On nerves of fine coverings of acyclic spaces

TL;DR

This paper investigates the properties of nerves of fine coverings in acyclic spaces, establishing embedding results and revealing complex behaviors of coverings in various topological contexts.

Contribution

It provides new embedding theorems for acyclic and cell-like spaces and demonstrates the existence of acyclic sets with nonacyclic fine coverings.

Findings

Spaces embeddable as cellular subspaces have nerves homeomorphic to cubes

Cell-like compacta can be embedded as cellular subsets in higher-dimensional Euclidean spaces

Existence of acyclic planar sets with all nonacyclic fine coverings

Abstract

The main results of this paper are: (1) If a space $X$ can be embedded as a cellular subspace of $\mathbb{R}^n$ then $X$ admits arbitrary fine open coverings whose nerves are homeomorphic to the $n$-dimensional cube $\mathbb{D}^n$; (2) Every $n$-dimensional cell-like compactum can be embedded into $(2n+1)$-dimensional Euclidean space as a cellular subset; and (3) There exists a locally compact planar set which is acyclic with respect to \v{C}ech homology and whose fine coverings are all nonacyclic.