Low-dimensional topology and the complex of curves

Topology in dimension three is the most accessible field ofmathematics; this is because three-dimensional space is the realm ofeveryday experience. Established by Poincare in a series of inspiringarticles in the late 1800's much of the work in low-dimensionaltopology was combinatorial in nature. In 1979 Thurston revolutionizedthe field by revealing deep connections to many other, more geometric,areas of mathematics.The geometric theme in low-dimensional topology has expanded to include coarse geometry. One striking success of this theme was the work of Masur and Minsky [1999, 2000]; they introduced the idea of using coarse geometry to understand the complex of curves. The complex of curves, defined by Harvey, is a combinatorial analogue of Teichmuller space.The novelty proposed here is to use the methods of Masur and Minsky tostudy and solve combinatorial problems, some first encountered byPoincare, in low-dimensional topology. The themes running through theproposal include handlebodies and Heegaard diagrams, the recognitionproblem for the three-sphere, the structure of the mapping classgroup, and the cobordism group of automorphisms of surfaces.

The direct impact of this project will be on the UK and international community of low-dimensional, geometric, and three-dimensional topologists. The project is also highly relevant to coarse geometers and researchers in the fields of Kleinian groups, Teichmuller theory and the mapping class group. To ensure this impact, all completed work resulting from the grant will be posted on the PI's website, uploaded to the arXiv, and published in journals. The PI will give conference and seminar talks on the completed work, both in the UK and internationally. Furthermore, all software written for the project will be open source and posted on the PI's website as well as to CompuTop.org.