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A slice-branched-cover gluing produces a closed definite four-manifold

Statement

Assume the Axiom of Choice. Let $K \subset S^{3}$ be a smooth oriented knot such that its double branched cover $M = Σ_{2} (K)$ is a rational homology sphere. Let $D \subset B^{4}$ be a smooth proper embedded disk with boundary $K$ , and let $B_{D}$ be the connected double cover of $B^{4}$ branched over $D$ . Let $W$ be a compact connected smooth simply connected oriented four-manifold with boundary identified orientation-compatibly with $M$ , and assume its boundary intersection form is positive definite. Assume also that the image of $π_{1} (M) \to π_{1} (B_{D})$ normally generates $π_{1} (B_{D})$ , meaning its normal closure is the whole group. Then the oriented gluing $X = W \cup_{M} (- B_{D})$ is a closed smooth simply connected positive-definite four-manifold, and inclusion induces an isometric full-rank embedding $Q_{W} ↪ Q_{X}$ of integral intersection lattices with $b_{2} (X) = b_{2} (W)$ .

Statement references

The oriented double branched cover of a knot

A smooth slice disk has a rational-homology-ball double branched cover

Facts & Assumptions

Given: Full AC; the knot, slice disk, rational-homology-sphere boundary, positive-definite simply connected filling, boundary identification, and normal-generation hypothesis in the statement.

[F1]

For a smooth proper disk, the double cover branched over the disk is a compact connected oriented smooth four-manifold with rational homology of a point, and its boundary is the double branched cover of the boundary knot. The result does not assert a fundamental-group conclusion. The double cover branched over a slice disk is a rational homology ball

[F2]

The double branched cover is defined by the connected cover of the knot exterior classified by the positive-meridian map to $Z / 2$ , filled along the closed lift of the squared meridian. Cyclic branched cover of a knot

[F3]

Smooth four-ball genus is the minimum genus of a smooth proper oriented surface in $B^{4}$ ; a smooth slice disk is the genus-zero case. Smooth and topological slice genus

[F4]

The boundary form is the signed intersection pairing on the torsion-free quotient, and is rationally nondegenerate when the boundary is a rational homology sphere. Intersection lattice of a four-manifold with rational-homology-sphere boundary The boundary intersection pairing is well defined and rationally nondegenerate

[F5]

Seifert–van Kampen computes the fundamental group of a union along a path-connected collar as the pushout of the two inclusion homomorphisms. Seifert–van Kampen identifies the fundamental group with a group pushout

[F6]

Mayer–Vietoris gives the singular-homology exact sequence for the collar-thickened decomposition along $M$ . Mayer–Vietoris sequence in singular homology

[F7]

Smooth collars on the two boundaries allow the orientation-compatible gluing and its collar-thickened open cover. Collar neighborhood theorem

[F8]

For any abelian group $A$ , $A \otimes_{Z} Q$ is its positive-integer localization, and $a \otimes 1 = 0$ exactly when some positive integer kills $a$ . The natural coefficient map identifies $H_{j} (Y; Z) \otimes Q$ with $H_{j} (Y; Q)$ . Rationalization is exact and commutes with singular homology

[A1]

For this particular branched cover $B_{D}$ , the boundary inclusion normally generates $π_{1} (B_{D})$ . This is an explicit additional hypothesis; it is not a consequence of the rational-homology-ball supplier [F1].

[A2]

Full AC is assumed by the branched-cover and homology suppliers in [F1]–[F3].

Proof

Given: The data and assumptions above.

1.1givenF1F2F3F7A2construct

By [F3], $D$ is a smooth proper slice disk; [F1] and [F2] then give a compact rational homology four-ball $B_{D}$ with boundary $M = Σ_{2} (K)$ . The boundary identification in the statement permits the smooth oriented gluing $X = W \cup_{M} (- B_{D})$ ; collars [F7] give a smooth closed oriented four-manifold.

1.2givenF5A1algebra

Applying [F5] to the collar-thickened union gives $π_{1} (X) = π_{1} (W) *_{π_{1} (M)} π_{1} (B_{D})$ . Since $π_{1} (W) = 1$ , this pushout is the quotient of $π_{1} (B_{D})$ by the normal closure of the image of $π_{1} (M)$ . That normal closure is all of $π_{1} (B_{D})$ by [A1], so $π_{1} (X) = 1$ .

1.3givenF1F6F8A2algebra

In the rational Mayer–Vietoris sequence of [F6], $H_{2} (M; Q) = H_{1} (M; Q) = 0$ and $H_{i} (B_{D}; Q) = 0$ for $i > 0$ by [F1]. Thus inclusion induces an isomorphism $H_{2} (W; Q) ≅ H_{2} (X; Q)$ , so $b_{2} (X) = b_{2} (W)$ . By [F8], $H_{2} (W; Z) \otimes Q ≅ H_{2} (W; Q)$ , and the kernel of the integral-to-rational map is exactly the torsion subgroup. If an integral class maps to torsion in $H_{2} (X; Z)$ , its rational image vanishes; the isomorphism forces its rational class in $W$ to vanish, so it was torsion. Therefore the induced map on torsion-free quotients is injective and has full rank.

2.1F4step 1.3algebra∎

Represent classes of $W$ by two-cycles in its interior and make two representatives transverse. Their signed intersection points are unchanged in the glued manifold, so the map of step 1.3 preserves the intersection form. Its rational extension is an isomorphism and transports the positive-definite form of $W$ to $X$ ; hence $Q_{X}$ is positive definite.

Unresolved proof obligations

The displayed gluing lemma is proved only with [A1]. A proof that every smooth slice-disk branched cover in this setting has boundary-normal-generated fundamental group has not been found in the inspected local suppliers. The rational homology conclusion [F1] does not supply [A1]. Consequently this lemma does not yet cover every smooth slice disk and cannot by itself certify unrestricted nonsliceness.

Depends on

Used by

Failure of a full-rank standard lattice embedding obstructs smooth sliceness under the gluing hypotheses Corollary

Dependency tree · two levels

67 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

R. H. Fox and J. W. Milnor, Singularities of 2-spheres in 4-space and cobordism of knots (standard reference, not scraped)
Simon K. Donaldson, An Application of Gauge Theory to the Topology of 4-Manifolds (standard reference, not scraped)