\operatorname{sh}X$ in the definition of
fundamental dimension (Definition 9) can be replaced by
$\operatorname{sh}Y=\operatorname{sh}X$.

The following theorem is due to Borsuk [75].

######  Theorem 63.

Let $X$ and $Y$ be compact metrizable spaces. Then

(13) $\operatorname{sd}(X\times
Y)\leqslant\operatorname{sd}X+\operatorname{sd}Y.$

Novák [318] showed that there exist compact connected polyhedra $P$ and $Q$
with $\dim P=m$ and $\dim Q=n$, where $m,n\geqslant 3$, such that
$\dim(P\times Q)=\max\\{m,n\\}$. It follows that inequality (13) cannot
generally be replaced by the equality

(14) $\operatorname{sd}(X\times Y)=\operatorname{sd}X+\operatorname{sd}Y.$

Spież [367], [369] proved that there exists a metrizable continuum $X$ of
shape dimension $\operatorname{sd}X=2$ such that

$\operatorname{sd}(X\times S^{n})<\operatorname{sd}X+n.$

In particular, for $n=1$, we obtain

(15) $\operatorname{sd}(X\times S^{1})=\operatorname{sd}X=2.$

Moreover, if a continuum $X$ satisfies condition (15), then

(16) $\operatorname{sd}(X\times Y)=\operatorname{sd}(S^{1}\times Y)$

for any continuum $Y$ with $\operatorname{sd}Y>0$.

Nevertheless, the following theorem of Novák is valid [323].

######  Theorem 64.

Let $X$ and $Y$ be compact metrizable spaces such that $\operatorname{sd}X\neq
2$, $\dim Y=n$, and the cohomology group $H^{n}(Y,G)$ is nontrivial for any
nontrivial Abelian group $G$. Then equality (14) holds.

Keesling [230] studied the shape dimension of topological groups; in
particular, he proved the following theorem.

######  Theorem 65.

If $X$ is a connected compact group, then $\operatorname{sd}X=\dim X$.

The shape dimension of the Stone–Čech compactification $\beta X$ was studied
in [234]. We mention the following result.

######  Theorem 66.

Let $X$ be a Lindelöf space, and let $K\subset\beta X\backslash X$ be a
continuum in the Stone–Čech remainder of $X$. Then $\operatorname{sd}K=\dim
K$.

In relation to this theorem, we give the following interesting result of
Winslow [414].

######  Theorem 67.

The Stone–Čech remainder $\beta\mathbf{R}^{3}\backslash\mathbf{R}^{3}$
contains $\mathfrak{c}=2^{\aleph_{0}}$ continua of different shapes and
$2^{\mathfrak{c}}$ continua of different topological types.

The notion of the shape dimension of shape morphisms and continuous maps
between topological spaces was introduced in [189].

We say that the _dimension $\dim(\mathbf{f}_{\beta},\varphi)$_ of a morphism
$(\mathbf{f}_{\beta},\varphi)\colon\mathbf{X}=(X_{\alpha},\mathbf{p}_{\alpha\alpha^{\prime}},A)\rightarrow\mathbf{Y}=(Y_{\beta},\mathbf{q}_{\beta\beta^{\prime}},B)$
of inverse ANR-systems is $\leqslant n$ if, for any $\beta\in B$, there exists
an $\alpha\geqslant\varphi(\beta)$ such that the map
$f_{\beta\alpha}\colon=f_{\beta}p_{\varphi(\beta)\alpha}\colon
X_{\alpha}\rightarrow Y_{\beta}$ can be homotopically factored through an ANR
$P$ of dimension $\dim P\leqslant n$, i.e., there exist maps $u\colon
X_{\alpha}\rightarrow P$ and $v\colon P\rightarrow Y_{\beta}$ such that
$f_{\beta\alpha}\simeq v\circ u$. This property is preserved under the passage
to equivalent morphisms of inverse systems. Therefore, we can define the
_dimension of morphisms in the category pro-H-CW_ and the _shape dimension_ of
a shape morphism. We say that the _shape dimension $\operatorname{sd}F$_ of a
shape morphism $F\colon X\to Y$ is $\leqslant n$ if the corresponding morphism
$\mathbf{f}\colon\mathbf{X}\to\mathbf{Y}$ in the category pro-H-CW has
dimension $\dim\mathbf{f}\leqslant n$. For a continuous map $f\colon X\to Y$,
the condition $\operatorname{sd}f\leqslant n$ means that
$\operatorname{sd}S(f)\leqslant n$.

######  Theorem 68.

A space $X$ has shape dimension $\operatorname{sd}X\leqslant n$ if and only if
the identity map $1_{X}$ has shape dimension $\operatorname{sd}1_{X}\leqslant
n$.

######  Theorem 69.

Let $F\colon X\rightarrow Y$ be a shape morphism. Then
$\operatorname{sd}F\leqslant\min(\operatorname{sd}X,\operatorname{sd}Y)$.

In particular, for a continuous map $f\colon X\rightarrow Y$, we have
$\operatorname{sd}f\leqslant\min(\operatorname{sd}X,\operatorname{sd}Y)$.

It follows from Theorem 69 that maps between shape finite-dimensional spaces
have finite shape dimension. There arises the natural question of whether the
image of a shape finite-dimensional space under a shape finite-dimensional map
can be shape infinite-dimensional. In [189], this question was answered in the
affirmative, and an example of a shape finite-dimensional surjective map
$f\colon X\to Y$ between shape infinite-dimensional spaces $X$ and $Y$ was
constructed.

######  Theorem 70.

Let $F\colon X\rightarrow Y$ be a shape equivalence of topological spaces with
$\operatorname{sd}F=n$. Then $\operatorname{sd}X=\operatorname{sd}Y=n$.

The following theorem gives an important characterization of the shape
dimension of a map $f\colon X\to Y$.

######  Theorem 71.

A map $f\colon X\rightarrow Y$ has shape dimension
$\operatorname{sd}f\leqslant n$ if and only if, given any map $h\colon
Y\rightarrow Q$ to an ANR-space $Q$, the composition $h\circ f$ can be
homotopically factored through an ANR $P$ of dimension $\dim P\leqslant n$,
i.e., there exist maps $u\colon X\rightarrow P$ and $v\colon P\rightarrow Q$
such that $h\circ f\simeq v\circ u$.

######  Theorem 72.

Let $F\colon X\rightarrow Y$ be a shape morphism of dimension
$\operatorname{sd}F\leq n$. Then, for any Abelian group $G$ and any $k>n$, the
homomorphisms
$\check{F}_{k}\colon\check{H}_{k}(X;G)\rightarrow\check{H}_{k}(Y;G)$ are
trivial.

##  9\. Embeddings in Shape Theory

The notion of a _shape embedding_ is defined by analogy with topological
embeddings and embeddings up to homotopy type. We say that a compact
metrizable space $X$ is _shape embedded_ in a space $Y$ if there exists a
compact metrizable subspace $X^{\prime}$ of $Y$ for which
$\operatorname{sh}X=\operatorname{sh}X^{\prime}$.

First, we recall a classical embedding theorem for finite-dimensional compact
metrizable spaces (see [1]).

######  Theorem 73 (Menger-Nöbeling-Pontryagin).

Any $n$-dimensional compact metrizable space $X$ is embedded in Euclidean
space $\mathbf{R}^{2n+1}$.

Much effort has been devoted to reducing the dimension of the ambient
Euclidean space $\mathbf{R}^{2n+1}$. Flores [167] proved that this cannot be
done without additional assumptions, having constructed an example of a
compact $n$-dimensional polyhedron not embedded in $\mathbf{R}^{2n}$ for each
$n$. However, any smooth $n$-manifold is smoothly embedded in
$\mathbf{R}^{2n}$ (see Whitney’s paper [413]).

Below we recall the well-known theorem of Stallings on homotopy embeddings of
finite CW-complexes, which has turned out to be very useful in the study of
shape embeddings. A simple proof of Stallings’ theorem can be found in
Dranishnikov and Repovš’ paper [121].

######  Theorem 74 (Stallings [370]).

Any finite $n$-dimensional ($n>0$) CW-complex $K$ is embedded in
$\mathbf{R}^{2n}$ up to homotopy type, i.e., there exists a polyhedron $M$
having the homotopy type of the CW-complex $K$ which is embedded in
$\mathbf{R}^{2n}$.

In this theorem the dimension of $\mathbf{R}^{2n}$ cannot be reduced (at least
for $n=2^{k}$, $k\geqslant 1$), because it is known [325] that the real
projective plane $\mathbf{P}^{n}$ cannot be embedded in $\mathbf{R}^{2n-1}$ up
to homotopy type for such $n$.

In shape theory, the following natural question arises (it was asked by Borsuk
in [10]).

######  Problem 1.

Is it true that any $n$-dimensional metrizable continuum is shape embedded in
$\mathbf{R}^{2n}$?

Clearly, the answer is “no” already for $n=1$, because solenoids, which are
one-dimensional continua, cannot be shape embedded in $\mathbf{R}^{2}$.
Indeed, solenoids are not movable, while all compact sets in the plane are
movable. Borsuk [10] proved a more general theorem, according to which a one-
dimensional metrizable continuum $X$ is shape embedded in $\mathbf{R}^{2}$ if
and only if $X$ is movable.

Problem 1 also has a negative solution in the case of metrizable continua $X$
of shape dimension $\operatorname{sd}X=n$. Duvall and Husch [124] constructed
a metrizable continuum $X$ of shape dimension $\operatorname{sd}X=n$,
$n=2^{k}$, which is not shape embedded in $\mathbf{R}^{2n}$ for each $k>1$.
Nevertheless, under certain additional constraints, Problem 1 has a positive
solution. The first results in this direction were obtained by Ivanšić [216].

######  Theorem 75.

Let $X$ be a pointed 1-movable metrizable continuum with
$\operatorname{sd}X=n\geqslant 3$. Then $X$ is shape embedded in
$\mathbf{R}^{2n}$.

This theorem is a corollary of the following more general theorem (see Husch
and Ivanšić’s paper [212]).

######  Theorem 76.

Let $X$ be a shape $r$-connected $(r+1)$-movable metrizable continuum with
$\operatorname{sd}X=n\geqslant 3$. Then $X$ is shape embedded in
$\mathbf{R}^{2n-r}$.

In the last theorem, the $(r+1)$-movability assumption can be removed at the
expense of increasing the dimension of $\mathbf{R}^{2n-r}$ by 1 (Husch and
Ivanšić [212]).

######  Theorem 77.

Let $X$ be a shape $r$-connected metrizable continuum with
$\operatorname{sd}X=n$ and $n-r\geqslant 2$. Then $X$ is shape embedded in
$\mathbf{R}^{2n-r+1}$.

Of interest is also the following problem, which was posed by Borsuk in [10].

######  Problem 2.

Let $X$ and $Y$ be compact metrizable spaces such that
$\operatorname{sh}X\leqslant\operatorname{sh}Y$ and $Y\subset\mathbf{R}^{n}$.
Is it true that $X$ is shape embedded in $\mathbf{R}^{n}$?

Husch and Ivanšić [213] obtained a positive solution of this problem under
fairly strong constraints.

######  Theorem 78.

Let $X$ and $Y$ be metrizable continua such that $Y\subset\mathbf{R}^{n}$,
$\dim Y=k$, $X$ has the shape of a finite complex, and
$\operatorname{sh}X\leqslant\operatorname{sh}Y$. If $3k<2(n-1)$ and
$k\geqslant 3$, then $X$ is shape embedded in $\mathbf{R}^{n}$.

An example constructed by Kadlof [219] shows that Problem 2 has a negative
answer. To be more precise, there exists a compact connected two-dimensional
CW-complex $X$ which is homotopy dominated by a compact connected polyhedron
$Y\subset\mathbf{R}^{3}$ but is not shape embedded in $\mathbf{R}^{3}$.

Recall that a space $X$ is said to be _$\Pi$ -similar_, where $\Pi$ is a
family of finite polyhedra, if each open finite cover of $X$ has an open
refinement whose nerve is homeomorphic to a polyhedron $P$ in $\Pi$. If
$\Pi=\\{T^{n},n\geqslant 0\\}$, where each $T^{n}=S^{1}\times\ldots\times
S^{1}$ is the $n$-torus, then $X$ is said to be _$T^{n}$ -similar_. Any
$\Pi$-similar continuum is the limit of an inverse system consisting of
elements of the family $\Pi$. Questions concerning shape embeddings of
$T^{n}$-similar continua in Euclidean spaces were studied by Keesling and
Wilson [239]. They proved the following theorems.

######  Theorem 79.

An arbitrary $n$-dimensional compact connected Abelian topological group $A$
is (topologically) embedded in $\mathbf{R}^{n+2}$.

######  Theorem 80.

Let $X$ be a $T^{n}$-similar continuum. Then $X$ is shape embedded in
$\mathbf{R}^{n+2}$.

The proofs of these theorems essentially use McCord’s theorem [295] on an
embedding of the limit of an inverse system in $\mathbf{R}^{n}$ and a theorem
of Keesling [230] (see Theorem 9) characterizing $T^{n}$-similar continua.

In [239] is was also proved that any $T^{n}$-similar continuum is embedded in
$\mathbf{R}^{2n}$. For $I^{n}$-similar continua, this was proved by Isbell
[214].

##  10\. Cell-Like Maps and Shape Theory

Cell-like maps (or, briefly, CE-maps) play a very important role in geometric
topology. They were defined by Armentrout [52] in terms of $UV$-properties and
have already been extensively studied when shape theory was discovered. Here
we discuss those aspects of cell-like maps which are of particular interest
for shape theory. Information about other questions can be found in the fairly
extensive literature on CE-maps, which includes, in particular, the survey
papers [264] by Lacher, [161] by Edwards, [175] by Geoghegan, [342] by Rubin,
and [18] by Dranishnikov and Shchepin.

######  Definition 12.

A map $f\colon X\to Y$ is said to be cell-like if all fibers $f^{-1}(y)$,
$y\in Y$, have trivial shape, i.e., $\operatorname{sh}(f^{-1}(y))=0$.

If $X$, $Y$, and $Z$ are metrizable ANRs and $f\colon X\to Y$ and $g\colon
Y\to Z$ are cell-like maps, then the composition $g\circ f\colon X\to Z$ is
cell-like as well. Therefore, the compact metrizable ANRs and cell-like maps
form a category. This is not the case for compact sets not being ANRs (see
Taylor’s paper [377]).

The following version of Smale’s theorem in shape theory was proved by Dydak
[142] (see also papers [262] by Kuperberg, [243] by Kodama, and [6] by
Bogatyi).

######  Theorem 81.

Let $X$ and $Y$ be compact metrizable sets, and let $f\colon X\to Y$ be a
cell-like map. Then the induced homomorphisms

$\text{$\mathrm{pro}$-}\pi_{n}(f)\colon\text{$\mathrm{pro}$-}\pi_{n}(X,*)\to\text{$\mathrm{pro}$-}\pi_{n}(Y,*)$

of homotopy pro-groups are isomorphisms for all $n$.

The following theorem is a homotopy characterization of cell-like maps.

######  Theorem 82.

(Lacher [264]). A map $f\colon X\to Y$ between compact ANRs $X$ and $Y$ is
cell-like if and only if, given any open set $V\subset Y$, the map
$f|f^{-1}(V)\colon f^{-1}(V)\to V$ is a homotopy equivalence.

This theorem directly implies the following assertion.

######  Theorem 83.

Any cell-like map $f\colon X\to Y$ between compact ANRs $X$ and $Y$ is a
homotopy equivalence.

However, cell-like maps between arbitrary compact spaces (not necessarily
being absolute neighborhood retracts) do not have this property, they are not