Universal nowhere dense subsets of locally compact manifolds

TL;DR

This paper constructs universal nowhere dense subsets, called spongy sets, in manifolds modeled on cubes, demonstrating their universality for all nowhere dense subsets via homeomorphisms.

Contribution

It introduces the concept of spongy sets in manifolds and proves their universality, utilizing a new theorem on topological equivalence of certain manifold decompositions.

Findings

Existence of spongy sets in various manifolds.

Spongy sets are universal for all nowhere dense subsets.

A new theorem on the topological equivalence of decompositions.

Abstract

In each manifold $M$ modeled on a finite or infinite dimensional cube $[0,1]^n$ we construct a closed nowhere dense subset $S\subset M$ (called a spongy set) which is a universal nowhere dense set in $M$ in the sense that for each nowhere dense subset $A\subset M$ there is a homeomorphism $h:M\to M$ such that $h(A)\subset S$. The key tool in the construction of spongy sets is a theorem on topological equivalence of certain decompositions of manifolds. A special case of this theorem says that two vanishing cellular strongly shrinkable decompositions $\mathcal A,\mathcal B$ of a Hilbert cube manifold $M$ are topologically equivalent if any two non-singleton elements $A\in\mathcal A$ and $B\in\mathcal B$ of these decompositions are ambiently homeomorphic.