Non-trivial automorphisms of the Turing degrees

By an automorphism of the Turing degrees, we mean an automorphism of the Turing degrees considered as a partial order.

• Jockusch and Solovay [JS77] showed that if $f$ is an automorphism of the Turing degrees which preserves the Turing jump (i.e. for all $x$, $f(x') = f(x)'$) then $f$ is equal to the identity on the cone above $0^{(4)}$.
• Nerode and Shore [NS80] proved that every automorphism is equal to the identity on some cone. Slaman and Woodin later showed that every automorphism is equal to the identity on the cone above $0^{(2)}$ [SW05].
• Slaman and Woodin showed that there are only countably many automorphisms [SW05]. Furthermore, they showed that every automorphism can be represented by an arithmetic function—i.e. for any automorphism $f$ there is an arithmetic function $g\colon 2^\mathbb{N} \to 2^\mathbb{N}$ such that for all $x$, $[g(x)]_T = f([x]_T)$.
• Slaman and Woodin also showed that any automorphism of the Turing degrees is fully determined by what is does on any $5$-generic real. In particular, the set of degrees below $0^{(5)}$ is an automorphism base for the Turing degrees [SW05].
• Kjos-Hanssen [Kjo18] showed that no permutation of the natural numbers induces a non-trivial automorphism of the Turing degrees.

The question has been open since at least the 1970s. In the 1990s, Cooper claimed to have constructed a non-trivial automorphsim. However, his claims were not widely accepted.