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Higher-dimensional obstructions for star reductions

Volume 255 / 2021

Alex Kruckman, Aristotelis Panagiotopoulos Fundamenta Mathematicae 255 (2021), 209-230 MSC: Primary 03E15; Secondary 54H05. DOI: 10.4064/fm35-2-2021 Published online: 20 April 2021

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.

Authors

  • Alex KruckmanDepartment of Mathematics and Computer Science
    Wesleyan University
    Science Tower 655, 265 Church Street
    Middletown, CT 06459, U.S.A.
    e-mail
  • Aristotelis PanagiotopoulosDepartment of Mathematics
    Caltech
    1200 E California Blvd
    Pasadena, CA 91125, U.S.A.
    e-mail

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