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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$π_{1} (T^{2}) ≅ Z \times Z$

Statement

For $T^{2} = (R / Z)^{2}$ pointed at $([0], [0])$ ,

$π_{1} (T^{2}, ([0], [0])) ≅ (Z, +) \times (Z, +) .$

Facts & Assumptions

Given: The pointed torus $T^{2}$ of The two-dimensional torus $T^{2} = (R / Z)^{2}$ .

[L1]

The fundamental group of a product is naturally the direct product of the two fundamental groups ( $π_{1} (X \times Y, (x_{0}, y_{0})) ≅ π_{1} (X, x_{0}) \times π_{1} (Y, y_{0})$ ).

[F1]

The degree map is an isomorphism $π_{1} (R / Z, [0]) ≅ (Z, +)$ ( $Deg : π_{1} (R / Z, [0]) \to (Z, +)$ is an isomorphism).

Proof

technique · direct

1.1

By the torus definition and [L1], $π_{1} (T^{2}, ([0], [0])) ≅ π_{1} (R / Z, [0]) \times π_{1} (R / Z, [0]) .$

2.1

Applying [F1] in both coordinates gives the displayed isomorphism with $Z \times Z$ .

step 1.1F1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

J. Peter May, A Concise Course in Algebraic Topology, Chapter 2, Section 8 (standard reference, not scraped)

π1(T2)≅Z×Zπ1​(T2)≅Z×Z