  The difference $[L_1]-[L_2]$ of a pair of skew lines on a cubic threefold
defines a vanishing cycle on the cubic surface as the hyperplane section
spanned by the two lines. By deforming the hyperplane, the flat translation of
such vanishing cycle forms a 72-to-1 covering space $T_v$ of a Zariski open
subspace of $(\mathbb P^4)^*$. Based on a lemma of Stein on the
compactification of finite analytic covers, we found a compactification of
$T_v$ to which the topological Abel-Jacobi map extends. Moreover, the boundary
points of the compactification can be interpreted in terms of local monodromy
and the singularities on cubic surfaces. We prove the associated map on
fundamental groups of topological Abel-Jacobi map is surjective.
