Universal nowhere dense subsets of locally compact manifolds

Abstract

In each manifold $M$ modeled on a finite- or infinite-dimensional cube ${\left[0,1\right]}^n$, $n\le \omega$, 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 $h\left(A\right)\subset S$. The key tool in the construction of spongy sets is a theorem on the 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 nonsingleton elements $A\in \mathcal{A}$ and $B\in \mathcal{ℬ}$ of these decompositions are ambiently homeomorphic.