Differential topology, like differential geometry, is the study of smooth (or ‘differential’) manifolds. There are several equivalent versions of the definition: a common one is the existence of local charts mapping open sets in the manifold Mm to open sets in ℝm, with the requirement that coordinate changes are smooth, i.e. infinitely differentiable.

If M and N are smooth manifolds, a map f : M → N is called smooth if its expressions by the local coordinate systems are smooth. This leads to the concept of smooth embedding. If f : M → N and g : N → M are smooth and inverse to each other, they are called diffeomorphisms: we can then regard M and N as copies of the same manifold. If f and g are merely continuous and inverse to each other, they are homeomorphisms. Thus homeomorphism is a cruder means of classification than diffeomorphism.

The notion of smooth manifold gains in concreteness from the theorem of Whitney that any smooth manifold Mm may be embedded smoothly in Euclidean space ℝn for any n ≥ 2m + 1, and so may be regarded as a smooth submanifold of ℝn, locally defined by the vanishing of (n − m) smooth functions with linearly independent differentials. An important example is the unit sphere Sn−1 in ℝn. The disc Dn bounded by Sn−1 is an example of the slightly more general notion of manifold with boundary.

Whitney's result is more precise: it states that (if M is compact) embeddings are dense in the space of all maps f : Mm → ℝn, suitably topologised, provided n ≥ 2m + 1, and more generally the same holds for maps Mm → Nn for any manifold N of dimension n. Other ‘general position’ results include the fact that if m > p + q, a map f : Pp → Mm will in general avoid any union of submanifolds of M of dimension ≤ q. These results can be deduced from the general transversality theorem, which also applies to permit detailed study of the local forms of singularities of smooth maps.