Can polynomial many-to-one reduction be done to a specific problem instance? Let's say I reduce the problem $A \in L$ to $B \in K$ , with a function $f: \Sigma^{*} \rightarrow \Gamma^{*}$ such that $w \in L \Leftrightarrow f(w) \in K$ . I know if I want to solve $A$, given some polynomial time algorithm for $B$, I just have to transform $A$ to $B$ and solve $B$. So it can be thought as: 


  The reduction must be done from arbitrary instance of $A$ to a legal instance of $B$


My question is, do I have to reduce to arbitrary instance of $B$ or some instance of $B$? I.e. reduction from TQBNF to Generalized Geography is done to some valid graph instance, but there exist many more valid instances of Generalized Geography.
The mapping does not have to be surjective (onto) nor injective (one-to-one). In fact, any problem that can be solved in polynomial time can be polynomial time many-one reduced to any problem that has at least one accepting instance and at least one rejecting instance: just solve the original problem in polynomial time, then return the accepting instance if the original problem was an accepting instance, or the rejecting instance if the original problem was a rejecting instance.

That said, the Berman–Hartmanis conjecture states that all NP-complete problems are polynomial time isomorphic, meaning that there is a bijective polynomial-time many-one reduction between them with a polynomial-time inverse. This is currently an unproven conjecture, and only refers to NP-complete problems.