Whitney's lifelong interest in geometric properties of functions also began around this time. His earliest work in this subject was on the possibility of extending a function defined on a closed subset of Rn to a function on all of Rn with certain smoothness properties. A complete solution to this problem was only found in 2005 by Charles Fefferman.
In a 1936 paper, Whitney gave a definition of a "smooth manifold of class Cr", and proved that, for high enough values of r, a smooth manifold of dimension n may be embedded in R2n+1, and immersed in R2n. (In 1944 he managed to reduce the dimension of the ambient space by 1, so long as n > 2, by a technique that has come to be known as the "Whitney trick.") This basic result shows that manifolds may be treated intrinsically or extrinsically, as we wish. The intrinsic definition had only been published a few years earlier in the work of Oswald Veblen and J.H.C. Whitehead. These theorems 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.
He was one of the major developers of cohomology theory, and characteristic classes, as these concepts emerged in the late 1930s, and his work on algebaic topology continued into the 40s. He also returned to the study of functions in the 1940s, continuing his work on the extension problems formulated a decade earlier, and answering a question of Schwarz in a 1948 paper On Ideals of Differentiable Functions.
Whitney had, throughout the 1950s, an almost unique interest in the topology of singular spaces and in singularities of smooth maps. An old idea, even implicit in the notion of a simplicial complex, was to study a singular space by decomposing it into smooth pieces (nowadays called "strata"). Whitney was the first to see any subtlety in this definition, and pointed out that a good "stratification" should satisfy conditions he termed "A" and "B". The work of René Thom and John Mather in the 1960s showed that these conditions give a very robust definition of stratified space. The singularities in low dimension of smooth mappings, later to come to prominence in the work of René Thom, were also first studied by Whitney.
These aspects of Whitney’s work have looked more unified, in retrospect and with the general development of singularity theory. Whitney’s purely topological work (Stiefel-Whitney class, basic results on vector bundles) entered the mainstream more quickly.
He was Instructor of Mathematics at Harvard University, 1930-31, 1933-35; NRC Fellow, Mathematics, 1931-33; Assistant Professor, 1935-40; Associate Professor, 1940-46, Professor, 1946-52; Professor Instructor, Institute for Advanced Study, Princeton University, 1952-77; Professor Emeritus, 1977-89; Chairman of the Mathematics Panel, National Science Foundation, 1953-56; Exchange Professor, College de France, 1957; Memorial Committee, Support of Research in Mathematical Sciences, National Research Council, 1966-67; President, International Commission of Mathematical Instruction, 1979-82; Research Mathematicians, National Defense Research Committee, 1943-45; Construction of the School of Mathematics. Recipient, National Medal of Science, 1976, Wolf Prize, Wolf Foundation, 1983; and a Steele Prize in 1985.
He was a member of the National Academy of Science; Colloquium Lecturer, American Mathematical Society, 1946; Vice President, 1948-50 and Editor, American Journal of Mathematics, 1944-49; Editor, Mathematical Reviews, 1949-54; Chairman of the Committee vis. lectureship, 1946-51; Committee Summer Instructor, 1953-54; Steele Prize, 1985, American Mathematical Society; American National Council Teachers of Mathematics, Swiss Mathematics Society (Honorary), Académie des Sciences (Foreign Associate); New York Academy of Sciences.
Married Margaret R. Howell, May 30, 1930; children: James Newcomb, Carol, Marian; married Mary Barnett Garfield, January 16, 1955; children: Sarah Newcomb, Emily Baldwin; and married Barbara Floyd Osterman, February 8, 1986.