Modern Geometry: Methods and Applications, Volume 3

Author: B.A. Dubrovin et al.
Publisher:
Publication Date: 1999-11-01
Features:
In expositions of the elements of topology, it is customary for homology to be given a fundamental role. Since Poincaré, who laid the foundations of topology, homology theory has been regarded as the appropriate primary basis for an introduction to the methods of algebraic topology. From homotopy theory, on the other hand, only the fundamental group and covering-space theory have traditionally been included among the basic initial concepts. Essentially, all elementary classical textbooks of topology (the best of which, in the opinion of the present authors, is Seifert and Threlfall's A Textbook of Topology) begin with the homology theory of one or another class of complexes. Only at a later stage (and then still from a homological point of view) do fibre-space theory and the general problem of classifying homotopy classes of maps (homotopy theory) come into consideration. However, methods developed in investigating the topology of differentiable manifolds, and intensively elaborated from the 1930s onwards by Whitney and others, now permit a wholesale reorganization of the standard exposition of the fundamentals of modern topology. In this new approach, which resembles more that of classical analysis, these fundamentals turn out to consist primarily of the elementary theory of smooth manifolds, homotopy theory based on these, and smooth fibre spaces. Furthermore, over the decade of the 1970s, it became clear that exactly this complex of topological ideas and methods were proving to be fundamentally applicable in various areas of modern physics. This book is in English.