AI 中文总结
本文定义$\mathbb{Z}_p$-Thurston范数并研究嵌入$3$-流形的$\mathbb{Z}_p$-折叠曲面，推广Bredon-Wood结果确定$L(3n,1)$的$\mathbb{Z}_3$-Thurston范数，并证明其广义三角剖分至少需$n$个四面体。
AI 中文摘要
$\mathbb{Z}_p$-折叠曲面是一类具有基本$H_2(\mathbb{Z}/p\mathbb{Z})$类的有向折叠曲面。本文研究嵌入$3$-流形中的$\mathbb{Z}_p$-折叠曲面。这一视角自然引出了$\mathbb{Z}_p$-Thurston范数的定义，该范数推广了Jaco、Rubinstein和Tillmann的$\mathbb{Z}_2$-Thurston范数，并适配了Turaev对$\theta$的定义。我们定义了典范$\mathbb{Z}_p$-折叠曲面，将$\mathbb{Z}_p$-折叠曲面理论与单顶点$3$-流形三角剖分的研究联系起来，并将其欧拉示性数与$\mathbb{Z}_p$-Thurston范数相关联。我们推广了Bredon和Wood的经典结果，证明了嵌入$L(3n,1)$中的非分离非空$\mathbb{Z}_3$-折叠曲面的最大欧拉示性数为$2-n$（其中$n\in\mathbb{Z}^{>0}$），从而确定了这些情况下的$\mathbb{Z}_3$-Thurston范数。最后，我们通过证明透镜空间$L(3n,1)$的任何广义三角剖分必须至少有$n$个四面体，展示了该理论在广义三角剖分研究中的激励性应用。这一最终结果是与Spreer合作工作的预览，在该合作中我们使用更精细的技术确定了$L(3n,1)$的精确复杂度。
英文摘要
$\mathbb{Z}_p$-folded surfaces are a class of oriented folded surfaces with a fundamental $H_2(\mathbb{Z}/p\mathbb{Z})$ class. In this paper we study the $\mathbb{Z}_p$-folded surfaces that embed in $3$-manifolds. This perspective naturally leads to the definition of the $\mathbb{Z}_p$-Thurston norm, which generalises the $\mathbb{Z}_2$-Thurston norm of Jaco, Rubinstein and Tillmann and adapts Turaev's definition of $θ$. We define canonical $\mathbb{Z}_p$-folded surfaces, which connect the theory of $\mathbb{Z}_p$-folded surfaces to the study of one-vertex $3$-manifold triangulations, and we relate their Euler characteristic to the $\mathbb{Z}_p$-Thurston norm. We generalise a classical result of Bredon and Wood by proving that the maximum Euler characteristic of a non-separating non-empty $\mathbb{Z}_3$-folded surface embedded in $L(3n,1)$ is $2-n$ for $n\in\mathbb{Z}^{>0}$, determining the $\mathbb{Z}_3$-Thurston norm in these cases. We conclude by demonstrating the motivating application of this theory to the study of generalised triangulations by proving that any generalised triangulation of the lens space $L(3n,1)$ must have at least $n$ tetrahedra. This final result is a preview of joint work with Spreer in which we determine the exact complexity of $L(3n,1)$ using more sophisticated techniques.
Comments102 pages, 30 figures
