The Stable Homotopy Groups of Spheres

A fundamental problem in algebraic topology is to understand the higher homotopy groups of spheres, $\pi_n(S^k)$. Freudenthal showed that, for large $k$, these groups depend only on the difference $n-k$. The corresponding group for $m=n-k$, denoted $\pi_m^S$, is finite for $m>0$, and the sequence of finite abelian groups $\pi_m^S$ exhibits fascinating patterns. We shall survey how the study of these groups is related not only to algebraic topology, but also to differential topology and algebra, and discuss some of the modern methods developed to understand them.

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