TOPOLOGY, ALGEBRAIC

application of algebraic methods to the study of certain questions about topological spaces. The questions studied are usually of a qualitative nature, and the phenomena studied are usually globalthat is, they belong to the space as a whole but are not inherent in any small part of it. Certain intuitive notions can be analyzed, clarified, and classified topologically. A plane closed curve that does not cross itself has an inside and an outside; this intuitive idea can be made precise and can even be proved. The properties of knottedness and unknottedness of closed curves in three-dimensional space can be defined and analyzed, and this analysis is far from trivial. As indicated above, topology deals not only with spaces but also with mapsthat is, continuous functions from one space to another. The notation f : X Y means that f is a map taking points of X to points of Y. Consider, for example, the Brouwer fixed-point theorem, asserted by the 20th-century Dutch mathematician L.E.J. Brouwer: if Dn is the unit disc in coordinate n-space Rn, it can be defined by an inequality (see 88). The theorem states that any map from the disc to itself has a fixed point (see 89). This asserts that a given equation has a solution, a useful conclusion. More can be proved. With any map f : X X (intuitively a function that redistributes the points of X) satisfying suitable assumptions such as continuity, the Lefschetz number L(f) can be introduced. On the one hand, this is the total number of fixed points of f, counted with appropriate multiplicities; in particular, if L(f) is nonzero, then f has a fixed point. On the other hand, L(f) can be calculated without detailed knowledge of f; for example, if X = Dn, then L(f) = 1 for all f. The Lefschetz number is a first example of an algebraic invariant to be discussed immediately below. Additional reading Works on algebraic topology include Edwin H. Spanier, Algebraic Topology (1966, reissued 1981); and Dale Husemoller, Fibre Bundles, 3rd ed. (1994), two of the most generally useful standard texts: the scope of Spanier is wider; and, while it is not easy for a student to read Spanier as a first text, mathematically it is excellent; the scope of Husemoller is more limited. J. Frank Adams, Algebraic Topology: A Student's Guide (1972), has a much longer bibliography, with a survey article and a guide to the literature. H. Seifert and W. Threlfall, Seifert and Threlfall: A Textbook of Topology (1980; originally published in German, 1934), is a classic introduction to geometric and algebraic topology. More recent works include Gregory L. Naber, Topological Methods in Euclidean Spaces (1980); James R. Munkres, Elements of Algebraic Topology (1984); William S. Massey, A Basic Course in Algebraic Topology (1991); and Glen E. Bredon, Topology and Geometry (1993). The Editors of the Encyclopdia Britannica