TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The fundamental group of a topological group is abelian

Statement

If $G$ is a topological group with identity $e$ , then $π_{1} (G, e)$ is an abelian group.

Facts & Assumptions

Given: A topological group $G$ with identity $e$ and based loops $α, β$ at $e$ .

[L1]

Pointwise multiplication of based loops in a topological group descends to loop classes and agrees there with loop concatenation; the pointwise product loop is homotopic to both concatenation orders (Pointwise multiplication and concatenation of loops in a topological group agree up to homotopy).

[L2]

Loop concatenation makes $π_{1} (G, e)$ a group whose identity is the class of the constant loop at $e$ (Loop classes form the group $π_{1} (X, x_{0})$ under concatenation).

[F1]

A group is abelian when its operation is commutative (Group and abelian group).

Proof

technique · direct

1.1

The classes $[α]$ and $[β]$ have concatenation product $[α * β]$ , while their pointwise product is represented by $t \mapsto α (t) β (t)$ ; [L1] identifies these two classes.

givenL1L2

2.1

The same pointwise product is also homotopic to $β * α$ by [L1]. Therefore $[α] [β] = [α * β] = [β * α] = [β] [α]$ .

step 1.1L1L2

3.1

Since $[α]$ and $[β]$ were arbitrary, multiplication in $π_{1} (G, e)$ is commutative, so the fundamental group is abelian.

step 2.1F1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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

J. Peter May, A Concise Course in Algebraic Topology, Chapter 1, Problem 3 (standard reference, not scraped)