[Paper Review] Null, recursively starlike-equivalent decompositions shrink

Jeffery L. Meier, Patrick Orson+1 more|arXiv (Cornell University)|Sep 13, 2019

Geometric Analysis and Curvature Flows33 citations

TL;DR

The paper proves that null, recursively starlike-equivalent decompositions of a compact metric space shrink, i.e., the quotient map is approximable by homeomorphisms, with applications to Freedman’s disc embedding theorem.

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 $X o X/\mathcal{D}$ 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.

Motivation & Objective

• Motivate the study of decomposition spaces and shrinkability in the context of topological 4-manifolds.
• Define and analyze recursively starlike-equivalent decompositions and their filtration length.
• Establish a general shrinkability theorem for null, recursively starlike-equivalent decompositions of compact metric spaces.
• Demonstrate applications to Freedman’s disc embedding theorem and related 4-manifold results.

Proposed method

• Introduce and recall key notions from decomposition space theory, including upper semi-continuous decompositions and shrinkability.
• Prove a starlike shrinking result (Starlike shrinking lemma) that controls shrinking of E while handling null collections.
• Extend the shrinking argument to starlike-equivalent sets and to null decompositions containing such sets (Lemmas 3.1 and 3.2).
• Use induction on the filtration length N of recursively starlike-equivalent sets to prove the main shrinkability Theorem 1.1.
• Apply the main theorem to obtain a corollary for D^4 with red blood cell–like decompositions (Theorem 3.4).

Experimental results

Research questions

• RQ1 When does a null, recursively starlike-equivalent decomposition of a compact metric space shrink?
• RQ2 Can the shrinking results for starlike-equivalent and recursively starlike-equivalent decompositions be extended to yield approximability by homeomorphisms for higher filtration lengths?
• RQ3 How can shrinking results be used to support Freedman’s disc embedding theorem and related 4-manifold results?

Key findings

• A null, recursively starlike-equivalent decomposition of length N in a compact metric space yields a quotient map that is approximable by homeomorphisms (shrinks) under appropriate open set conditions.
• The shrinking is established by inductively reducing the problem to lower filtration lengths and applying starlike-equivalent shrinking lemmas.
• A key technical tool is controlling the shrink on a null collection when shrinking a starlike-equivalent set (Lemmas 3.1 and 3.2).
• The main result generalizes prior shrinkings by Denman–Starbird and Freedman, enabling applications to Freedman’s disc embedding theorem and 4-manifold topology (Theorem 3.4).
• Theorem 3.4 provides a concrete shrinkage homeomorphism for a D^4 decomposition into recursively starlike-equivalent elements of filtration length one, under boundary conditions.