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

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 $\mathbb R^{n+1}$.