higher homotopy group

同伦群

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

Other names / variants: pi-n

Translation evidence: Literature usage

Meaning in context

For n at least two, the abelian group of based maps from S^n to a based space modulo based homotopy, with addition from pinching the sphere into a wedge of two spheres.

当 n≥2n\geq2 时，保基点映射 Sn→XS^n\to X 的保基点同伦类形成的阿贝尔群；加法由球面的捏合映射给出。

Example

π2(S2)≅Z\pi_2(S^2)\cong\mathbb Z.

Usage and distinctions

The Chinese source uses the general name 同伦群; this record restricts to n≥2. Such groups can be nonzero above the dimension of the target.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Allen Hatcher — Algebraic Topology§4.1, Definitions and Basic Constructions, printed pp. 340–342 · Checked 2026-10-14

Translation sources

  1. UCAS — 代数拓扑 course syllabus第一章第5节 同伦群的定义 · Checked 2026-10-14
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