  In this paper, we want to construct a one-to-one correspondence from the set
of diffeomorphism classes of spin $d$-twisted homology $\mc P^3$ to the set of
isotopy classes of the embedding from $S^3$ to $S^6$, which is a generalization
of the Montgomery-Yang correspondence. Furthermore, we will apply this
generalized correspondence to prove the existence of free involution on these
$d$-twisted homology $\mc P^3$.
