Topological K-theory and the Adams operations

Abstract: 

Often given a topological space one can assign to it an algebraic invariant which gives us information on the space. One well known example is the fundamental group. In this talk I will present another algebraic invariant K(X) for a compact Hausdorff space. This is a ring which comes with a class of ring endomorphisms called Adams operations. We will look at how these Adams operations give us more structure on the ring K(X), by showing that there is no compact Hausdorff space such that K(X)=Z[i]. We will also see how these Adams operations can be applied to show the nonexistence of division algebras other than those of dimension 1,2,4,8, which are R,C, the quaternions and the octonions respectively.