Higher dimensional obstructions for star reductions

Abstract: A $$-reduction between two equivalence relations is a Baire measurable reduction which preserves generic notions, i.e., preimages of meager sets are meager. We show that a $$-reduction between orbit equivalence relations induces generically an embedding between the associated Becker graphs. We introduce a notion of dimension for Polish $G$-spaces which is generically preserved under $$-reductions. For every natural number $n$ we define a free action of $S_{\infty}$ whose dimension is $n$ on every invariant Baire measurable non-meager set. We also show that the $S_{\infty}$-space which induces the equivalence relation $=^{+}$ of countable sets of reals is $\infty$-dimensional on every invariant Baire measurable non-meager set. We conclude that the orbit equivalence relations associated to all these actions are pairwise incomparable with respect to $$-reductions.