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Invariance in $ \boldsymbol{\mathcal{E}^*}$ and $ \boldsymbol{\mathcal{E}_\Pi}$


Author: Rebecca Weber
Journal: Trans. Amer. Math. Soc. 358 (2006), 3023-3059
MSC (2000): Primary 03D25, 03D28
DOI: https://doi.org/10.1090/S0002-9947-06-03984-5
Published electronically: March 1, 2006
MathSciNet review: 2216257
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Abstract: We define $ G$, a substructure of $ \mathcal{E}_\Pi$ (the lattice of $ \Pi^0_1$ classes), and show that a quotient structure of $ G$, $ G^\diamondsuit$, is isomorphic to $ \mathcal{E}^*$. The result builds on the $ \Delta^0_3$ isomorphism machinery, and allows us to transfer invariant classes from $ \mathcal{E}^*$ to $ \mathcal{E}_\Pi$, though not, in general, orbits. Further properties of $ G^\diamondsuit$ and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance.


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Additional Information

Rebecca Weber
Affiliation: Department of Mathematics, 255 Hurley Hall, University of Notre Dame, Notre Dame, Indiana 46556
Address at time of publication: Department of Mathematics, 6188 Bradley Hall, Dartmouth College, Hanover, New Hampshire 03755
Email: rweber@math.dartmouth.edu

DOI: https://doi.org/10.1090/S0002-9947-06-03984-5
Received by editor(s): June 16, 2004
Published electronically: March 1, 2006
Additional Notes: This work is the author’s Ph.D. research under the direction of Peter Cholak, University of Notre Dame, to whom many thanks are due. The author was partially supported by a Clare Boothe Luce graduate fellowship and National Science Foundation Grant No. 0245167.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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