Half-lives-half-dies theorem ($\dim \ker i_{\ast }=\frac{1}{2}\dim H_1(\partial W;{\mathbb{F}}_2)$)

For a compact three-manifold $W$, let $L=\ker (H_1(\partial W;{\mathbb{F}}_2)\to H_1(W;{\mathbb{F}}_2))$. The boundary intersection pairing satisfies $L^{\perp }=L$: the long exact sequence in relative homology identifies $L$ with boundaries of relative surfaces, and the boundary-interior adjoint identity identifies its orthogonal complement with the same kernel. Hence $2\dim L=\dim H_1(\partial W;{\mathbb{F}}_2)$. This also implies that a closed surface with odd first mod-two Betti number, such as ${\mathbb{RP}}^2$, cannot be the entire boundary of a compact three-manifold.