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First homology of knot fillings

Statement

Assume ${A C}_{ω}$ . Let $K$ be a smooth oriented knot in the oriented $3$ -sphere, choose a closed tubular neighborhood $N (K)$ , and set $E_{K} = S^{3} ∖ int N (K)$ and $T_{K} = \partial E_{K} = \partial N (K)$ ; assume $E_{K}$ is connected. Let $μ$ be the oriented boundary of a normal disk whose orientation, preceded by the oriented tangent to $K$ , agrees with the orientation of $S^{3}$ . Let $λ$ be a simple closed curve on $T_{K}$ whose degree under the normal-tube projection $T_{K} \to K$ is $+ 1$ and whose image in $H_{1} (E_{K}; Z)$ is zero; such a curve is a preferred longitude. Assume $([μ], [λ])$ is an ordered basis of $H_{1} (T_{K}; Z)$ . For coprime integers $p, q$ not both zero, let $E_{K} (p / q)$ be the filling whose attached solid-torus meridian maps to $p [μ] + q [λ]$ . Then $H_{1} (E_{K} (p / q); Z) ≅ {\begin{cases} Z / ∣ p ∣ Z, & p \neq 0, \\ Z, & p = 0. \end{cases}$

Statement references

Countable-choice hypothesis for the chosen exterior and peripheral data

Chosen knot exterior and boundary torus

Positive meridian and degree-one preferred longitude

Slope and filling convention used in this proof

Separate exterior-homology theorem; the needed peripheral content is derived locally below

Saveliev, Lecture 2 §2.2, surgery on links in $S^{3}$

Ozsváth–Stipsicz–Szabó, §1.2.3, Mayer–Vietoris homology of fillings

Elhamdadi–Hajij–Istvan, §4.2, meridians and preferred longitudes

Facts & Assumptions

Given: A smooth oriented knot $K \subset S^{3}$ , its exterior $E_{K}$ , tubular neighborhood $N (K)$ , boundary torus $T_{K}$ , meridian $μ$ , preferred longitude $λ$ , and coprime integers $p, q$ not both zero.

[F1]

Mayer–Vietoris sequence in singular homology: for an open cover $X = U \cup V$ , the singular-homology Mayer–Vietoris sequence is exact, with the map from the overlap given by $c \mapsto (i_{*} c, - j_{*} c)$ for the two inclusions $i : U \cap V ↪ U$ and $j : U \cap V ↪ V$ , and the next map by $(u, v) \mapsto i_{*} u + j_{*} v$ .

[F2]

Homology of spheres: for $n \geq 1$ , the reduced singular homology of $S^{n}$ is $Z$ in degree $n$ and zero in all other degrees; in particular $H_{1} (S^{3}; Z) = H_{2} (S^{3}; Z) = 0$ and $H_{1} (S^{1}; Z) = Z$ .

[F3]

Mayer–Vietoris computation of first homology of the torus: $H_{1} (T^{2}; Z) ≅ Z \oplus Z$ , generated by the two factor circles.

[F4]

Singular homology is invariant under deformation retracts: a deformation-retract inclusion induces an isomorphism on singular homology.

[F5]

Oriented slopes and knot Dehn filling: the filling glues a solid torus to $E_{K}$ with its meridian identified with the class $p [μ] + q [λ]$ .

[A1]

${A C}_{ω}$ is the stated axiom The Axiom of Countable Choice ( ${A C}_{ω}$ ). The chosen positive meridian bounds its normal disk in $N (K)$ , and the preferred-longitude definition supplies both degree $+ 1$ over the core and $[λ] = 0$ in $H_{1} (E_{K}; Z)$ ; the pair being a boundary homology basis is an explicit hypothesis Meridian and preferred longitude. The proof derives that $[μ]$ generates $H_{1} (E_{K}; Z)$ in step 2.1 by Mayer–Vietoris and does not invoke the separate full-AC exterior-homology theorem.

[F6]

For any space, $H_{0} (-; Z)$ is free on its path components; in particular a path-connected space has $H_{0} ≅ Z$ , and an inclusion between path-connected spaces sends the point generator to the point generator Zero-th singular homology is free on path components.

Proof

1.1givenF1F2F4construct

Choose a bicollar of $T_{K}$ in $S^{3}$ and thicken $E_{K}$ and $N (K)$ slightly to an open cover. The two sets deformation-retract to $E_{K}$ and $N (K)$ , and their intersection deformation-retracts to $T_{K}$ . By [F1] and [F2], the segment $H_{2} (S^{3}; Z) \to H_{1} (T_{K}; Z) \to H_{1} (E_{K}; Z) \oplus H_{1} (N (K); Z) \to H_{1} (S^{3}; Z)$ has zero groups at both ends, so the middle map is an isomorphism.

1.2givenA1F1F2F3F4F5F6construct

Let $M = E_{K} (p / q)$ and choose a bicollar of the gluing torus. Thickening the exterior and attached solid torus across this collar gives an open cover of $M$ whose two sets retract to the pieces and whose intersection retracts to $T_{K}$ . Write $α = p μ + q λ$ , and choose $β = r μ + s λ$ with $p s - q r = 1$ , so $α, β$ form a basis of $H_{1} (T_{K}; Z)$ . In the Mayer–Vietoris map, $α$ maps to $(p [μ], 0)$ because it bounds the attached solid-torus meridian disk, and $β$ maps to $(r [μ], \pm v)$ where $v$ generates the first homology of the attached solid torus; the coefficient is $\pm 1$ because $β$ intersects $α$ once. The pieces and their overlap are path-connected: the exterior is a connected manifold and hence path-connected, and the solid torus and boundary torus are path-connected. By [F6], the following $H_{0}$ map sends $1$ to $(1, - 1)$ and is injective. Exactness therefore identifies $H_{1} (M; Z)$ with the cokernel of the $H_{1}$ map.

2.1step 1.1A1F2F3F4algebra

By [F3], $H_{1} (T_{K}; Z) ≅ Z^{2}$ , and by [F4] and [F2] the solid torus $N (K) ≃ S^{1}$ has $H_{1} (N (K); Z) ≅ Z$ . Since $H_{1} (E_{K}; Z) \oplus Z ≅ Z^{2}$ , the direct summand $H_{1} (E_{K}; Z)$ is free of rank one. Under the isomorphism in step 1.1, $μ$ maps to $([μ], 0)$ because it bounds a disk in $N (K)$ . The longitude maps to $(0, \pm 1)$ : [A1] gives its zero exterior image, and degree $+ 1$ on the core gives a generator of $H_{1} (N (K); Z)$ under the deformation-retraction identification. Since the map is an isomorphism, $[μ]$ generates $H_{1} (E_{K}; Z)$ .

3.1step 2.1step 1.2algebra∎

Thus $H_{1} (M; Z)$ has generators $[μ], v$ and relations $p [μ] = 0$ and $r [μ] \pm v = 0$ . Eliminating $v$ leaves $Z ⟨ [μ] ⟩ / ⟨ p [μ] ⟩$ . If $p \neq 0$ , this is $Z / ∣ p ∣ Z$ ; if $p = 0$ , coprimality gives $q = \pm 1$ and the group is $Z$ .

Depends on

Used by

Trefoil surgery with trivial first homology and nontrivial fundamental group Counterexample
Worked cases of Dehn surgery on the unknot Example
An S3 filling detects the meridian slope Lemma
Noncyclic SU(2) representations of numerator-one surgeries Lemma
Property P for nontrivial knots Theorem

Dependency tree · two levels

28 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

Nikolai Saveliev, Lectures on the Topology of 3-Manifolds, 2nd revised edition, Lecture 2 §2.2 (standard reference, not scraped)
Peter S. Ozsváth, András I. Stipsicz, and Zoltán Szabó, Heegaard Floer homology, version 2025-12-15, §1.2.3 (standard reference, not scraped)
Mohamed Elhamdadi, Mustafa Hajij, and Kyle Istvan, Framed Knots, arXiv:1910.10257v6, §4.2 (standard reference, not scraped)