Shadows of acyclic $4$–manifolds with sphere boundary

Abstract

In terms of Turaev’s shadows, we provide a sufficient condition for a compact, smooth, acyclic $4$–manifold with boundary the $3$–sphere to be diffeomorphic to the standard $4$–ball. As a consequence, we prove that if a compact, smooth, acyclic $4$–manifold with boundary the $3$–sphere has shadow-complexity at most $2$, then it is diffeomorphic to the standard $4$–ball.