Taut fillings

Abstract

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 $M$ with $\partial M = X$; a taut filling is one with minimal $L_1$-norm. We show that any taut filling arises from an extension of $\sigma$ to a simplicial complex homeomorphic to the 3-ball. The filling is clean: it has no repeated tetrahedron, and its support complex is a clean simplicial complex. This support complex is shellable and flag: every clique in its 1-skeleton occurs as a simplex. 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. We used AI to formalize and prove in Lean the splitting theorem and the resulting cleanness, shellability, and flagness results.