If A is mapping-reducible to B and is not mapping-reducible to co-B, is A Turing-reducible to co-B?
Hint: If $A \leq_m B$ then there is a computable function $f$ such that $x \in A$ iff $f(x) \in B$ iff it is not the case that $f(x) \in \text{co-}B$. Can you use this to Turing-reduce $A$ to $\text{co-}B$?