Hassler Whitney - Work - Research

Research

Whitney's earliest work, from 1930 to 1933, was on graph theory. Many of his contributions were to the graph-coloring, and the ultimate computer-assisted solution to the four-color problem relied on some of his results. His work in graph theory culminated in a 1935 paper, where he laid the foundations for matroids, a fundamental notion in modern combinatorics and representation theory.

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 found only in 2005 by Charles Fefferman.

In a 1936 paper, Whitney gave a definition of a smooth manifold of class C r, 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, provided that 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 been published only 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 Schwartz 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, implicit even 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.

His book Geometric Integration Theory gives a theoretical basis for Stokes' theorem applied with singularities on the boundary and later inspired the generalization found by Jenny Harrison.

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.