Shadows of 4-manifolds with complexity zero and polyhedral collapsing
Hironobu Naoe

TL;DR
This paper classifies acyclic 4-manifolds with shadow complexity zero by using combinatorial methods on simple polyhedra, showing they are diffeomorphic to 4-balls when with boundary.
Contribution
It introduces a canonical method to assign gleams to internal regions of collapsed polyhedra, proving all such 4-manifolds with boundary are 4-balls.
Findings
Acyclic simple polyhedra with circle singular sets collapse onto disks.
Any acyclic 4-manifold with shadow complexity zero and boundary is a 4-ball.
Existence of a canonical gleam assignment ensuring diffeomorphic reconstructed 4-manifolds.
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 and a simple polyhedron that is obtained by collapsing from . Then we prove that there exists a canonical way to equip internal regions of with gleams so that two 4-manifolds reconstructed from and 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Computational Geometry and Mesh Generation · Advanced Combinatorial Mathematics
