Smooth parametric primitives for a smooth exact family on a compact manifold

Use [F1] to fix one Riemannian metric. The set of all strongly convex open neighbourhoods is an open cover, so compactness extracts a finite subcover $\mathcal{U}=(U_1,\dots ,U_r)$. Every nonempty finite intersection $U_{i_0\cdots i_s}$ is strongly convex by [F1]. There are only finitely many such intersections; choose one point in each and contract the intersection to it along the unique smoothly endpoint-dependent geodesics. By [F3], these contractions give fixed linear Poincaré homotopy operators $K_{i_0\cdots i_s}$. In local coordinates their coefficients are finite-interval integrals of coefficients of the pulled-back form and the fixed smooth contraction. Differentiation under that compact integral therefore shows directly that each $K_{i_0\cdots i_s}$ preserves smooth dependence on the finite-dimensional parameter.

Use [F2] finitely many times to fix a partition of unity subordinate to $\mathcal{U}$. Start the Čech--de Rham descent with $b_i=K_i(\alpha {\mathrm{\mid }}_{U_i})$, so $d{b}_i=\alpha {\mathrm{\mid }}_{U_i}$. The alternating differences $\delta b$ are closed because $d\delta =\delta d$. On each nonempty double intersection apply its fixed $K_{ij}$ to obtain a primitive; subtracting it makes the next alternating discrepancy closed one degree lower. Repeat. After at most $k$ repetitions the remaining discrepancy is a Čech cocycle of locally constant functions on the finite good cover.

Regard the last cocycle as a vector in the finite-dimensional simplicial cochain complex of the nerve. Because $\alpha$ is globally exact, comparison with any global primitive shows that this cocycle lies in the image of the preceding Čech coboundary. Fix a linear right inverse of that coboundary on its image by choosing bases once. Solve there, then reverse the finite descent. At the final gluing step the fixed partition gives a global $(k-1)$-form $R\alpha$ with $dR\alpha =\alpha$. Thus $R$ is one fixed linear operator on the space of exact $k$-forms; no primitive of an individual input was selected.

Put ${\beta }_p=R{\alpha }_p$. Restriction, the finitely many homotopy integrals, Čech differences, multiplication by fixed partition functions, and the fixed finite-dimensional linear solver all commute with differentiation in the finite-dimensional parameter. Hence ${\beta }_p$ is jointly smooth and $d{\beta }_p={\alpha }_p$. The empty manifold is immediate. The sole nonfinite choice input is the metric supplied under ${\mathrm{A}\mathrm{C}}_{\omega }$; all subsequent selections are finite.