  We derive the partition function of 5d ${\cal N}=1$ gauge theories on the
manifold $S^3_b \times \Sigma_{\frak g}$ with a partial topological twist along
the Riemann surface, $\Sigma_{\frak g}$. This setup is a higher dimensional
uplift of the two-dimensional A-twist, and the result can be expressed as a sum
over solutions of Bethe-Ansatz-type equations, with the computation receiving
nontrivial non-perturbative contributions. We study this partition function in
the large $N$ limit, where it is related to holographic RG flows between
asymptotically locally AdS$_6$ and AdS$_4$ spacetimes, reproducing known
holographic relations between the corresponding free energies on $S^{5}$ and
$S^{3}$ and predicting new ones. We also consider cases where the 5d theory
admits a UV completion as a 6d SCFT, such as the maximally supersymmetric
${\cal N}=2$ Yang-Mills theory, in which case the partition function computes
the 4d index of general class ${\cal S}$ theories, which we verify in certain
simplifying limits. Finally, we comment on the generalization to ${\cal M}_3
\times \Sigma_{\frak g}$ with more general three-manifolds ${\cal M}_3$ and
focus in particular on ${\cal M}_3=\Sigma_{\frak g'}\times S^{1}$, in which
case the partition function relates to the entropy of black holes in AdS$_6$.
