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New Dichotomies for Borel Equivalence Relations

Published online by Cambridge University Press:  15 January 2014

Greg Hjorth  and
Alexander S. Kechris
Greg Hjorth
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90095, USAE-mail: greg@math.ucla.edu
Alexander S. Kechris
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USAE-mail: kechris@caltech.edu
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Extract

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 ≤BF 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.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 3 , Issue 3 , September 1997 , pp. 329 - 346
Copyright
Copyright © Association for Symbolic Logic 1997

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