Shadows of 4-manifolds with complexity zero and polyhedral collapsing

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 $X$ 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.