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We are dealing with the complexity of the homeomorphism equivalence relationon some classes of metrizable compacta from the viewpoint of invariantdescriptive set theory. We prove that the homeomorphism equivalence relation ofabsolute retracts in the plane is Borel bireducible with the isomorphismequivalence relation of countable graphs. In order to stress the sharpness ofthis result, we prove that neither the homeomorphism relation of locallyconnected continua in the plane nor the homeomorphism relation of absoluteretracts in $\mathbb R^3$ is Borel reducible to the isomorphism relation ofcountable graphs. We also improve the recent results of Chang and Gao byconstructing a Borel reduction from both the homeomorphism equivalence relationof compact subsets of $\mathbb R^n$ and the ambient homeomorphism equivalencerelation of compact subsets of $[0,1]^n$ to the homeomorphism equivalencerelation of $n$-dimensional continua in $\mathbb R^{n+1}$.

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