Taut fillings of the 2-sphere

Peter Doyle, Matthew Ellison, Zili Wang

Published 2025-05-14Version 1

Let $\sigma$ 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 $\sigma$ 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.