In expositions of the elements of topology it is customary for homology to be
given a fundamental role. Since Poincare, 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
is, in the opinion of the present authors, 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 in for 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,f 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.
It was for these reasons that the present authors regarded as absolutely