ABSTRACT
At its core, chromatic homotopy theory provides a natural approach to the computation of the stable homotopy groups of spheres π ∗ S 0 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math5_1.jpg"/> . 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 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math5_2.jpg"/> . 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 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math5_3.jpg"/> is a degreewise finitely generated abelian group with π 0 S 0 ≅ Z https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781351251624/8b1079f9-023e-42f1-82dd-5172537bcc8a/content/inline-math5_4.jpg"/> and that all higher groups are torsion.
