Author:
Meier Jeffrey,Orson Patrick,Ray Arunima
Abstract
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 four-dimensional Poincaré conjecture.
Publisher
Cambridge University Press (CUP)