A manifold is said to be differentiable if it can be given the structure of a differential manifold, viz if it can be given a compatible differential structure. The term smooth is also sometimes used for this, though the term smooth might also be used for a differential manifold (i.e., for the manifold along with the differential structure).
Relation with other properties
|Property||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|manifold admitting a PL structure||manifold that admits the structure of a PL manifold||(any differential manifold has a unique PL structure compatible with its differential structure)||PL not implies differentiable|||FULL LIST, MORE INFO|
|triangulable manifold||manifold that admits a triangulation, i.e., it is homeomorphic to the geometric realization of a simplicial complex||(via PL structure, since a PL structure is a particular kind of triangulation)||(via PL)|||FULL LIST, MORE INFO|
|polyhedron||topological space that admits a triangulation||(via PL structure)||(via PL structure)||Triangulable manifold|FULL LIST, MORE INFO|
|manifold||a topological manifold; no structure beyond that||(by definition)||the E8 manifold is a counterexample in 4 dimensions||Triangulable manifold|FULL LIST, MORE INFO|