An Overview of Localization in Stable Homotopy Theory

I start by telling a bit about stable homotopy theory, to give a flavor of the world we’ll be working in today. One of the key ideas in algebraic topology is to use algebraic invariants to distinguish topological spaces. This leads to H∗, H ∗, and π∗. The first two are relatively easy to compute, but they are coarse invariants: they don’t capture enough of the information of the space. The last is really too difficult to compute in practice. So Stable Homotopy Theory aims to compute something a bit simpler, namely πs ∗ the stable homotopy groups. To define this we use the key theorem of stable homotopy theory.