Shadows of 4-manifolds with complexity zero and polyhedral collapsing

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 X₀ that is obtained by collapsing from X. Then we prove that there exists a canonical way to equip internal regions of X₀ with gleams so that two 4-manifolds reconstructed from X₀ 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.