Since the early part of the 20th century, topology has gradually spread to many other branches of mathematics, and this book demonstrates how the subject continues to play a central role in the field. Written by a world-renowned mathematician, this classic text traces the history of algebraic topology beginning with its creation in the early 1900s and describes in detail the important theories that were discovered before 1960. Through the work of Poincar, de Rham, Cartan, Hureqicz, and many others, this historical book also focuses on the emergence of new ideas and methods that have led 21st-century mathematicians towards new research directions. ***************************** This book is a well-informed and detailed analysis of the problems and development of algebraic topology, from Poincar and Brouwer to Serre, Adams, and Thom. The author has examined each significant paper along this route and describes the steps and strategyof its proofs and its relation to other work. Previously, the history of the many technical developments of 20th-century mathematics had seemed to present insuperable obstacles to scholarship. This book demonstrates in the case of topology how these obstacles can be overcome, with enlightening results.... Within its chosen boundaries the coverage of this book is superb. Read it! MathSciNet [The author] traces the development of algebraic and differential topology from the innovative work by Poincar at the turn of the century to the period around 1960. [He] has given a superb account of the growth of these fields. The details are interwoven with the narrative in a very pleasant fashion. [The author] has previous written histories of functional analysis and of algebraic geometry, but neither book was on such a grand scale as this one. He has made it possible to trace the important steps in the growth of algebraic and differential topology, and to admire the hard work and major advances made by the founders. Zentralblatt MATH.