CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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$π_{1} (S^{1} \lor S^{1})$ is the free group on two generators

Statement

For the wedge of two quotient circles, the fundamental group at the wedge point is the free group $F (a, b)$ on the two standard once-around loop classes $a$ and $b$ .

Facts & Assumptions

Given: The wedge $W_{2} = (R / Z) \lor (R / Z)$ and its two standard loop classes $a, b$ .

[L1]

The fundamental group of a wedge of $r$ quotient circles is free on its $r$ standard circle loops (The fundamental group of a finite wedge of circles is free of that rank).

Proof

technique · direct

1.1

Apply [L1] with $r = 2$ to obtain that $π_{1} (W_{2}, w)$ is free on its standard circle loops.

L1algebra

2.1

Those standard loops are precisely $a$ and $b$ , so $π_{1} (W_{2}, w) ≅ F (a, b)$ .

step 1.1∎

Depends on

The fundamental group of a finite wedge of circles is free of that rank

Used by

Dependency tree · two levels

8 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, Example 1.21 (standard reference, not scraped)

π1(S1∨S1)π1​(S1∨S1) is the free group on two generators