van Kampen theorem

Van Kampen 定理

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

Other names / variants: Seifert–van Kampen theorem

Translation evidence: Literature usage

Meaning in context

For a space covered by two path connected open sets with nonempty path connected intersection and a basepoint in that intersection, its fundamental group is the pushout of the two fundamental groups over that of the intersection.

空间由两个道路连通开集覆盖，交集非空且道路连通，基点位于交集中时，总空间基本群是两开集基本群沿交集基本群的推出。

Example

For two circles meeting at a point, suitable open neighborhoods give $\pi_1\cong\mathbb Z*\mathbb Z$.

Usage and distinctions

A disconnected intersection needs a groupoid version or additional bookkeeping; the stated group formulation cannot simply ignore it.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Allen Hatcher — Algebraic Topology§1.2, The van Kampen Theorem, printed pp. 43–45 · Checked 2026-10-14

Translation sources

1. Bao Zhiqiang — Peking University topology syllabus第12周11月28日，主要概念：Van Kampen 定理 · Checked 2026-10-14

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