homotopy equivalence

同伦等价

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

Translation evidence: Literature usage

Meaning in context

A continuous map f:X to Y admitting a continuous g:Y to X such that both composites are homotopic to the corresponding identities.

连续映射 f:X→Yf:X\to Y 存在连续的同伦逆 g:Y→Xg:Y\to X，使两个复合分别同伦于恒等映射。

Example

The inclusion S1↪R2∖{0}S^1\hookrightarrow\mathbb R^2\setminus\{0\} is a homotopy equivalence.

Usage and distinctions

A homotopy equivalence need not be a bijection or a homeomorphism.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Allen Hatcher — Algebraic TopologyChapter 0, Homotopy and Homotopy Type, printed pp. 3–4 · Checked 2026-10-14

Translation sources

  1. Bao Zhiqiang — Peking University topology syllabus第11周11月23日，主要概念：同伦等价 · Checked 2026-10-14
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