[论文解读] Recursively squeezable sets are squeezable

Fredric D. Ancel|arXiv (Cornell University)|Sep 6, 2020

Logic, Reasoning, and Knowledge参考文献 4被引用 33

一句话总结

论文显示，在紧凑度量空间子集上的 squeezability 概念对齐：squeezable 当且仅当 squashable，并且递归 squeezable 元素意味着 shrinkable 的零分解，而不需要 uniform upper bound 假设。

ABSTRACT

In work by Freedman [F2] and Freedman-Quinn [FQ] on the topology of 4-manifolds, null decompositions whose non-singleton elements are, in the terminology of [MOR], recursively starlike-equivalent sets of filtration length 1 arise and are shown to be shrinkable. The main result of [MOR] is a general theorem covering these types of decompositions. It establishes the shrinkability of null decompositions whose non-singleton elements are recursively starlike-equivalent sets whose filtration lengths have a uniform finite upper bound. That result is the inspiration for this article. Here it is shown that the hypothesis of a uniform finite upper bound on filtration lengths is unnecessary. In outline: notions of squeezable subsets and squashable subsets of a compact metric space are defined. It is observed that starlike-equivalent sets are squeezable, and that any null decomposition of a compact metric space whose non-singleton elements are squeezable is shrinkable. It is also proved that a set is squeezable if and only if it is squashable, and that every recursively squashable set is squashable. It follows that any null decomposition of a compact metric space whose non-singleton elements are recursively squeezable is shrinkable. The latter theorem has as a corollary the main result of [MOR] with the hypothesis of a uniform finite upper bound on filtration lengths removed.

研究动机与目标

• Define squeezable and squashable subsets of a compact metric space.
• Show starlike-equivalent sets are squeezable.
• Prove that a null decomposition with squeezable non-singleton elements is shrinkable.
• Establish equivalence between squeezable and squashable sets.
• Deduce that recursively squeezable elements are squashable and apply to remove uniform bound assumptions in existing theorems.

提出的方法

• Introduce and compare notions of squeezable and squashable subsets.
• Prove that starlike-equivalent sets are squeezable.
• Prove: a null decomposition with non-singleton elements that are squeezable is shrinkable.
• Prove: a set is squeezable if and only if it is squashable.
• Prove: every recursively squashable set is squashable.
• Apply these results to null decompositions to recover MOR-type conclusions without a finite upper bound on filtration lengths.

实验结果

研究问题

• RQ1 What is the precise relationship between squeezable and squashable sets?
• RQ2 Do the known shrinkability results extend when the uniform finite upper bound on filtration lengths is removed?
• RQ3 Are recursively squeezable sets necessarily squashable, and what implications does this have for null decompositions?

主要发现

• Squeezable and squashable sets are equivalent notions.
• Starlike-equivalent sets are squeezable.
• A null decomposition of a compact metric space with non-singleton elements that are squeezable is shrinkable.
• A set is squeezable if and only if it is squashable.
• Every recursively squashable set is squashable.
• Removing the uniform finite upper bound on filtration lengths preserves shrinkability conclusions for the corresponding null decompositions.