DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Integral intersection lattice of a simply connected four-manifold

Definition

Let $X$ be a closed connected oriented smooth four-manifold. Its integral intersection lattice is the abelian group $L_{X} = H_{2} (X; Z) / Tor H_{2} (X; Z)$ with the integer-valued symmetric bilinear pairing $Q_{X} (\overset{ˉ}{x}, \overset{ˉ}{y}) = ⟨ {PD}^{- 1} (x) ⌣ {PD}^{- 1} (y), [X] ⟩$ . Here $Tor$ is the subgroup of elements of finite order, $[X] \in H_{4} (X; Z)$ is the orientation class, and $PD : H^{2} (X; Z) \to H_{2} (X; Z)$ is the Poincaré duality isomorphism. The rank is $b_{2} (X) = \dim_{Q} H_{2} (X; Q)$ . The lattice is positive definite when $Q_{X} (x, x) > 0$ for every nonzero $x \in L_{X}$ ; it is unimodular when a Gram matrix in an integral basis has determinant $1$ or $- 1$ .

Definition references

Integral middle-dimensional intersection pairing and orientation convention

Depends on

The middle-dimensional intersection form of a closed oriented 4k-manifold

Used by

Dependency tree · two levels

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Sources

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