A history of algebraic and differential topology, 1900 – 1960

Jean Dieudonné (auth.)0817649069, 9780817649067

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 Poincaré, 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.

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This book is a well-informed and detailed analysis of the problems and development of algebraic topology, from Poincaré and Brouwer to Serre, Adams, and Thom. The author has examined each significant paper along this route and describes the steps and strategy of 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 Poincaré 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

Table of contents :
Front Matter….Pages i-xxii
Front Matter….Pages 1-1
Introduction….Pages 3-14
The Work of Poincaré….Pages 15-35
The Build-Up of “Classical” Homology….Pages 36-59
The Beginnings of Differential Topology….Pages 60-66
The Various Homology and Cohomology Theories….Pages 67-157
Front Matter….Pages 1-1
Introduction….Pages 161-166
The Concept of Degree….Pages 167-181
Dimension Theory and Separation Theorems….Pages 182-196
Fixed Points….Pages 197-203
Local Homological Properties….Pages 204-213
Quotient Spaces and Their Homology….Pages 214-231
Homolagy of Groups and Homogeneous Spaces….Pages 232-248
Applications of Homology to Geometry and Analysis….Pages 249-269
Front Matter….Pages 1-1
Introduction….Pages 273-292
Fundamental Group and Covering Spaces….Pages 293-310
Elementary Notions and Early Results in Homotopy Theory….Pages 311-384
Fibrations….Pages 385-420
Homology of Fibrations….Pages 421-452
Sophisticated Relations between Homotopy and Homology….Pages 453-509
Cohomology Operations….Pages 510-554
Front Matter….Pages 1-1
Generalized Homology and Cohomology….Pages 555-611
Back Matter….Pages 1-37