# Shadows of acyclic 4-manifolds with sphere boundary

## TL;DR

This paper establishes conditions under which certain acyclic 4-manifolds with sphere boundary are diffeomorphic to the standard 4-ball, using Turaev's shadows and shadow-complexity constraints.

## Contribution

It provides a new sufficient condition based on shadow complexity for identifying when such 4-manifolds are standard 4-balls.

## Key findings

- Acyclic 4-manifolds with boundary S^3 and shadow-complexity ≤ 2 are diffeomorphic to the 4-ball.
- A sufficient condition for a 4-manifold to be a 4-ball is given in terms of Turaev's shadows.
- The paper links shadow complexity with the topological classification of 4-manifolds.

## 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.