On the shrinkability of decompositions of $3$-manifolds

Abstract:

An upper semicontinuous decomposition $G$ of a metric space $M$ is said to be shrinkable in case for each covering $\mathcal {U}$ of the union of the nondegenerate elements, for each $\varepsilon > 0$, and for an arbitrary homeomorphism $h$ from $M$ onto $M$, there exists a homeomorphism $f$ from $M$ onto itself such that (1) if $x \in M - ( \cup \{ U:U \in \mathcal {U}\} )$, then $f(x) = h(x)$, (2) for each $g \in G$, (a) $\operatorname {diam} f[g] < \varepsilon$ and (b) there exists $D \in \mathcal {U}$ such that $h[D] \supset h[g] \cup f[g]$. Our main result is that if $G$ is a cellular decomposition of a $3$-manifold $M$, then $M/G = M$ if and only if $G$ is shrinkable. We also define concepts of local and weak shrinkability, and we show the equivalence of the various types of shrinkability for certain cellular decompositions. Some applications of these notions are given, and extensions of theorems of Bing and Price are proved.