We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.
§1. Introduction. For X a Polish (i.e., separable, completely metrizable) space and E a Borel equivalence relation on X, a (complete) classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c(x) = c(y). To be of any value we would expect I and c to be “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.
The following partial (pre-)ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbols
if there is a Borel map f : X → Y with xEy ⇔ f(x)Ff(y). Then if is an embedding of X/E into Y/F, which is “Borel” (in the sense that it has a Borel lifting).
Intuitively, E ≤ B F might be interpreted in any one of the following ways:
(i) The classi.cation problem for E is simpler than (or can be reduced to) that of F: any invariants for F work as well for E (after composing by an f as above).
(ii) One can classify E by using as invariants F-equivalence classes.
(iii) The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F.