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Null, recursively starlike-equivalent decompositions shrink

Jeffrey Meier, Patrick Orson, Arunima Ray

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 Poincaré conjecture.

Null, recursively starlike-equivalent decompositions shrink

Abstract

A subset of a metric space is said to be starlike-equivalent if it has a neighbourhood which is mapped homeomorphically into for some , sending to a starlike set. A subset is said to be recursively starlike-equivalent if it can be expressed as a finite nested union of closed subsets such that is starlike-equivalent for each and is a point. A decomposition of a metric space is said to be recursively starlike-equivalent, if there exists such that each element of is recursively starlike-equivalent of filtration length . We prove that any null, recursively starlike-equivalent decomposition of a compact metric space shrinks, that is, the quotient map 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 -manifolds, including the -dimensional Poincaré conjecture.

Paper Structure

This paper contains 6 sections, 8 theorems, 15 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Upper semi-continuous decompositions
  4. Shrinking
  5. Null collections and starlike sets
  6. Shrinking null, recursively starlike-equivalent decompositions

Key Result

Theorem \oldthetheorem

Let $\mathcal{D}$ be a null, recursively starlike-equivalent decomposition of length $N\geq 0$ of a compact metric space $X$, and let $U\subset X$ be an open set such that all the non-singleton elements of $\mathcal{D}$ lie in $U$. Then the quotient map $\pi\colon X\to X/\mathcal{D}$ is approximable

Figures (2)

  • Figure 1: A model red blood cell consists of a copy of $S^1\times D^3$ along with a disc glued in along $S^1\times \{*\}$, where $*$ is a point in $\partial D^3$. The fourth dimension is suppressed in the figure. A red blood cell in $D^4$ is the image of a flat, injective, continuous map of a model red blood cell.
  • Figure 2: Examples of (a) starlike, (b) starlike-equivalent and (c) recursively starlike-equivalent sets. In (b), the origin is shown in red. In (c), $E_4$ is shown in dark blue, $E_3$ is the union of $E_4$ and the space shown in green, $E_2$ is the union of $E_3$ and the space shown in purple, and $E_1$ is the union of $E_2$ and the space shown in orange. The entire space is $E$; the black dot in $E_4$ is $e$.

Theorems & Definitions (19)

  • Theorem \oldthetheorem
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1: Daverman
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.2
  • proof
  • Definition 2.5
  • Definition 2.6
  • ...and 9 more