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
background material.
In order to emphasize the geometrical and intuitive aspects of
differential topology, I have avoided the use of algebraic topolog>\ 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 Uke homology theor>^ 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.