TOPOLOGICAL GROUP

in mathematics, a group of points of a topological space for which operations are continuous. Topological group theory has developed within topology, algebra, and analysis. The interaction of classical and modern mathematical methods has been most productive in that area of analysis in which topological methods play a role. The result has stimulated much work in topology and other branches of mathematics and has important applications in theoretical physics. It is advantageous to consider topological methods in analysis under three headings and in rather different manners. The first section, the theorems of Tikhonov and Ascoli, considers two basic theorems that can be proved in a quite general setting and the importance of which cannot be overstressed. After some preliminary remarks, the second section, continuous groups, starts from the definition of a topological group and develops a theory that is gradually seen to encompass a whole range of general results. The third section, analysis on manifoldswhich is concerned largely with working from some important problems in classical analysisdiscusses the methods topologists have employed in tackling them, and shows how the body of theory so developed has built itself into a coherent whole. In this way a broad view of the areas of topological groups is presented. Before proceeding, however, some elementary and general remarks on the role of topology in analysis will be helpful. A distinctive feature of topological arguments in analysis is that they are qualitative and nonnumerical. A simple instance of topological reasoning is the following. A ball is thrown vertically upward, and a few moments later it falls back to its starting point; one deduces that at some point on the way it must have been stationary. This observation depends, of course, on certain preconceptions about gravity and the continuity properties of space and time. When suitably formulated, they form the foundation of analysis and topology. Now, in such a simple example as a falling ball, an elementary application of the Newtonian calculus will give full information about the whole motion, while the topological argument merely asserts the obvious. If, however, a single ball is now replaced by a large number of balls moving in more complicated ways, then direct analytical computation may become too lengthy or difficult. In such a situation one may settle for some qualitative aspects of the various motions, which may be derived by topological arguments but which may be far from self-evident because of the complexity of the situation. In fact, if the various balls are replaced by planets and one proceeds to study the Newtonian motion of several bodies under their mutual gravitational attraction, extremely difficult and unsolved mathematical problems are met. Additional reading The reader interested in topological groups may wish to consult some of the following works: L.S. Pontryagin (L.S. Pontriagin), Topological Groups, 3rd ed. (1986; originally published in Russian, 4th ed., 1984); Deane Montgomery and Leo Zippin, Topological Transformation Groups (1955, reprinted 1974); Claude Chevalley, Theory of Lie Groups, vol. 1 (1946, reissued 1957), and Thorie des groupes de Lie, vol. 23 (195155, reissued 1968); Sminaire Sophus Lie, vol. 1 (1955); Hermann Weyl, The Classical Groups, 2nd ed. with supplement (1946, reissued 1966), and The Theory of Groups and Quantum Mechanics (1931, reissued 1950; originally published in German, 2nd rev. ed., 1931); Lynn H. Loomis, An Introduction to Abstract Harmonic Analysis (1953); Andre Weil, L'Intgration dans les groupes topologiques et ses applications, 2nd ed. (1951, reissued 1965); Eugene P. Wigner, Group Theory and Its Application to the Quantum Mechanics of the Atomic Spectra, expanded and improved ed. (1959; originally published in German, 1931); Luther Pfahler Eisenhart, Continuous Groups of Transformations (1933, reissued 1961); E.T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed. (1937, reissued 1988); Elie Cartan, Oeuvres compltes, 3 vol. in 6 (195255, reprinted 3 vol. in 4, 1984), Leons sur les invariants intgraux (1922, reissued 1971), and Les Systmes differentiels extrieurs et leur applications gomtriques (1945, reissued 1971); Sophus Lie and Friedrich Engel, Theorie der Transformationsgruppen, 2nd ed., 3 vol. (1970); Louis Auslander and Calvin C. Moore, Unitary Representations of Solvable Lie Groups (1966); Dikran N. Dikranjan, Ivan R. Prodanov, and Luchezar N. Stoyanov, Topological Groups: Characters, Dualities, and Minimal Group Topologies (1989); and W.W. Comfort, Topological Groups, in Kenneth Kunen and Jerry E. Vaughan (eds.), Handbook of Set-Theoretic Topology (1984), pp. 11431263. George Daniel Mostow The Editors of the Encyclopdia BritannicaThe following works are references dealing with analysis on manifolds: Michael Spivak, Calculus on Manifolds (1965), an introductory text on theorems of advanced calculus in the context of manifolds; Friedrich Hirzebruch, Topological Methods in Algebraic Geometry, 3rd enlarged ed. (1966, reissued 1995; originally published in German, 2nd ed., 1962), a good survey of modern topological techniques as applied to algebraic geometry; John Milnor, Morse Theory (1963), a simple and good introduction to the theory of critical points; Jean Mawhin and Michel Willem, Critical Point Theory and Hamiltonian Systems (1989), a survey of applications of critical point techniques to Hamiltonian systems; Richard S. Palais and Chuu-lian Terng, Critical Point Theory and Submanifold Geometry (1988); W.V.D. Hodge, The Theory and Applications of Harmonic Integrals, 2nd ed. (1952, reissued 1989), a classic work; Georges de Rham, Differentiable Manifolds: Forms, Currents, Harmonic Forms (1984; originally published in French, 1955), a good treatment of the Hodgede Rham theory; Birger Iversen, Cauchy Residues and de Rham Homology, L'Enseignement Mathmatique, series 2, 35(12):117 (1989); Robert C. Gunning and Hugo Rossi, Analytic Functions of Several Complex Variables (1965), for sheaf theory applied to complex analysis; Masaki Kashiwara and Pierre Schapira, Sheaves on Manifolds (1990), for sheaf theory on real manifolds; Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (1978), a treatment of curvature and related topics, in particular spaces connected with Lie groups; and Robert Osserman, Curvatures in the Eighties, The American Mathematical Monthly, 97(8):731756 (1990). Sir Michael Francis Atiyah The Editors of the Encyclopdia Britannica