Cayley complexes embedded in three dimensions

该项目主要在代数组合学领域，EPSRC致力于继续支持该领域。但它也使用了几何拓扑学中的重要机械，可以预期也会在那里产生影响。学生将首先彻底理解几何拓扑学关于作用于球面上的群的经典结果，以及图论中的一些现代结果。然后他将尝试将这些结果从2维扩展到3维：关于作用于球体上的群的结果需要扩展到3维球体上的群作用。关于嵌入到曲面上的图的结果需要推广到二维复形到三维空间的嵌入。具体地说，该项目旨在证明以下定理：一个有限群在三维球面上通过同胚作用当且仅当它允许一个很好地嵌入在三维球面中的Cayley复形。

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.