TOPOLOGY, DIFFERENTIAL

study of differentiable manifolds and of differentiable maps between them. The field also is concerned with geometric objects (such as vector fields, foliations, and group actions), which one can associate with manifolds. Differential topology is thus in a position midway between differential geometry on the one hand and classical differential calculus on the other. The chief divergence from the spirit of differential geometry is that the latter only considers manifolds equipped with more precise structures (e.g., Riemannian metrics, connections) and has thus a basically quantitative aspect, whereas differential topology is purely qualitative in its aims. It borders on analysis in the study of differentiable maps and their singularities. This is the subject that one could call differential analysis and is the study of maps that are required to be only differentiable, rather than analytic. Differential topology has been studied as such only since the mid-1930s. Mathematicians have from the start made use of low-dimensional manifolds (such as points, curves, and surfaces), but the idea of developing a general theory of objects defined by conditions that are merely differentiable, as opposed to analytic or algebraic, was slow to appear. The French mathematician Joseph-Louis Lagrange in his Mcanique analytique (1788; Analytic Mechanics) considers such manifolds as the phase spaces of a dynamic system, and the canonical transformations met with in dynamics contain the idea of structure-preserving coordinate transformations. It is only in the work of Riemann, however, that the general notion of an n-dimensional manifold was first formulated. He called it an m-fach ausgedehnte Mannigfaltigkeit (m-fold extended manifold). The impetus of Italian algebraic geometry and the theories of algebraic and analytic surfaces led Weyl, in The Concept of a Riemann Surface (1913), to the first precise definition, within the particular context of an algebraic curve, of a manifold defined by coordinate charts and transformations and thus to the idea of analytic structure. Only a little later, the theory of general relativity, with its insistence on the concept of curved space-time, gave a tremendous impulse to tensorial differential geometry. At the same time, the topological theory of Lie groups, due to Cartan, also drew attention to the notion of manifolds. The first modern definition of a manifold, by local coordinate systems and transformations, is due to the mathematicians Oswald Veblen and Henry Whitehead. It is with the fundamental work of Hassler Whitney in 1935 that differential topology as such was born. From that time on, differential topology took its inspiration from two opposite sources. On the one hand, differential analysis brought very powerful tools to bear from 193540 onward: the theorem on the zero measure of the set of critical values of a differentiable map (since referred to as Sard's theorem, although Anthony Paul Morse had previously proved it in special cases), the extension and spectral theorems of Whitney, the generalization of the implicit function theorem known as the transversality lemma (a lemma is a theorem, usually proved to use in the proof of another theorem), and, finally, a C generalization of the Weierstrass preparation theorem for analytic maps (a C mapping is a mapping that is continuously differentiable an infinite number of times). On the other hand, algebraic topology was developing at the same time, with the foundations of singular homology and cohomology theories, the theory of fibre spaces and of their characteristic classes, spectral sequences, and homotopy theory. So by about 1950 the techniques necessary for a detailed investigation of differentiable manifolds were available. By 1935 Marston Morse had shown by means of his theory of critical points of functions and the inequalities that bear his name what sort of purely topological information about a manifold M could be deduced just from an inspection of the singularities of a real-valued function f : M . A more detailed study of his methods, making use of gradient trajectories, has succeeded in an extremely direct fashion in associating to a generic function f : M a partition of M into cells, each cell of dimension k having its centre in a critical point of f, of index k; in fact this cellular decomposition satisfies the homotopy-theoretical properties of a C-W complex. This representation of any manifold as a union of balls was to play a basic role in the appearance of surgery techniques (see below Surgery techniques), which study the possibility of transforming a given manifold M into another M of the same dimension, generalizing the process by which one passes from a sphere to a torus by adding a handle. Additional reading References on differential topology include David B. Gauld, Differential Topology (1982), an introduction at the advanced undergraduate level that assumes some calculus and linear algebra but no topology; Theodor Brcker and Klaus Jnich, Introduction to Differential Topology (1982; originally published in German, 1973); Lawrence Conlon, Differentiable Manifolds: A First Course (1993); and Juan Margalef-Roig and Enrique Outerelo Dominguez, Differential Topology (1992), a monograph on Banach manifolds with corners. Serge Lang, Differential and Riemannian Manifolds, 3rd ed. (1995); and Shlomo Sternberg, Lectures on Differential Geometry, 2nd ed. (1983), are two books on differential geometry that treat differentiable manifoldssome of the original articles remain highly recommended reading. Also useful are the following: Hassler Whitney, Differentiable Manifolds, Annals of Mathematics, 37:645680 (1936); R. Thom, Quelques proprits globales des varits diffrentiables, Commentarii mathematici Helvetici, 28:1786 (1954), on cobordism; John Milnor, On Manifolds Homeomorphic to the 7-Sphere, Annals of Mathematics, 64:399405 (1956), on exotic differentiable structures; Stephen Smale, Generalized Poincar's Conjecture in Dimensions Greater Than Four, Annals of Mathematics, 74:391406 (1961), and The Story of the Higher-Dimensional Poincar Conjecture, The Mathematical Intelligencer, 12(2):4451 (1989). As for differential analysis and singularity theory, Ralph Abraham and Joel Robbin, Transversal Mappings and Flows (1967); and Frdric Pham, Introduction l'tude topologique des singularits de Landau (1967), introduce the subject. V.I. Arnold, The Theory of Singularities and Its Applications (1991), briefly presents developments in the field. Bernard Malgrange, Ideals of Differentiable Functions (1966), covers the foundations of differential analysis. Additional references are Eduard Cech, On Bicompact Spaces, Annals of Mathematics, 38:823844 (1937); and M.H. Stone, Applications of the Theory of Boolean Rings to General Topology, Transactions of the American Mathematical Society, 41:321364 (1937). Ren Frdric Thom The Editors of the Encyclopdia Britannica