Movable space

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