The smooth ideal (Q2795902)

This remarkable article is an important step towards the classification and the understanding of equivalence relations with Borel graphs on Polish spaces. Let \(E\) be a Borel equivalence relation on a Polish space \(X\). We say that \(E\) is \textit{smooth} if it is Borel-reducible to the equality on \(2^\mathbb{N}\). \textit{L. A. Harrington} et al. [J. Am. Math. Soc. 3, No. 4, 903--928 (1990; Zbl 0778.28011)] proved that the equivalence relation \(\mathbb{E}_0\) on \(2^\mathbb{N}\), given by \((c,d)\in \mathbb{E}_0 \Leftrightarrow \exists n \in \mathbb{N}\, \forall m \geq n\, c(m)=d(m)\), is continuously embeddable into every Borel equivalence relation on a Polish space which is not smooth. In the paper under review, the authors prove the following generalization of the result above:NEWLINENEWLINE Suppose that \(X\) is a Polish space and \(E\) and \(F\) are Borel equivalence relations on \(X\). Then exactly one of the following holds. {\parindent=0.7cm\begin{itemize}\item[(1)] There is an \(E\)-smooth Borel set \(B\subset X\) off of which \(E\) has \(\sigma\)-bounded finite index over \(E\cap F\). \item[(2)] There is a continuous embedding of \(E_0\) into the restriction of \(E\) to a partial transversal of \(F\).' NEWLINENEWLINE\end{itemize}} Here, a set \(Y\subset X\) is a \textit{partial transversal} of \(E\) if it intersects every \(E\)-class in at most one point, and a set \(Y\subset X\) is \(E\)-smooth if the restriction of \(E\) to \(Y\) is smooth. \(E\) has bounded \textit{finite index} over \(F\) if, for some \(n\in\mathbb{N}\), every class of \(E\) is the union of at most \(n\) classes of \(F\), and \(E\) has \textit{\(\sigma\)-bounded finite index} over \(F\) if \(X\) is the union of countably many Borel sets on which \(E\) has bounded finite index over \(F\).NEWLINENEWLINEIn the next parts of the article, the authors establish some rigidity results connecting each Borel equivalence relation with the corresponding \(\sigma\)-ideal \(\mathcal{I}_E\) generated by the family of \(E\)-smooth Borel sets. In particular, they show that appropriate morphisms between Borel equivalence relations are also morphisms between the corresponding smooth ideals. At the end of the paper, the authors characterize homogeneity of smooth \(\sigma\)-ideals and prove that the following conditions are equivalent:{\parindent=0.7cm\begin{itemize}\item[(1)] The equivalence relation \(E\) is essentially hyperfinite. \item[(2)] The ideal \(\mathcal{I}_E\) is reduction homogeneous. \item[(3)] The ideal \(\mathcal{I}_E\) is cohomomorphism homogeneous. NEWLINENEWLINE\end{itemize}}