The recognition of the domain of mathematics called fibre bundles
took place in the period 1935-1940. The first general definitions were
given by H. Whitney. His work and that of H, Hopf and E, Stiefel
demonstrated the importance of the subject for the applications of
topology to differential geometry. Since then, some seventy odd
papers dealing with bundles have appeared. The subject has attracted
general interest, for it contains some of the finest applications of
topology to other fields, and gives promise of many more. It also
marks a return of algebraic topology to its origin; and, after many
years of introspective development, a revitalization of the subject
from its roots in the study of classical manifolds.
No exposition of fibre bundles has appeared. The literature is in a
state of partial confusion, due mainly to the experimentation with
a variety of definitions of "fibre bundle." It has not been clear that
any one definition would suffice for all results. The derivations of
analogous conclusions from differing hypotheses have produced much
overlapping. Many "known" results have not been published. It
has been realized that certain standard theorems of topology are special
cases of propositions about bundles, but the generalized forms have not
been given.