Algebraic Topology: A First Course (Repost)

This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the relations 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, differential topology, etc.), we concentrate our attention on concrete problems in low dimensions, introducing only as much algebraic machinery 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 topologists-without, we hope, discouraging budding topologists. We also feel that this approach is in better harmony with the historical development 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: understanding the relation between homology and integration, first on plane domains, later on Riemann surfaces and in higher dimensions; winding numbers and degrees of mappings, fixed-point theorems; applications such as the Jordan curve theorem, invariance of domain; indices of vector fields and Euler characteristics; fundamental groups.

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