Current methods in stable homotopy theory

Harvard FAS

A classical problem in homotopy theory is to understand the homotopy groups of spheres, $\pi_{n+k}(S^k)$. A classical theorem of Freudenthal says that these groups are independent of $k$ once $k>n+1$. The groups one obtains in this stable range are known as the stable homotopy groups of spheres, denoted $\π_n S$. By a theorem of Serre, these groups are finite abelian groups for $n>0$. Understanding the groups $\π_n S$ remains one of the central motivating…