Complex K-theory KU

Summary

Complex K-theory assigns to a compact space $X$ the group $K^0(X)$ built from complex vector bundles over $X$. It also assigns groups $K^{-n}(X)$ obtained from suspensions. These groups form a generalized cohomology theory, denoted $KU$, and by Bott periodicity they repeat with period $2$. It is the first and most used example of topological K-theory. It has applications to index theory and to the Hopf invariant one problem.