On low-codimensional decompositions

Decompositions on manifolds appear in various geometric structures. Necessary and sufficient conditions for quotient spaces of decompositions to be manifolds are widely characterized. We characterize necessary and sufficient conditions to be $k$-manifolds $(k = 1, 2)$, which generalize characterizations in the codimension-$k$ cases for the leaf spaces of foliations, the orbit spaces of group-actions, decomposition spaces of upper semi-continuous decompositions, and leaf class spaces of Riemannian foliation with regular closure. To prove such characterizations, we generalize a characterization of upper semi-continuity for decomposition into one for a class decomposition. In addition, we completely characterize upper semi-continuity for class decompositions of homeomorphisms on orientable compact surfaces and of homeomorphisms isotopic to identity on non-orientable compact surfaces. Furthermore, a flow on a connected compact $3$-manifold whose class decomposition is upper semi-continuous is "almost $k$ dimensional" $(k=0, 1, 2, 3)$ or has "complicated" minimal sets.