Cayley complexes embedded in three dimensions

该项目主要是在代数组合学领域，EPSRC致力于继续支持。但它也使用了几何拓扑的重要机制，并且可以预期也会产生影响。学生将首先深入了解几何拓扑中关于作用于球体的群的经典结果，以及图论中的一些现代结果。然后，他将试图扩大这些结果从2维到3维：结果有关团体的作用领域需要扩大到集团的行动3维领域。有关图嵌入曲面的结果需要推广到二维复形嵌入三维空间的情形。具体地说，该项目旨在证明以下定理：一个有限群通过同胚作用在三维球面上当且仅当它允许一个凯莱复形嵌入三维球面。

This project is mainly in the area of algebraic Combinatorics, which the EPSRC is committed to continue supporting. But it also uses important machinery from Geometric Topology, and can be expected to have an impact there too.The student will first gain a thorough understanding of classical results of Geometric Topology about groups acting on spheres, as well as some modern results in graph theory. He will then try to extend these results from 2-dimensions to 3-dimensions: results about groups acting on the sphere need to be extended to group actions on the 3-dimensional sphere. Results about graphs embedded on a surface need to be extended into embeddings of 2-dimensional complexes into 3-dimensional space. Concretely, the project aims at proving the following theorem:A finite group acts by homeomorphisms on the 3-dimensional sphere if and only if it admits a Cayley complex embedded well in the 3-dimensional sphere.