ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-21

How statement and proof provenance work

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.

Two paths can induce distinct change-of-basepoint isomorphisms on $S^{1} \lor S^{1}$

Example

Let $W = S^{1} \lor S^{1}$ have wedge point $w$ , and let $z \neq w$ lie on the first circle. Write $a, b \in π_{1} (W, w)$ for the standard loops in the first and second circles. Choose a simple path $ρ$ in the first circle from $w$ to $z$ , and put $σ = b * ρ$ . Then $ρ$ and $σ$ have the same endpoints but induce distinct isomorphisms

$c_{ρ}, c_{σ} : π_{1} (W, w) ⟶ π_{1} (W, z) .$

Facts & Assumptions

Given: The points, paths, and standard loop classes in the Example.

[L1]

The group $π_{1} (W, w)$ is the free group $F (a, b)$ on the two standard circle loops ( $π_{1} (S^{1} \lor S^{1})$ is the free group on two generators).

[F1]

Loop concatenation is the fundamental-group operation, and path reversal represents inversion (Loop classes form the group $π_{1} (X, x_{0})$ under concatenation).

[F2]

The reduced words on a basis and its formal inverses form the free group on that basis (Reduced words form the free group on an alphabet).

Verification

technique · direct

1.1F1construct

For a path $η$ from $w$ to $z$ , define $c_{η} ([α]) = [\overset{ˉ}{η} * α * η]$ . Endpoint-fixed homotopies are preserved by concatenating fixed paths, while the standard cancellation homotopies for $η * \overset{ˉ}{η}$ and $\overset{ˉ}{η} * η$ show that $c_{η}$ is a homomorphism with inverse $c_{\overset{ˉ}{η}}$ . Thus both $ρ$ and $σ$ define basepoint-change isomorphisms.

2.1step 1.1L1F1

Since $σ = b * ρ$ , reversal of concatenation gives $c_{σ} ([α]) = [\overset{ˉ}{ρ} * \overset{ˉ}{b} * α * b * ρ] = c_{ρ} (b^{- 1} [α] b) .$ Hence $c_{\overset{ˉ}{ρ}} c_{ρ} (a) = a$ , whereas $c_{\overset{ˉ}{ρ}} c_{σ} (a) = b^{- 1} a b$ in the identification [L1].

3.1step 2.1F2∎

The words $a$ and $b^{- 1} a b$ are distinct reduced words by [F2]. Therefore their images under the isomorphism $c_{ρ}$ are distinct, so $c_{ρ} \neq c_{σ}$ .

Depends on

Used by

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Two paths can induce distinct change-of-basepoint isomorphisms on S1∨S1S1∨S1