Decomposition space theory

TL;DR

This paper introduces decomposition space theory, focusing on how upper semi-continuous decompositions can produce homeomorphic quotient spaces, with significant applications in 4-dimensional topology and manifold theory.

Contribution

It summarizes classical results and highlights the Bing shrinkability criterion, connecting decomposition theory with major topological theorems and manifold embedding approximations.

Findings

Upper semi-continuous decompositions can produce homeomorphic quotient spaces if shrinkable.

Every cell-like subset in a 2D manifold is cellular.

Antoine's necklace is a wild Cantor set.

Abstract

In these notes we give a brief introduction to decomposition theory and we summarize some classical and well-known results. The main question is that if a partitioning of a topological space (in other words a decomposition) is given, then what is the topology of the quotient space. The main result is that an upper semi-continuous decomposition yields a homeomorphic decomposition space if the decomposition is shrinkable (i.e. there exist self-homeomorphisms of the space which shrink the partitions into arbitrarily small sets in a controllable way). This is called Bing shrinkability criterion. It is applied in major 4-dimensional results: in the disk embedding theorem and in the proof of the 4-dimensional topological Poincare conjecture. It is extensively applied in constructing approximations of manifold embeddings in dimension at least 5, see for example Edwards's cell-like approximation theorem. We prove for example that every cell-like subset in a 2-dimensional manifold is cellular, that Antoine's necklace is a wild Cantor set, that in a complete metric space a usc decomposition is shrinkable if and only if the decomposition map is a near-homeomorphism and that every manifold has collared boundary.