DescriptionOne of the biggest open questions in mathematics is the following:Given a smooth compact 4-dimensional manifold X with the same algebraic topology as the 4-sphere, does it follow that X is diffeomorphic to the 4-sphere? This is a higher-dimensional analogue of the Poincare conjecture known as the smooth Poincare conjecture (4dPC) , and is the remaining open case of the h-cobordism hypothesis (partial results towards this hypothesis have so far been worth four Fields medals). In contrast to the 3-dimensional Poincare conjecture, mathematicians are very unsure of the truth of the 4dPC and there is no very convincing idea of how to prove or disprove it. In this project we will aim to understand how to describe smooth 4-dimensional manifolds in various ways and then look at recent approaches towards the 4dPC and analogues of it.
PrerequisitesTopology III.Co-requisitesAlgebraic Topology IV.ResourcesYou can start on wikipedia here and keep following the links to get a nice overview of how things stand in this area. You could also look up the article on exotic spheres to get some idea of its importance.
|
email: Andrew Lobb