Bidouble covers π : S → Q := P¹×P¹ of the quadric and their natural deformations are parametrized by connected families _a,b,c,d depending on four positive integers a,b,c,d. We shall call these surfaces abcd-surfaces. In the special case where b = d we call them abc-surfaces.

Such a Galois covering π admits a small perturbation yielding a general 4-tuple covering of Q with branch curve Δ, and two natural Lefschetz fibrations obtained from a small perturbation of the composition p_i°π, i = 1,2, p_i being the i-th projection of Q onto P¹.

The first and third author showed that the respective mapping class group factorizations corresponding to the first Lefschetz fibration are equivalent for two abc-surfaces with the same values of a + c, b, a result which implies the diffeomorphism of two such surfaces.

We reporton a more general resultof the threeauthors implying thatthe firstbraid monodromy factorization corresponding to Δ determines the three integers a,b,c in the case of abc-surfaces. We provide in this article a new proof of the non equivalence of two such factorizations for different values of a,b,c.

We finally show that, under certain conditions, although the first Lefschetz fibrations are equivalent for two abc-surfaces with the same values of a + c, b, the second Lefschetz fibrations need not be equivalent.

These results rally around the question whether abc-surfaces with fixed values of a + c, b, although diffeomorphic but not deformation equivalent, might be not canonically symplectomorphic.