Abstract

At its core, chromatic homotopy theory provides a natural approach to the computation of the stable homotopy groups of spheres π ∗ S 0 . Historically, the first few of these groups were computed geometrically through the classification of stably framed manifolds, using the Pontryagin–Thom isomorphism π ∗ S 0 ≅ Ω ∗ fr . However, beginning with the work of Serre, it soon turned out that algebraic tools were more effective, both for the computation of specific low-degree values as well as for establishing structural results. In particular, Serre proved that π ∗ S 0 is a degreewise finitely generated abelian group with π 0 S 0 ≅ Z and that all higher groups are torsion.