Null, recursively starlike-equivalent decompositions shrink

TL;DR

This paper proves that null, recursively starlike-equivalent decompositions of compact metric spaces shrink, generalizing previous results and supporting foundational theorems in topological 4-manifold theory, including the proof of the 4D Poincaré conjecture.

Contribution

It establishes that such decompositions always shrink, extending earlier work and aiding in the proof of Freedman's disc embedding theorem.

Findings

Null, recursively starlike-equivalent decompositions shrink.

Supports proof of Freedman's disc embedding theorem.

Facilitates foundational results in topological 4-manifolds.

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.