Recursively squeezable sets are squeezable

TL;DR

This paper proves that null decompositions with recursively squeezable non-singleton elements are shrinkable, removing previous restrictions on filtration length, thus advancing understanding of decompositions in 4-manifold topology.

Contribution

It introduces the concepts of squeezable and squashable sets, showing their equivalence and extending shrinkability results to recursively squeezable sets without filtration length bounds.

Findings

Null decompositions with squeezable elements are shrinkable.

Recursively squashable sets are squashable, unifying the concepts.

Main theorem extends previous results by removing filtration length restrictions.

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.