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