Higher-dimensional obstructions for star reductions

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_∞ whose dimension is n on every invariant Baire measurable non-meager set. We also show that the S_∞-space which induces the equivalence relation =+ of countable sets of reals is ∞-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.