He was awarded a Wolf Prize in mathematics in 1983, and a Steele Prize in 1985.

In 1935 Whitney proved that any differential manifold of dimension *n* may be embedded in **R**^{2n}, and immersed in **R**^{2n-1}. This basic result shows that manifolds may be treated intrinsically or extrinsically, as we wish. It is often said that this paper contains the first accurate definition of 'manifold'; it certainly built on the work of Veblen and J.H.C. Whitehead published a few years earlier. It opened the way for much more refined studies: of embedding, immersion and also of smoothing, that is, the possibility of having various smooth structures on a given topological manifold. The argument of Whitney is necessarily of general position type.

A few years later, Whitney wrote the foundational paper on matroids. This is apparently a chapter of combinatorics; it has in recent years been seen increasingly as related to the fine structure of Grassmannians. In fact the idea of stratification, used for that application and many others, was also introduced by Whitney, in a precise form (his conditions A and B).

The singularities in low dimension of smooth mappings, later to come to prominence in the work of Thom, were also first studied by Whitney.

His book *Geometric Integration Theory* gives a theoretical basis for Stokes' theorem applied with singularities on the boundary.

These aspects of Whitney’s work have looked more unified, in retrospect and with the general development of singularity theory in its aspect of the failure of smoothness. Whitney’s purely topological work (Stiefel-Whitney class, basic results on vector bundles) entered the mainstream more quickly.