The complexity of homeomorphism relations on some classes of compacta with bounded topological dimension

TL;DR

This paper investigates the complexity of homeomorphism relations on certain classes of compacta, revealing their Borel reducibility to graph isomorphism in some cases and establishing boundaries for this complexity.

Contribution

It demonstrates that homeomorphism of absolute retracts in the plane is Borel bireducible with graph isomorphism, and shows limitations for other classes, advancing the understanding of topological classification complexity.

Findings

Homeomorphism of absolute retracts in the plane is Borel bireducible with graph isomorphism.

Homeomorphism relations of locally connected continua in the plane are not Borel reducible to graph isomorphism.

Constructed Borel reductions from compact subsets in n to n-dimensional continua in n+1.

Abstract

We are dealing with the complexity of the homeomorphism equivalence relation on some classes of metrizable compacta from the viewpoint of invariant descriptive set theory. We prove that the homeomorphism equivalence relation of absolute retracts in the plane is Borel bireducible with the isomorphism equivalence relation of countable graphs. In order to stress the sharpness of this result, we prove that neither the homeomorphism relation of locally connected continua in the plane nor the homeomorphism relation of absolute retracts in $\mathbb R^3$ is Borel reducible to the isomorphism relation of countable graphs. We also improve the recent results of Chang and Gao by constructing a Borel reduction from both the homeomorphism equivalence relation of compact subsets of $\mathbb R^n$ and the ambient homeomorphism equivalence relation of compact subsets of $[0,1]^n$ to the homeomorphism equivalence relation of $n$-dimensional continua in $\mathbb R^{n+1}$.