Let $M$ be a compact smooth manifold and let $\{U_i\}_{i\in I}$ be an open cover. We say a handle decomposition of $M$ is subordinate to the open cover if each handle is contained in a $U_i$. Do such handle decompositions always exist?

If instead of handle decompositions one considers simplicial structures, then this can of course be done by barycentric subdivisions. This raises a related question: Can one associate to a simplicial structure a Morse function, where the trivial points are precisely the barycenters? In pictures this looks reasonable, but I am not sure about the technicalities.