Invariance in ℰ* and ℰ_{Π}

Weber, Rebecca

doi:10.1090/S0002-9947-06-03984-5

Invariance in $\boldsymbol {\mathcal {E}^*}$ and $\boldsymbol {\mathcal {E}_\Pi }$
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by Rebecca Weber PDF

Trans. Amer. Math. Soc. 358 (2006), 3023-3059 Request permission

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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