Intersection lattice of a four-manifold with rational-homology-sphere boundary

Assume the Axiom of Choice. Let $W$ be a compact connected oriented smooth four-manifold whose boundary $M$ is a rational homology three-sphere, meaning $H_{\ast }(M;\mathbb{Q})\cong H_{\ast }(S^3;\mathbb{Q})$. The boundary intersection lattice has underlying group $L_W=H_2(W;\mathbb{Z})\mathrm{/}\mathrm{Tor}{H}_2(W;\mathbb{Z})$. Poincaré–Lefschetz duality identifies each $x\in H_2(W;\mathbb{Z})$ with a class ${\alpha }_x\in H^2(W,M;\mathbb{Z})$. Define $Q_W(\bar{x},\bar{y})=⟨{\alpha }_x⌣{\alpha }_y,[W,M]⟩$, where $⌣$ is the relative cup product and $[W,M]\in H_4(W,M;\mathbb{Z})$ is the relative orientation class; equivalently, this is the signed intersection number of interior two-cycle representatives. Its rank is $b_2(W)={\dim }_{\mathbb{Q}}{H}_2(W;\mathbb{Q})$. A nondegenerate positive-definite $Q_W$ is called a positive-definite filling lattice. It is not required to be unimodular; the boundary linking form can make its determinant have absolute value greater than one.

Integral intersection pairing convention on a closed four-manifold

Poincaré–Lefschetz duality for an oriented manifold with boundary