Homeomorphism relation of locally connected continua is complete

Every Borel action of a Polish group G on a Polish space X gives rise to a relation ~ given by x ~ y if and only if x and y are in the same orbit of the action. Such relations are called orbit equivalence relations. Let ~ and ~' be orbit equivalence relations on Polish spaces X and Y, respectively. We say that ~ is Borel reducible to ~' if there exists a Borel map f : X -> Y such that for every elements x, y of X we have x ~ y if and only if f(x) ~' f(y). We say that an orbit equivalence relation ~ is complete if every orbit equivalence relation is Borel reducible to ~. In this talk I will sketch a proof that the homeomorphism relation of locally connected continua is complete. This answers a question posed by Chang and Gao.