To the Teacher. This book is designed to introduce a student to some of
the important ideas of algebraic topology by emphasizing the re- lations
of these ideas with other areas of mathematics. Rather than choosing one
point of view of modem topology (homotopy theory, simplicial complexes,
singular theory, axiomatic homology, differ- ential topology, etc.), we
concentrate our attention on concrete prob- lems in low dimensions,
introducing only as much algebraic machin- ery as necessary for the
problems we meet. This makes it possible to see a wider variety of
important features of the subject than is usual in a beginning text. The
book is designed for students of mathematics or science who are not
aiming to become practicing algebraic topol- ogists-without, we hope,
discouraging budding topologists. We also feel that this approach is in
better harmony with the historical devel- opment of the subject. What
would we like a student to know after a first course in to- pology
(assuming we reject the answer: half of what one would like the student
to know after a second course in topology)? Our answers to this have
guided the choice of material, which includes: under- standing the
relation between homology and integration, first on plane domains, later
on Riemann surfaces and in higher dimensions; wind- ing numbers and
degrees of mappings, fixed-point theorems; appli- cations such as the
Jordan curve theorem, invariance of domain; in- dices of vector fields
and Euler characteristics; fundamental groups
