Shadows of 4-manifolds with complexity zero and polyhedral collapsing

Hironobu Naoe

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)4561-4572
Number of pages12
JournalProceedings of the American Mathematical Society
Issue number10
Publication statusPublished - 2017
Externally publishedYes


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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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