= Ambiguity of harmonic wedge-pairing representatives
{title2=$\operatorname{Ann}(\mathcal H^p)=\operatorname{im}d\oplus\operatorname{im}\delta$}
On a compact oriented <Riemannian manifold> without boundary, a smooth $(n-p)$-form $\chi$ annihilates all harmonic $p$-forms under wedge integration exactly when its harmonic projection vanishes. Indeed $\int_M\varphi\wedge\chi=\langle*\varphi,\chi\rangle_{L^2}$, and the <Hodge star> maps $\mathcal H^p$ isomorphically onto $\mathcal H^{n-p}$. By the <Hodge decomposition theorem>, this annihilator is
$$
d\Omega^{n-p-1}\oplus\delta\Omega^{n-p+1}.
$$
Thus both exact and coexact additions are allowed; a harmonic representative is unique. The potentials of these additions need not be unique, and out-of-range degree spaces are zero. For a concrete coexact ambiguity, take the <torus> $(\mathbb R/2\pi\mathbb Z)^2$ with metric $dx^2+dy^2$. The form $\chi=\delta(\sin y\,dx\wedge dy)=\cos y\,dx$ annihilates every harmonic one-form by orthogonality, yet $d\chi=\sin y\,dx\wedge dy$ is nonzero. Thus an annihilating representative need not even be closed, let alone exact.
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