The boundary kernel of a compact oriented 3-manifold is Lagrangian

Given: AC; a compact oriented smooth 3-manifold $M$, its induced boundary orientation, and the inclusion $i:\mathrm{\Sigma }=\mathrm{\partial }M\hookrightarrow M$.

The homology sequence of $(M,\mathrm{\Sigma })$ is exact at $H_1(\mathrm{\Sigma })$, so $\ker i_{\ast }=\mathrm{im}(\mathrm{\partial }:H_2(M,\mathrm{\Sigma };\mathbb{R})\to H_1(\mathrm{\Sigma };\mathbb{R}))$ (Long exact sequence of a pair).

Under AC, cap product with $[M,\mathrm{\Sigma }]$ is the Poincaré–Lefschetz isomorphism $H^1(M;\mathbb{R})\to H_2(M,\mathrm{\Sigma };\mathbb{R})$; the cap boundary identity identifies the boundary of this cap with the restriction of the cohomology class capped with $[\mathrm{\Sigma }]$, up to a fixed sign (Poincaré–Lefschetz duality, Cap product boundary identity, Relative cap products with quotient domains displayed, Relative fundamental class and boundary orientation).

Poincaré duality on each closed oriented boundary component identifies $P{D}_{\mathrm{\Sigma }}(\beta )=\beta \cap [\mathrm{\Sigma }]$ and the intersection pairing satisfies ${\omega }_{\mathrm{\Sigma }}(P{D}_{\mathrm{\Sigma }}\alpha ,P{D}_{\mathrm{\Sigma }}\beta )=⟨\beta ,P{D}_{\mathrm{\Sigma }}\alpha ⟩$ in the first-factor convention (Poincaré duality for oriented topological manifolds, The intersection form on the homology of a closed oriented surface, Kronecker evaluation pairing).

The cohomology map $i^{\ast }:H^1(M;\mathbb{R})\to H^1(\mathrm{\Sigma };\mathbb{R})$ is the transpose of $i_{\ast }$ under the perfect field-valued Kronecker pairings; finite-dimensional annihilator identities apply (Cohomology over a field is dual to homology over that field, Compact oriented manifolds with boundary have finite-dimensional field cohomology, The kronecker pairing is independent of cocycle and cycle representatives).

Full AC is used only through the Poincaré–Lefschetz and Poincaré duality suppliers in [F2]–[F4] (The Axiom of Choice).