The collection of complex vector bundles on a compact Hausdorff space $X$ forms a commutative monoid under fibrewise direct sum. Applying Grothendieck's group completion to this monoid yields the abelian group $K(X)$. In fact, if we apply the completion procedure to the monoid with the multiplicative operation of fibrewise tensor product, we have that $K$ is a contravariant functor from compact Hausdorff spaces to rings, with continuous maps inducing pullbacks of vector bundles. This is eerily reminiscent of singular cohomology.

The Bott Periodicity Theorem states the beautiful result that the (reduced) $K$ group of $S^n$ is $\mathbb{Z}$ for $n$ even and $0$ for $n$ odd. In this talk I will outline an elementary proof of this theorem by Atiyah and Bott published in 1964. I also hope to give an interesting qualitative characterization of the tangent bundle of $S^2$.