Higher dimensional obstructions for star reductions

TL;DR

This paper explores how $*$-reductions between orbit equivalence relations preserve a notion of dimension, revealing that certain relations are incomparable under these reductions due to their differing dimensions.

Contribution

It introduces a dimension concept for Polish $G$-spaces and demonstrates its invariance under $*$-reductions, establishing incomparability results among specific orbit equivalence relations.

Findings

Dimension is preserved under $*$-reductions.

Constructed actions of $S_{ty}$ with arbitrary finite dimension.

The $=^{+}$ relation is infinite-dimensional on non-meager sets.

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.