\(\mathbb{Z}_3\)-folded surfaces and the complexity of lens spaces

Will Hobkirk
University of Sydney

Abstract

In this talk, we introduce $\mathbb{Z}_3$-folded surfaces -- geometric representatives of classes in $H_2(M;\mathbb{Z}_3)$ for a $3$-manifold $M$. We define the $\mathbb{Z}_3$-Thurston “norm”, which measures the minimum complexity of these representatives. This function generalises the $\mathbb{Z}_2$-Thurston norm and provides information about triangulations of $M$. We determine the $\mathbb{Z}_3$-Thurston norm for the family of lens spaces $L(3n,1)$, uncovering some interesting properties of $\mathbb{Z}_3$-folded surfaces along the way. As a corollary, we show that any triangulation of $L(3n,1)$ must have at least $n$ tetrahedra. A refinement of this approach determines the exact minimum number of tetrahedra required to triangulate these spaces, in forthcoming joint work with Jonathan Spreer.