Taut fillings of the 2-sphere
Peter Doyle, Matthew Ellison, Zili Wang
TL;DR
The paper studies minimal $L_1$-norm fillings of 2-sphere cycles arising from triangulations $\sigma$ of $S^2$ and proves that taut fillings correspond exactly to extensions of $\sigma$ to shellable triangulations of the 3-ball, i.e., to triangulations of $B^3$. A central technical advance is that $\mathrm{Zvol}$, the minimal filling size, is additive under almost disjoint unions for $n\ge 2$, and that taut fillings split under such unions; these properties propagate to the 2-sphere setting via a detailed combinatorial and topological framework. The authors connect $\mathrm{Zvol}(\sigma)$ to the minimal number of tetrahedra needed to extend $\sigma$ to a triangulation of $B^3$, showing $\mathrm{Zvol}(\sigma)=\mathrm{tetvol}(\sigma)$, and demonstrate that taut fillings of $X(\sigma)$ yield freely monotone shellable ball triangulations. While the results sit firmly in dimension 2, they illuminate the structure of fillings and their decomposition, though they do not extend to 3- or higher-dimensional spheres. The work integrates coning, almost disjoint unions, and shelling as key tools to bridge combinatorial optimization with constructive topology on triangulated spheres.
Abstract
Let $σ$ be a simplicial triangulation of the 2-sphere, $X$ the associated integral 2-cycle. A filling of $X$ is an integral 3-chain $Y$ with $\partial Y = X$; a taut filling is one with minimal $L_1$-norm. We show that any taut filling arises from an extension of $σ$ to a shellable simplicial triangulation of the 3-ball. The key to the proof is the general fact that any taut filling of an $n$-cycle splits under disjoint union, connected sum, and more generally what we call almost disjoint union, where summands are supported on sets that overlap in at most $n+1$ vertices. Despite the generality of this result, we have nothing to say about optimal fillings of spheres of dimension 3 or higher.