Distributional Lusternik-Schnirelmann category of manifolds

We obtain several sufficient conditions for the distributional LS-category (dcat) of closed manifolds to be maximum, i.e., equal to their classical LS-category (cat). This gives us many new computations of dcat, especially for essential manifolds and (generalized) connected sums. In the process, we also determine the dcat of closed 3-manifolds having torsion-free fundamental groups and some closed geometrically decomposable 4-manifolds. Finally, we extend some of our results to closed Alexandrov spaces and discuss their cat and dcat in dimension 3.