= Half-lives-half-dies theorem
{title2=$\dim\ker i_*=\tfrac12\dim H_1(\partial W;\mathbb F_2)$}
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>.
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