I want to find out whether, assuming a language $L_1$ being mapping reducible (i.e., $L_1$ maps to $L_2$ and the complement of $L_1$ maps to the complement of $L_2$) to a language $L_2$ and $L_2$ being recursively enumerable, $L_1$ is recursive or not.
I tried creating a recursive Turing machine for $L_1$ by using the recursively enumerable machine of $L_2$, but if the input belongs to the complement of $L_1$, its mapping will also be in the complement of $L_2$ and we cannot say anything about it.
So I tried proving by contradiction instead: Assuming $L_1$ is recursive, show that our assumption that $L_2$ is recursively enumerable is wrong. However, this will require an inverse mapping. Alternatively, we could also try assuming $L_1$ is recursive and show that such a mapping cannot exist, but I can't think of any approach to it.
Can someone help me?
Mapping reducibility of two languages $L_1$ and $L_2$ is defined as a function which, when given a string in $L_1$, gives as output a string in $L_2$ and, when given as input a string in $L_1$ complement, gives as output a string in $L_2$ complement.