Movable space
A compact space 
, embedded in the Hilbert cube 
, is movable provided every neighbourhood 
of 
in 
admits a neighbourhood 
of 
such that, for any other neighbourhood 
of 
, there exists a homotopy 
with 
, 
. In other words, sufficiently small neighbourhoods of 
can be deformed arbitrarily close to 
[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 
is a pointed shape morphism between pointed movable metric continua (cf. also Pointed space; Continuum; Shape theory), which induces isomorphisms of the shape groups 
, for all 
and if the spaces 
, 
are finite-dimensional, then 
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 
induces isomorphisms of homotopy pro-groups 
[a6], [a5].
Borsuk also introduced the notion of 
-movability. A compactum 
is 
-movable provided every neighbourhood 
of 
in 
admits a neighbourhood 
of 
in 
such that, for any neighbourhood 
of 
, any compactum 
of dimension 
and any mapping 
, there exists a mapping 
such that 
and 
are homotopic in 
. Clearly, if a compactum 
is 
-movable and 
, then 
is movable. Moreover, every 
-compactum is 
-movable [a3]. The notion of 
-movability was the beginning of the 
-shape theory, which was especially developed by A.Ch. Chigogidze [a4] (cf. also Shape theory). The 
-shape theory is an important tool in the theory of 
-dimensional Menger manifolds, developed by M. Bestvina [a1].
References
| [a1] | M. Bestvina, "Characterizing  -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  -movability" Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. , 20 (1972) pp. 859–864 |
| [a4] | A.Ch. Chigogidze, "Theory of  -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 |
Movable space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Movable_space&oldid=50151