  Our purpose is to classify acyclic 4-manifolds having shadow complexity zero.
In this paper, we focus on simple polyhedra and discuss this problem
combinatorially. We consider a shadowed polyhedron $X$ and a simple polyhedron
$X_0$ that is obtained by collapsing from $X$. Then we prove that there exists
a canonical way to equip internal regions of $X_0$ with gleams so that two
4-manifolds reconstructed from $X_0$ and $X$ are diffeomorphic. We also show
that any acyclic simple polyhedron whose singular set is a union of circles can
collapse onto a disk. As a consequence of these results, we prove that any
acyclic 4-manifold having shadow complexity zero with boundary is diffeomorphic
to a 4-ball.
