Differentiable manifold
This article defines a property of manifolds and hence also of topological spaces
Definition
Symbol-free definition
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
Weaker 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 |