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PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Vertex groups recover the fundamental group

Statement

For every x∈X, reversal gives a canonical group isomorphism ρx:π1(X,x)→∼Aut⁡Π1(X)(x),ρx([α])=[αˉ]. Here π1(X,x) has the published first-loop-first multiplication, while the automorphism group has categorical composition. The identity map on the underlying loop classes is an anti-isomorphism, not an isomorphism.

Facts & Assumptions

Given: A topological space X and a point x∈X.

[F1]

Based loops and the fundamental group defines the loop-class set, the proposed product [α][β]=[α∗β], the constant loop cx, and reversal αˉ.

[F2]

Loop classes form the group π1(X,x0) under concatenation proves that this product is well defined and is a group law with identity [cx] and inverse [αˉ].

[F3]

Fundamental groupoid of a space has the same endpoint-fixed loop classes at x, but [γ]∘[δ]=[δ∗γ] in the vertex automorphism group.

Proof

technique · direct
1.1

Reversal respects endpoint-fixed path homotopy, is its own inverse on classes, and sends [cx] to itself. Hence ρx is a canonical bijection that preserves the identity and inverses.

F1F2F3
1.2

Reversing a concatenation gives α∗β‾=βˉ∗αˉ up to the standard endpoint-fixed reparametrization. By [F3], ρx([α])∘ρx([β])=[αˉ]∘[βˉ]=[βˉ∗αˉ]=ρx([α][β]). Thus ρx is a homomorphism.

F1F2F3
2.1

The bijective homomorphism in steps 1.1–1.2 is the asserted group isomorphism. Without reversal, [F3] gives [α]∘[β]=[β∗α], which proves the final anti-isomorphism warning as well.

F2F3step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

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Sources