Based on the work of Grothendieck, in the 1960's Atiyah and Hirzebruch developed K-theory as a tool for algebraic geometry. Adapted to the topological setting K-theory can be regarded as the study of a ring generated by vector bundles. In the 1970's it was introduced as a tool in C*-algebras. C*-algebras are often considered to be "noncommutative topology", additionally they are an algebra over the complex numbers. In this setting the algebraic and topological definitions of K-theory overlap giving us a powerful tool. Essential for the Elliott classification program, for certain classes of C*-algebras, K-theory is a complete invariant. K-theory is also a natural setting for higher index theory.
We will begin by looking at different types of equivalence for projections. Then we will build a monoid where these types of equivalences are equivalent. We then use the Grothendieck construction to turn our monoid into an abelian group. This group is called the $K_0$ group of our algebra and can be thought of as the "connected components" of projections in our C*-algebra.
Next, in a similar manner, we construct the $K_1$ group using unitaries from our C*-algebra.
Once we have the $K_0$ and $K_1$ groups we will discuss Bott periodicity and the six-term exact sequence, a tool used to calculate K-theory.