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A [[Compact space|compact space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202501.png" />, embedded in the [[Hilbert cube|Hilbert cube]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202502.png" />, is movable provided every neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202503.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202504.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202505.png" /> admits a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202506.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202507.png" /> such that, for any other neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202508.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m1202509.png" />, there exists a [[Homotopy|homotopy]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025010.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025012.png" />. In other words, sufficiently small neighbourhoods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025013.png" /> can be deformed arbitrarily close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025014.png" /> [[#References|[a2]]]. K. Borsuk proved that movability is a shape invariant. The solenoids (cf. [[Solenoid|Solenoid]]) are examples of non-movable continua.
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A [[Compact space|compact space]] $X$, embedded in the [[Hilbert cube|Hilbert cube]] $Q$, is movable provided every neighbourhood $U$ of $X$ in $Q$ admits a neighbourhood $U ^ { \prime }$ of $X$ such that, for any other neighbourhood $U ^ { \prime \prime } \subseteq U$ of $X$, there exists a [[Homotopy|homotopy]] $H : U ^ { \prime } \times I \rightarrow U$ with $H _ { 0 } | _ { U ^ { \prime } } = \operatorname{id}$, $H _ { 1 } ( U ^ { \prime } ) \subseteq U ^ { \prime \prime }$. In other words, sufficiently small neighbourhoods of $X$ can be deformed arbitrarily close to $X$ [[#References|[a2]]]. K. Borsuk proved that movability is a shape invariant. The solenoids (cf. [[Solenoid|Solenoid]]) are examples of non-movable continua.
  
 
The question whether movable continua are always pointed movable is still (1998) open.
 
The question whether movable continua are always pointed movable is still (1998) open.
  
For movable spaces various shape-theoretic results assume simpler form. E.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025015.png" /> is a pointed shape morphism between pointed movable metric continua (cf. also [[Pointed space|Pointed space]]; [[Continuum|Continuum]]; [[Shape theory|Shape theory]]), which induces isomorphisms of the shape groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025016.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025017.png" /> and if the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025019.png" /> are finite-dimensional, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025020.png" /> is a pointed shape equivalence. This is a consequence of the shape-theoretic Whitehead theorem (cf. also [[Homotopy type|Homotopy type]]; [[Homotopy group|Homotopy group]]) and the fact that such an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025021.png" /> induces isomorphisms of homotopy pro-groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025022.png" /> [[#References|[a6]]], [[#References|[a5]]].
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For movable spaces various shape-theoretic results assume simpler form. E.g., if $f : ( X , * ) \rightarrow ( Y , * )$ is a pointed shape morphism between pointed movable metric continua (cf. also [[Pointed space|Pointed space]]; [[Continuum|Continuum]]; [[Shape theory|Shape theory]]), which induces isomorphisms of the shape groups $f _ { \# } : \check{\pi} _ { k } ( X , * ) \rightarrow \check{\pi} _ { k } ( Y , * )$, for all $k$ and if the spaces $X$, $Y$ are finite-dimensional, then $f$ is a pointed shape equivalence. This is a consequence of the shape-theoretic Whitehead theorem (cf. also [[Homotopy type|Homotopy type]]; [[Homotopy group|Homotopy group]]) and the fact that such an $f$ induces isomorphisms of homotopy pro-groups $\pi _ { k } ( \mathcal{X} , * ) \rightarrow \pi _ { k } ( \mathcal{Y} , * )$ [[#References|[a6]]], [[#References|[a5]]].
  
Borsuk also introduced the notion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025023.png" />-movability. A compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025024.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025026.png" />-movable provided every neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025029.png" /> admits a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025032.png" /> such that, for any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025034.png" />, any compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025035.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025036.png" /> and any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025037.png" />, there exists a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025038.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025040.png" /> are homotopic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025041.png" />. Clearly, if a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025042.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025043.png" />-movable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025044.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025045.png" /> is movable. Moreover, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025047.png" />-compactum is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025048.png" />-movable [[#References|[a3]]]. The notion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025049.png" />-movability was the beginning of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025051.png" />-shape theory, which was especially developed by A.Ch. Chigogidze [[#References|[a4]]] (cf. also [[Shape theory|Shape theory]]). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025052.png" />-shape theory is an important tool in the theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025053.png" />-dimensional Menger manifolds, developed by M. Bestvina [[#References|[a1]]].
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Borsuk also introduced the notion of $n$-movability. A compactum $X \subseteq  { Q }$ is $n$-movable provided every neighbourhood $U$ of $X$ in $Q$ admits a neighbourhood $U ^ { \prime }$ of $X$ in $Q$ such that, for any neighbourhood $U ^ { \prime \prime } \subseteq U$ of $X$, any compactum $K$ of dimension $\operatorname { dim} K \leq n$ and any mapping $f : K \rightarrow U ^ { \prime }$, there exists a mapping $g : K \rightarrow U ^ { \prime \prime }$ such that $f$ and $g$ are homotopic in $U$. Clearly, if a compactum $X$ is $n$-movable and $\operatorname{dim} X \leq n$, then $X$ is movable. Moreover, every $L C ^ { n - 1 }$-compactum is $n$-movable [[#References|[a3]]]. The notion of $n$-movability was the beginning of the $n$-shape theory, which was especially developed by A.Ch. Chigogidze [[#References|[a4]]] (cf. also [[Shape theory|Shape theory]]). The $n$-shape theory is an important tool in the theory of $n$-dimensional Menger manifolds, developed by M. Bestvina [[#References|[a1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Bestvina,  "Characterizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025054.png" />-dimensional universal Menger compacta"  ''Memoirs Amer. Math. Soc.'' , '''71''' :  380  (1988)  pp. 1–110</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Borsuk,  "On movable compacta"  ''Fund. Math.'' , '''66'''  (1969)  pp. 137–146</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Borsuk,  "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025055.png" />-movability"  ''Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys.'' , '''20'''  (1972)  pp. 859–864</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.Ch. Chigogidze,  "Theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120250/m12025056.png" />-shape"  ''Uspekhi Mat. Nauk'' , '''44''' :  5  (1989)  pp. 117–140  (In Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Dydak,  "The Whitehead and the Smale theorems in shape theory"  ''Dissert. Math.'' , '''156'''  (1979)  pp. 1–55</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.E. Keesling,  "On the Whitehead theorem in shape theory"  ''Fund. Math.'' , '''92'''  (1976)  pp. 247–253</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  M. Bestvina,  "Characterizing $k$-dimensional universal Menger compacta"  ''Memoirs Amer. Math. Soc.'' , '''71''' :  380  (1988)  pp. 1–110</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  K. Borsuk,  "On movable compacta"  ''Fund. Math.'' , '''66'''  (1969)  pp. 137–146</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  K. Borsuk,  "On the $n$-movability"  ''Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys.'' , '''20'''  (1972)  pp. 859–864</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  A.Ch. Chigogidze,  "Theory of $n$-shape"  ''Uspekhi Mat. Nauk'' , '''44''' :  5  (1989)  pp. 117–140  (In Russian)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  J. Dydak,  "The Whitehead and the Smale theorems in shape theory"  ''Dissert. Math.'' , '''156'''  (1979)  pp. 1–55</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  J.E. Keesling,  "On the Whitehead theorem in shape theory"  ''Fund. Math.'' , '''92'''  (1976)  pp. 247–253</td></tr></table>

Latest revision as of 16:56, 1 July 2020

A compact space $X$, embedded in the Hilbert cube $Q$, is movable provided every neighbourhood $U$ of $X$ in $Q$ admits a neighbourhood $U ^ { \prime }$ of $X$ such that, for any other neighbourhood $U ^ { \prime \prime } \subseteq U$ of $X$, there exists a homotopy $H : U ^ { \prime } \times I \rightarrow U$ with $H _ { 0 } | _ { U ^ { \prime } } = \operatorname{id}$, $H _ { 1 } ( U ^ { \prime } ) \subseteq U ^ { \prime \prime }$. In other words, sufficiently small neighbourhoods of $X$ can be deformed arbitrarily close to $X$ [a2]. K. Borsuk proved that movability is a shape invariant. The solenoids (cf. Solenoid) are examples of non-movable continua.

The question whether movable continua are always pointed movable is still (1998) open.

For movable spaces various shape-theoretic results assume simpler form. E.g., if $f : ( X , * ) \rightarrow ( Y , * )$ is a pointed shape morphism between pointed movable metric continua (cf. also Pointed space; Continuum; Shape theory), which induces isomorphisms of the shape groups $f _ { \# } : \check{\pi} _ { k } ( X , * ) \rightarrow \check{\pi} _ { k } ( Y , * )$, for all $k$ and if the spaces $X$, $Y$ are finite-dimensional, then $f$ is a pointed shape equivalence. This is a consequence of the shape-theoretic Whitehead theorem (cf. also Homotopy type; Homotopy group) and the fact that such an $f$ induces isomorphisms of homotopy pro-groups $\pi _ { k } ( \mathcal{X} , * ) \rightarrow \pi _ { k } ( \mathcal{Y} , * )$ [a6], [a5].

Borsuk also introduced the notion of $n$-movability. A compactum $X \subseteq { Q }$ is $n$-movable provided every neighbourhood $U$ of $X$ in $Q$ admits a neighbourhood $U ^ { \prime }$ of $X$ in $Q$ such that, for any neighbourhood $U ^ { \prime \prime } \subseteq U$ of $X$, any compactum $K$ of dimension $\operatorname { dim} K \leq n$ and any mapping $f : K \rightarrow U ^ { \prime }$, there exists a mapping $g : K \rightarrow U ^ { \prime \prime }$ such that $f$ and $g$ are homotopic in $U$. Clearly, if a compactum $X$ is $n$-movable and $\operatorname{dim} X \leq n$, then $X$ is movable. Moreover, every $L C ^ { n - 1 }$-compactum is $n$-movable [a3]. The notion of $n$-movability was the beginning of the $n$-shape theory, which was especially developed by A.Ch. Chigogidze [a4] (cf. also Shape theory). The $n$-shape theory is an important tool in the theory of $n$-dimensional Menger manifolds, developed by M. Bestvina [a1].

References

[a1] M. Bestvina, "Characterizing $k$-dimensional universal Menger compacta" Memoirs Amer. Math. Soc. , 71 : 380 (1988) pp. 1–110
[a2] K. Borsuk, "On movable compacta" Fund. Math. , 66 (1969) pp. 137–146
[a3] K. Borsuk, "On the $n$-movability" Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. , 20 (1972) pp. 859–864
[a4] A.Ch. Chigogidze, "Theory of $n$-shape" Uspekhi Mat. Nauk , 44 : 5 (1989) pp. 117–140 (In Russian)
[a5] J. Dydak, "The Whitehead and the Smale theorems in shape theory" Dissert. Math. , 156 (1979) pp. 1–55
[a6] J.E. Keesling, "On the Whitehead theorem in shape theory" Fund. Math. , 92 (1976) pp. 247–253
How to Cite This Entry:
Movable space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Movable_space&oldid=50151
This article was adapted from an original article by S. Mardešić (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article