There is a technique in differential topology called turbulization that essentially twists the leaves of the foliation along the ideal boundary of a manifold, say the open 3-ball, rendering the foliation complete and this is because the leaves all attain infinite length because they spiral asymptotically along the ideal boundary, which in this case, is the 2 sphere that bounds the open 3-ball.
Now with the 3-ball in the real smooth case, the cool part is that the leaves can all have punctures/missing points and turbulization can still render the foliation complete after this twisting procedure.
My question is about the possible physical applications/implications of this turbulization technique. My mind goes directly to black holes which are singularities of a spacetime.
Is turbulization leveraged in foliations of spacetime in which the leaves each have points missing? What does it mean physically for turbulization to render the foliation of the spacetime complete when it used to have holes/singularities?