fundamental group

基本群

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

Other names / variants: pi-one

Translation evidence: Literature usage

Meaning in context

The group of loops at a chosen point modulo homotopy fixing the endpoints, with multiplication given here by traversing the first loop then the second.

以指定点为基点的闭道路在定端同伦下的等价类组成的群；乘积约定先走第一条再走第二条。

Example

π1(S1,1)≅Z\pi_1(S^1,1)\cong\mathbb Z via winding number.

Usage and distinctions

Changing basepoint along different paths can change the comparison by an inner automorphism.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Allen Hatcher — Algebraic Topology§1.1, Paths and Homotopy; The Fundamental Group of the Circle · Checked 2026-10-14

Translation sources

  1. Bao Zhiqiang — Peking University topology syllabus第11周11月21日，主要概念：基本群 · Checked 2026-10-14
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