This is a question about two reducibility notions in computability theory. I suspect the answer is a fairly simple construction, and I'm just not seeing it.
For sets $X, Y\subseteq\omega$, we say $X$ is Turing reducible to $Y$ if there is a Turing machine $\Phi$ which, when given $Y$ as an oracle, yields (the characteristic function of) $X$, and we write $X\le_T Y$. We say $X$ is many-one reducible to $Y$ if there is some computable function $f$ such that $f(x)\in Y\iff x\in X$, and we write $X\le_mY$. Clearly many-one reducibility strictly implies Turing reducibility. A bit less clearly, the many-one degrees within a single Turing degree form a surprisingly large and interesting structure (see e.g. http://www.jstor.org/stable/2695042?seq=1, which among other things taught me that "objective" and "subjective" are technical terms in computability theory).
My question is the following:
Suppose $X\le_T Y$. Is there a set $A\equiv_T X$ such that $A\le_mY$?
Note that it is important that $Y$ is kept fixed, here: we of course always have $X\le_m Y\oplus X$, so allowing $Y$ to vary would make the question trivial. This observation amounts to saying that the relation $\le_m\subseteq deg(X)\times deg(Y)$ is total; and the question I am asking is whether it is surjective as well.