nLab
stem

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Stable Homotopy theory

Contents

Definition

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

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

References

The “stem”-terminology is due to:

(which otherwise introduced the Freudenthal suspension theorem).

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

Last revised on February 26, 2021 at 11:58:58. See the history of this page for a list of all contributions to it.