Shrinking of toroidal decomposition spaces (Q2921814)

Let \(X\) be a compact metric space. A collection \(\mathcal{D}= \{ \Delta _i\}\) of pairwise disjoint closed subsets of \(X\) such that \(\cup \mathcal{D} = X\) is called a decomposition of \(X\). A decomposition \(\mathcal{D}\) of a compact manifold \(X\) is shrinkable if the quotient map \(q: X \rightarrow X/\mathcal{D}\) can be approximated by homeomorphisms, that is, there exists a sequence of homeomorphisms which converges to \(q\) in the supremum norm. In particular this implies that \(X\) is homeomorphic to \(X/\mathcal{D}\). Approximating a map by homeomorphisms is possible if and only if the Bing shrinking criterion holds, see \textit{R. D. Edwards} [in: Proc. int. Congr. Math., Helsinki 1978, Vol. 1, 111--127 (1980; Zbl 0428.57004)].NEWLINENEWLINELet \(\mathcal{L}= L^1, L^2, L^3,...\) be a sequence of oriented links in \(S^3\), where \(L^i = L_0^i \cup L_1{}^i\cup \cdots \cup L_ {n_i }^i\) is an oriented \((n_i+1)\)-component link with a specified component \(L_0^i\) unknotted. The aim of the paper is to give general conditions to decide whether the following decomposition \(\mathcal{D}\) of \(S^3\) associated to \(\mathcal{L}\) shrinks. Each link \(L^i\) determines a link \(L_1^i\cup \cdots \cup L_{n_i}^i\) in \(S^3 - \nu L_0^i\), the complement of an open regular neighborhood of \(L_0^i\). The orientation of \(L_0^i\) and the embedding in \(S^3\) determine a canonical diffeomorphism \(S^3 - \nu L_0^i \cong S^1 \times D^2\). A closed regular neighborhood \(cl (\nu ( L^i - L_0^i))\) of \( L^i - L_0^i\) is in the same way canonically diffeomorphic to a disjoint union of solid tori. Therefore, every link \(L^i\) determines an embedding \(\hat {L}^i : \sqcup _{k=1}^{n_i} S^1 \times D^2 \hookrightarrow S^1 \times D^2\). These embeddings determine a sequence \(T_0 \supset T_1 \supset T_2 \cdots\) where \(T_0 \subset S^3\) is a single unknotted solid torus and each subsequent term \(T_s = \sqcup _{I_s} S^1 \times D^2\) is a disjoint union of solid tori, with \(I_s := \Pi_{i=1}^s n_i\) and \(I_0 := 1\). For \(s\in \mathbb {N}\), the subset \(T_s \subset T_{s-1}\) is obtained as \(T_s = \sqcup _{I_{s-1}} \sqcup_{k=1}^{n_s} S^1 \times D^2 \hookrightarrow \cup _{I_{s-1}} S^1 \times D^2 = T_{s-1}\). The decomposition \(\mathcal{D}\) is defined as the connected components of the intersection \(\cap_{s\in \mathbb N} T_s\) (as usual, only the subsets which are not singleton are specified). These decomposition spaces are called toroidal. To every link \(L\) in a solid torus a function \(D_L : \mathbb{N}\cup \{0\} \rightarrow \mathbb{N}\cup \{0\}\) is associated called the disc replicating function of \(L\). These functions provide a way to decide whether a decomposition obtained from a sequence of such links is shrinkable. The main theorem states that a decomposition \(\mathcal{D}\) of \(S^3\) obtained from a sequence of links \(\{ L^i\}_{i\in \mathbb {N}}\) is shrinkable if and only if \(lim_{p\rightarrow \infty} ( D_{L^{m+p}} \circ \cdots \circ D_{L^m}) (k) = 0\) for all \(k,m \in \mathbb{N}\). Also, the authors give new examples of decompositions for which they can determine whether they shrink. This generalizes previous results of \textit{R. B. Sher} [Fundam. Math. 61, 225--241 (1968; Zbl 0157.29902)], \textit{S. Armentrout} [Fundam. Math. 69, 15--37 (1970; Zbl 0209.54802)], \textit{F. D. Ancel} and \textit{M. P. Starbird} [Topology 28, No. 3, 291--304 (1989; Zbl 0679.57006)], and \textit{D. G. Wright} [Fundam. Math. 132, No. 2, 105--116 (1989; Zbl 0685.57008)].