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Introduction to Homotopy Theory
Paul Selick, University of Toronto, ON, Canada
A co-publication of the AMS and Fields Institute.
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Fields Institute Monographs
1997; 188 pp; softcover
Volume: 9
ISBN-10: 0-8218-4436-9
ISBN-13: 978-0-8218-4436-6
List Price: US$57
Member Price: US$45.60
Order Code: FIM/9.S
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This text is based on a one-semester graduate course taught by the author at The Fields Institute in fall 1995 as part of the homotopy theory program which constituted the Institute's major program that year. The intent of the course was to bring graduate students who had completed a first course in algebraic topology to the point where they could understand research lectures in homotopy theory and to prepare them for the other, more specialized graduate courses being held in conjunction with the program. The notes are divided into two parts: prerequisites and the course proper.

Part I, the prerequisites, contains a review of material often taught in a first course in algebraic topology. It should provide a useful summary for students and non-specialists who are interested in learning the basics of algebraic topology. Included are some basic category theory, point set topology, the fundamental group, homological algebra, singular and cellular homology, and Poincaré duality.

Part II covers fibrations and cofibrations, Hurewicz and cellular approximation theorems, topics in classical homotopy theory, simplicial sets, fiber bundles, Hopf algebras, spectral sequences, localization, generalized homology and cohomology operations.

This book collects in one place the material that a researcher in algebraic topology must know. The author has attempted to make this text a self-contained exposition. Precise statements and proofs are given of "folk" theorems which are difficult to find or do not exist in the literature.

Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

Reviews

"A comprehensive introduction to many topics in algebraic topology up to the tools currently used in research ... the author has pulled off a real tour de force ... could serve as an excellent route into some of the most exciting topics in mathematics."

-- Zentralblatt MATH

"Shows a well-marked trail to homotopy theory with plenty of beautiful scenery worth visiting, while leaving to the student the task of hiking along it. Most of us wish we had had this book when we were students."

-- Mathematical Reviews

Table of Contents

  • Prerequisites from category theory
  • Prerequisites from point set topology
  • The fundamental group
  • Homological algebra
  • Homology of spaces
  • Manifolds
  • Higher homotopy theory
  • Simplicial sets
  • Fibre bundles and classifying spaces
  • Hopf algebras and graded Lie algebras
  • Spectral sequences
  • Localization and completion
  • Generalized homology and stable homotopy
  • Cohomology operations and the Steenrod algebra
  • Bibliography
  • Index
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