Classification and Non-classification of Homeomorphism Relations

The paper deals with the program of determining the complexity of various homeomorphism relations. The homeomorphism relation on compact Polish spaces is known to be reducible to an orbit equivalence relation of a continuous Polish group action (Kechris-Solecki). It is shown that this result extends to locally compact Polish spaces, but does not hold for spaces in which local compactness fails at only one point. In fact it fails for those subsets of $\mathbb{R}^3$ which are unions of an open set and a point. In the end a list of open problems is given in this area of research.