A contamination carrying criterion for branched surfaces

Contact Structures, sigma-confoliations, and contaminations in 3-manifolds

We propose in this paper a method for studying contact structures in 3-manifolds by means of branched surfaces. We explain what it means for a contact structure to be carried by a branched surface embedded in a 3-manifold. To make the transition from contact structures to branched surfaces, we first define auxiliary objects called sigma-confoliations and pure contaminations, both generalizing contact structures. We study various deformations of these objects and show that the sigma-confoliations and pure contaminations obtained by suitably modifying a contact structure remember the contact str…

端期映射、分裂序列与分支表面

Endperiodic maps, splitting sequences, and branched surfaces

本文通过证明端期映射具有周期性分裂序列，加强了Gabai和Mosher的定理，展示了veering分支表面在缝合 manifold 中的应用及理论扩展。

Branched covers of the sphere and the prime-degree conjecture

To a branched cover between closed, connected and orientable surfaces one associates a "branch datum", which consists of the two surfaces, the total degree d, and the partitions of d given by the collections of local degrees over the branching points. This datum must satisfy the Riemann-Hurwitz formula. A "candidate surface cover" is an abstract branch datum, a priori not coming from a branched cover, but satisfying the Riemann-Hurwitz formula. The old Hurwitz problem asks which candidate surface covers are realizable by branched covers. It is now known that all candidate covers are realizable…

叶状结构欧拉类的一个公式

A formula for the Euler class of foliations

本文通过分支表面的对偶图定义了一个simplicial 1-环，用于表示叶状结构欧拉类的庞加莱对偶，同时推广了Lackenby和Dunfield之前的结论，并观察到某些分支表面满足组合横截面定理。