One of the most important objects associated to a smooth manifold is its tangent bundle; for example many ideas in differential geometry are phrased in terms of various structures on the tangent bundle (connections, curvature, parallel transport, differential forms, ...).

For this it quickly becomes apparent that a study of vector bundles is very helpful, the tangent bundle being one particular kind. There are many objects to deal with this, for example the deep and powerful theory of characteristic classes. There is another very important theory that is very helpful in their description, and this is K Theory.

K theory also brings to light many surprising phenomena: it is a cohomology theory along with the more usual singular cohomology for example, but it has many very distinct properties: for example it is periodic (this is Bott's periodicity theorem, although slightly rephrased), and hence often infinite in infinitely many different dimensions. It also provides interesting ways of proving many well known theorems, for example it can be used to prove the Frobenius theorem stating that every finite dimensional division algebra over is either , or ; or to show that only the spheres () , and are parallelisable (that is, they admit as many everywhere linearly independent vector fields as their dimension; we already know from the Hairy Ball theorem that this means the sphere must be of odd dimension).

Join us at 7:30pm in MS.04 for an accessible overview of K-Theory. Then we go to the pub!