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Shadows of 4-manifolds with complexity zero and polyhedral collapsing


Author: Hironobu Naoe
Journal: Proc. Amer. Math. Soc. 145 (2017), 4561-4572
MSC (2010): Primary 57N13, 57M20; Secondary 57R65
DOI: https://doi.org/10.1090/proc/13595
Published electronically: June 22, 2017
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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.


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Additional Information

Hironobu Naoe
Affiliation: Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan
Email: hironobu.naoe.p5@dc.tohoku.ac.jp

DOI: https://doi.org/10.1090/proc/13595
Keywords: 4-manifolds, shadows, polyhedra, collapse, complexity.
Received by editor(s): May 30, 2016
Received by editor(s) in revised form: September 27, 2016, and November 16, 2016
Published electronically: June 22, 2017
Communicated by: Ken Ono
Article copyright: © Copyright 2017 American Mathematical Society