#### Volume 20, issue 7 (2020)

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Shadows of acyclic $4$–manifolds with sphere boundary

### Yuya Koda and Hironobu Naoe

Algebraic & Geometric Topology 20 (2020) 3707–3731
##### 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.

##### Keywords
$4$–manifold, shadow, differentiable structure, handlebody, polyhedron
##### Mathematical Subject Classification 2010
Primary: 57N13
Secondary: 57M20, 57R55, 57R65