Differentiable manifold

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This article defines a property of manifolds and hence also of topological spaces


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