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

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