CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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A simply connected overlap turns the van Kampen pushout into a free product

Statement

Under the hypotheses of Seifert–van Kampen identifies the fundamental group with a group pushout, if $U \cap V$ is simply connected, then the inclusion-induced homomorphisms give an isomorphism

$π_{1} (U, x_{0}) * π_{1} (V, x_{0}) ≅ π_{1} (X, x_{0}) .$

Facts & Assumptions

Given: A two-set van Kampen cover $X = U \cup V$ with simply connected overlap.

[L1]

The fundamental group of $X$ is the group pushout of the two inclusion-induced maps from $π_{1} (U \cap V, x_{0})$ (Seifert–van Kampen identifies the fundamental group with a group pushout).

[F1]

The free product with amalgamation over the trivial group is canonically isomorphic to the ordinary free product (Amalgamation over the trivial group is the ordinary free product).

[F2]

A simply connected space has a one-element fundamental group at every basepoint (Simply connected topological spaces).

Proof

technique · direct

1.1L1F2

By [F2], $π_{1} (U \cap V, x_{0})$ is the trivial group, so the two maps in the pushout of [L1] are the unique homomorphisms from the trivial group.

2.1step 1.1L1F1∎

The two maps from the trivial group are injective, so [F1] identifies their pushout with $π_{1} (U, x_{0}) * π_{1} (V, x_{0})$ . Combining this with [L1] gives the displayed isomorphism.

Depends on

Used by

Equal homology does not imply homotopy equivalence Counterexample
The fundamental group of a finite wedge of circles is free of that rank Theorem

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Allen Hatcher, Algebraic Topology, Theorem 1.20 and Example 1.21 (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Chapter 2, Section 8 (standard reference, not scraped)