CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Trefoil surgery with trivial first homology and nontrivial fundamental group

Statement refuted

Every connected closed oriented $3$ -manifold $M$ with $H_{1} (M; Z) = 0$ is simply connected, meaning that $π_{1} (M)$ is trivial.

Statement references

Countable-choice hypothesis for the chosen exterior and peripheral data

Oriented trefoil in the standard three-sphere

Geometric braid and transported strand labels

Positive geometric half twist

Braid closure convention

Chosen knot exterior

Meridian and preferred longitude

Filling along the specified meridian slope

First homology of the filling

Trefoil $+ 1$ filling group presentation

Hilden–Lozano–Montesinos-Amilibia, p. 24, canonical longitude and zero-surgery group

Saveliev, Lecture 2 §2.2, knot surgery along a specified peripheral slope

Facts & Assumptions

Given: ${A C}_{ω}$ , the connected exterior $E_{T}$ of the trefoil $T$ , the positive meridian $μ = a$ at the first left arc and the degree-one preferred longitude $λ$ of this same exterior, their boundary basis $([μ], [λ])$ , the $+ 1$ filling of $T$ , and the two displayed supplier statements. The preferred-longitude definition supplies $[λ] = 0$ ; the homology lemma derives that $[μ]$ generates $H_{1} (E_{T}; Z)$ .

[A1]

${A C}_{ω}$ is the stated axiom for the chosen exterior and peripheral data. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

First homology of knot fillings: a $p / q$ filling has first homology $Z / ∣ p ∣ Z$ when $p \neq 0$ .

[F2]

Group presentation for positive trefoil $+ 1$ surgery: in its specified positive-braid convention, $μ = a$ is the positive meridian at the first left arc and $λ$ is the degree-one preferred longitude; the matching $+ 1$ filling has presentation $⟨ a, b ∣ a b a = b a b, b a b^{- 1} a b a^{- 2} = 1 ⟩$ .

[F3]

Oriented slopes and knot Dehn filling: filling glues an oriented solid torus to the knot exterior along the specified meridian slope.

Counterexample

Assume ${A C}_{ω}$ . Let $T$ be the trefoil given by the closure of the positive two-strand braid $σ_{1}^{3}$ , using the positive half-twist convention for $σ_{1}$ . Choose its connected exterior and the same peripheral system as in the trefoil group-presentation calculation: $μ = a$ is the positive normal-disk meridian at the first left arc, and $λ$ is the degree-one preferred longitude of this exterior, with $([μ], [λ])$ a basis of $H_{1} (\partial E_{T}; Z)$ . Let $M = E_{T} (1 / 1)$ , where $+ 1$ means filling along $[μ] + [λ]$ . Then $H_{1} (M; Z) = 0$ but $π_{1} (M)$ is nontrivial.

1.1givenA1F1F3constructalgebra

By [F3], $M$ is the connected closed oriented manifold obtained by filling the connected knot exterior along its boundary torus. Apply [F1] with $p = q = 1$ . Then $H_{1} (M; Z) ≅ Z / 1 Z = 0$ .

1.2F2constructalgebra

Let $A_{5}$ denote the subgroup of even permutations of ${1, 2, 3, 4, 5}$ ; every $5$ -cycle is even because it is a product of four transpositions. Set $a = (1 2 3 4 5)$ and $b = (1 3 5 4 2)$ , composing right to left. Applying the factors of $a b a$ and $b a b$ from right to left gives the same image tuple $(2, 1, 3, 5, 4)$ on inputs $(1, 2, 3, 4, 5)$ , so both words equal $(1 2) (4 5)$ . Thus the braid relator in [F2] is satisfied.

1.3F2constructalgebra

Applying the factors of $b a b^{- 1} a b a^{- 2}$ from right to left, the six successive images for inputs $1, 2, 3, 4, 5$ are respectively $(4, 2, 3, 1, 2, 1)$ , $(5, 4, 5, 3, 4, 2)$ , $(1, 3, 4, 5, 1, 3)$ , $(2, 1, 2, 4, 5, 4)$ , and $(3, 5, 1, 2, 3, 5)$ . Their final entries are $(1, 2, 3, 4, 5)$ , so the filling relator is the identity. The assignments therefore satisfy both relators in [F2] and induce a homomorphism $π_{1} (M) \to A_{5}$ .

2.1step 1.1step 1.2step 1.3algebra∎

The image of $a$ is a nonidentity $5$ -cycle, so this homomorphism has nontrivial image and $π_{1} (M) \neq 1$ . Together with step 1.1, this refutes the statement.

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Sources

Hugh M. Hilden, María Teresa Lozano, and José María Montesinos-Amilibia, On the affine representations of the trefoil knot group, arXiv:1003.3554v1, p. 24 (standard reference, not scraped)
Nikolai Saveliev, Lectures on the Topology of 3-Manifolds, 2nd revised edition, Lecture 2 §2.2 (standard reference, not scraped)