Null, recursively starlike-equivalent decompositions shrink

Abstract

A subset $E$ of a metric space $X$ is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into $\mathbb{R}^n$ for some $n$, sending $E$ to a starlike set. A subset $E\subset X$ is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed subsets $\{E_i\}_{i=0}^{N+1}$ such that $E_{i}/E_{i+1}\subset X/E_{i+1}$ is starlike-equivalent for each $i$ and $E_{N+1}$ is a point. A decomposition $\mathcal{D}$ of a metric space $X$ is said to be recursively starlike-equivalent, if there exists $N\geq 0$ such that each element of $\mathcal{D}$ is recursively starlike-equivalent of filtration length $N$. We prove that any null, recursively starlike-equivalent decomposition $\mathcal{D}$ of a compact metric space $X$ shrinks, that is, the quotient map $X\to X/\mathcal{D}$ is the limit of a sequence of homeomorphisms. This is a strong generalisation of results of Denman-Starbird and Freedman and is applicable to the proof of Freedman's celebrated disc embedding theorem. The latter leads to a multitude of foundational results for topological $4$-manifolds, including the $4$-dimensional Poincar\'{e} conjecture.