## Abstract

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.

Original language | English |
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Pages (from-to) | 4561-4572 |

Number of pages | 12 |

Journal | Proceedings of the American Mathematical Society |

Volume | 145 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2017 |

Externally published | Yes |

## Keywords

- 4-manifolds
- Collapse
- Complexity
- Polyhedra
- Shadows

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics