Show that the Halting problem is reducible to its complement
Suppose $HALT$ was many one reducible to $\overline{HALT}$. Let the mapping function be $f$. We construct a machine $D$. It takes input $\langle M,w \rangle$ obtains the string $f(\langle M,w \rangle)=\langle M^{'},w^{'} \rangle$. $D$ simulates $ M$ on $w$ and  $M^{'}$ on $w^{'}$. If the first simulation ends  $D$ accepts if second simulation ends/halts it rejects. $D$ is a decider for $HALT$ which is not possible.