nLab stem

Definition

For $k \in \mathbb{Z}$, the $k$-stem of the homotopy groups of spheres is the collection of homotopy groups of the form $\pi_{n+k}(S^n)$ for all $n \in \mathbb{N}$, together with the suspension maps between them.

For $n \gt k + 1$ these groups stabilize (“stable stems”) and yield the stable homotopy groups of spheres.

Related concepts

• Freudenthal suspension theorem

References

The “stem”-terminology is due to:

• Hans Freudenthal, first paragraph of: Über die Klassen der Sphärenabbildungen I. Große Dimensionen, Compositio Mathematica, Tome 5 (1938) , pp. 299-314 (numdam:CM_1938__5__299_0)

(which otherwise introduced the Freudenthal suspension theorem).

For more see the references at homotopy groups of spheres, such as

• Mark Mahowald, Doug Ravenel, Towards a Global Understanding of the Homotopy Groups of Spheres (pdf)