stable homotopy group

稳定同伦群

Geometry & Topology · Research · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: stable stem (for spheres)

Translation evidence: Literature usage

Meaning in context

For a based CW complex X, the colimit of pi_{k+r}(Sigma^r X) under suspension homomorphisms, with r large enough to use group-valued degrees.

对带基点 CW 复形 $X$，沿悬挂同态取 $\pi_{k+r}(\Sigma^rX)$ 的余极限，$r$ 从群结构已定义的充分大值开始。

Example

For sphere stems, $\pi_0^s\cong\mathbb Z$ and $\pi_1^s\cong\mathbb Z/2$.

Usage and distinctions

For spheres the target is S^r; adding a disjoint basepoint to X changes the reduced suspension-spectrum convention.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Allen Hatcher — Algebraic Topology§4.2, Stable Homotopy Groups, printed p. 384; Corollary 4.24 · Checked 2026-10-14

Translation sources

1. Du Qingnian — Chinese Annals of Mathematics, Series A, 31(6), 2010Article title 球面稳定同伦群的一族新元素(γ)th0b41 and 中文关键词：稳定同伦群 · Checked 2026-10-14

Original encyclopedia and terminology content: CC BY-SA 4.0. Contribute or report a correction.