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_∞ 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.
Additional Information
© 2021 IMPAN. Published online: 20 April 2021. This work greatly benefited from a visit of Alex Kruckman at the California Institute of Technology in the Spring 2018. The authors gratefully acknowledge the hospitality and the financial support of the Institute.Attached Files
Submitted - 1809.02239.pdf
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- Eprint ID
- 111292
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- CaltechAUTHORS:20211008-183537226
- Caltech
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2021-10-08Created from EPrint's datestamp field
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2021-10-08Created from EPrint's last_modified field