This book presents some of the basic topological ideas used in studying
differentiable manifolds and maps. Mathematical prerequisites have been
kept to a minimum; the standard course in analysis and general topology
is adequate preparation. An appendix briefly summarizes some of the
back- ground material. In order to emphasize the geometrical and
intuitive aspects of differen- tial topology, I have avoided the use of
algebraic topology, except in a few isolated places that can easily be
skipped. For the same reason I make no use of differential forms or
tensors. In my view, advanced algebraic techniques like homology theory
are better understood after one has seen several examples of how the raw
material of geometry and analysis is distilled down to numerical
invariants, such as those developed in this book: the degree of a map,
the Euler number of a vector bundle, the genus of a surface, the
cobordism class of a manifold, and so forth. With these as motivating
examples, the use of homology and homotopy theory in topology should
seem quite natural. There are hundreds of exercises, ranging in
difficulty from the routine to the unsolved. While these provide
examples and further developments of the theory, they are only rarely
relied on in the proofs of theorems.