LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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The integral intersection pairing is unimodular for a simply connected four-manifold

Statement

Assume the Axiom of Choice. Let $X$ be a closed connected oriented smooth four-manifold with $π_{1} (X) = 0$ . Its integral intersection lattice on $H_{2} (X; Z) / Tor$ is a finite-rank free abelian group and its intersection pairing is unimodular.

Statement references

The integral intersection lattice and its pairing convention

Facts & Assumptions

Given: Full AC and a closed connected oriented smooth four-manifold $X$ with trivial fundamental group.

[F1]

The first Hurewicz map identifies $H_{1} (X; Z)$ with the abelianization of $π_{1} (X)$ ; in particular it is zero here. The first Hurewicz map is abelianization

[F2]

Poincaré duality for a closed oriented four-manifold gives an isomorphism $PD : H^{2} (X; Z) \to H_{2} (X; Z)$ , whose inverse sends a homology class to its cohomological dual. Poincaré duality for oriented topological manifolds

[F3]

The universal-coefficient exact sequence for cohomology is $0 \to {Ext}^{1} (H_{1} (X; Z), Z) \to H^{2} (X; Z) \to Hom (H_{2} (X; Z), Z) \to 0$ . The universal coefficient theorem for cohomology over a PID

[F4]

A closed smooth manifold has finite CW homotopy type under full AC, so its integral homology groups are finitely generated. A closed smooth manifold has the homotopy type of a finite CW complex

[F5]

The definition of the integral intersection pairing is the Poincaré-duality cup-product pairing on $H_{2}$ modulo torsion. Integral intersection lattice of a simply connected four-manifold

[A1]

Full AC is assumed; it supplies the CW homotopy-type result [F4] and the stated cohomological duality/UCT branch.

Proof

Given: The data and suppliers [F1]–[F5].

1.1givenF1F3A1algebra

By [F1] and [A1], $H_{1} (X; Z) = 0$ , so the Ext term in [F3] is zero and evaluation identifies $H^{2} (X; Z)$ with $Hom (H_{2} (X; Z), Z)$ .

1.2F4A1algebra

By [F4] and [A1], $H_{2} (X; Z)$ is finitely generated. Its quotient $L = H_{2} (X; Z) / Tor$ is therefore free of finite rank, and every homomorphism from $H_{2}$ to $Z$ factors uniquely through $L$ . Thus $Hom (H_{2}, Z) = Hom (L, Z)$ .

2.1F2F3F5step 1.1step 1.2algebra∎

Under the isomorphism [F2], the map $L \to Hom (L, Z)$ sending $\overset{ˉ}{x}$ to $(\overset{ˉ}{y} \mapsto Q_{X} (\overset{ˉ}{x}, \overset{ˉ}{y}))$ is the composite of the inverse Poincaré-duality map $H_{2} \to H^{2}$ with the evaluation map of [F3]; [F5] identifies that composite with the intersection pairing. Both component maps are isomorphisms by steps 1.1–1.2 and [F2]. Hence the pairing map is an isomorphism.

Depends on

Used by

Donaldson diagonalization theorem for positive-definite four-manifolds Theorem

Dependency tree · two levels

42 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

Simon K. Donaldson, An Application of Gauge Theory to the Topology of 4-Manifolds (standard reference, not scraped)