DescriptionIf you take the Moebius band and draw the central circle along it, you will notice that each point of the circle is orthogonal to an interval. The union of all the intervals covers the Moebius band. This shows that the Moebius band is a interval fiber bundle over the circle - each point of the circle has an interval living above it. This is a generalization of the notion of product - the simplest kind of interval fiber bundle over the circle would just be the product of the circle with the interval (in other words an annulus).In higher dimensions one might think of the unit tangent bundle to the 2-sphere. This is a circle fiber bundle over the 2-sphere - each point of the 2-sphere has associated to it a circle's worth of tangent directions. The non-triviality of this fiber bundle is exhibited in something called the Hairy Ball Theorem! Fiber bundles are ubiquitous in modern topology and geometry. This is project will begin with the basic definitions and the invariants of fiber bundles - known as characteristic classes, and then can develop in various directions depending on taste of those involved. Just for example, one might investigate the exotic smooth structures on the 7-sphere, one might investigate obstruction theory more deeply, one might look into the differential topology (Chern-Weil) approach to characteristic classes, or one might make a deeper study of K-theory. This project will be supervised first by Andrew Lobb in Michaelmas, and then by John Hunton in Epiphany.
PrerequisitesTopology III.Co-requisitesAlgebraic Topology IV.Resources
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email: Andrew Lobb email: John Hunton