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© 2002

Algebraic Topology from a Homotopical Viewpoint

Textbook

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xxix
  2. Marcelo Aguilar, Samuel Gitler, Carlos Prieto
    Pages 1-7
  3. Marcelo Aguilar, Samuel Gitler, Carlos Prieto
    Pages 9-57
  4. Marcelo Aguilar, Samuel Gitler, Carlos Prieto
    Pages 59-88
  5. Marcelo Aguilar, Samuel Gitler, Carlos Prieto
    Pages 89-147
  6. Marcelo Aguilar, Samuel Gitler, Carlos Prieto
    Pages 149-188
  7. Marcelo Aguilar, Samuel Gitler, Carlos Prieto
    Pages 189-226
  8. Marcelo Aguilar, Samuel Gitler, Carlos Prieto
    Pages 227-258
  9. Marcelo Aguilar, Samuel Gitler, Carlos Prieto
    Pages 259-287
  10. Marcelo Aguilar, Samuel Gitler, Carlos Prieto
    Pages 289-307
  11. Marcelo Aguilar, Samuel Gitler, Carlos Prieto
    Pages 309-329
  12. Marcelo Aguilar, Samuel Gitler, Carlos Prieto
    Pages 331-382
  13. Marcelo Aguilar, Samuel Gitler, Carlos Prieto
    Pages 383-419
  14. Back Matter
    Pages 421-479

About this book

Introduction

The purpose of this book is to introduce algebraic topology using the novel approach of homotopy theory, an approach with clear applications in algebraic geometry as understood by Lawson and Voevodsky. This method allows the authors to cover the material more efficiently than the more common method using homological algebra. The basic concepts of homotopy theory, such as fibrations and cofibrations, are used to construct singular homology and cohomology, as well as K-theory. Throughout the text many other fundamental concepts are introduced, including the construction of the characteristic classes of vector bundles. Although functors appear constantly throughout the text, no knowledge about category theory is expected from the reader. This book is intended for advanced undergraduates and graduate students with a basic knowledge of point set topology as well as group theory and can be used in a two semester course.
Marcelo Aguilar and Carlos Prieto are Professors at the Instituto de Matemticas, Universidad Nacional Autónoma de México, and Samuel Gitler is a member of El Colegio Nacional and professor at the Centro de Investigación y Estudios Avanzados del IPN.

Keywords

Adams operation Brown representability Category theory Characteristic class Homotopy group algebraic invariant cofibration cohomology group fibrations function space group theory homological algebra homotopy extension homotopy theory point set topology

Authors and affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMéxico, DFMéxico
  2. 2.Departamento de MatemáticasCentro de Investigación y de Estudios AvanzadosMéxico, DFMéxico

Bibliographic information

  • Book Title Algebraic Topology from a Homotopical Viewpoint
  • Authors Marcelo Aguilar
    Samuel Gitler
    Carlos Prieto
  • Series Title Universitext
  • DOI https://doi.org/10.1007/b97586
  • Copyright Information Springer-Verlag New York 2002
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-0-387-95450-9
  • Softcover ISBN 978-1-4419-3005-7
  • eBook ISBN 978-0-387-22489-3
  • Series ISSN 0172-5939
  • Series E-ISSN 2191-6675
  • Edition Number 1
  • Number of Pages XXIX, 479
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Additional Information Original Spanish edition published by McGraw-Hill Interamericana Editores, 1998
  • Topics Topology
    Algebraic Topology
  • Buy this book on publisher's site

Reviews

From the reviews:

"This is a basic course on algebraic topology and … it is excellently written and composed and can be strongly recommended to anybody wishing to learn the field. The reader can find many examples, calculations, and also a number of exercises. … The book has two appendices, … references consisting of 83 items and a long list of symbols. There are many remarks and comments making the orientation of the reader in the field of algebraic topology easier … ." (EMS, June, 2004)

"The modus operandi of algebraic topology is to associate algebraic invariants, such as groups or rings, to a topological space in such a way that equivalent spaces exhibit isomorphic invariants; here, ‘equivalent’ may be chosen to fit the geometry of the problem. In this book, ‘equivalent’ means homotopy equivalent … . For its clarity and directness, a welcome addition to advanced mathematics collections. Upper-division undergraduates through faculty." (J. McCleary, CHOICE, December, 2002)

"The purpose of this book is to introduce algebraic topology using the novel approach of homotopy theory … . The basic concepts of homotopy theory … are used to construct singular homology and cohomology, as well as K-theory. … Throughout the text many other fundamental concepts are introduced … ." (L’Enseignement Mathematique, Vol. 48 (3-4), 2002)

“The book of Aguilar, Gitler and Prieto gives an interesting new approach for a first course in algebraic topology. … What is also very interesting is the fact that the book contains a detailed presentation of some deep results of algebraic topology not usually covered by a first book in algebraic topology … . The text is carefully well written. … the student in algebraic topology will find in the book a lot of interesting well-exposed material.” (Yves Félix, Bulletin of the Belgian Mathematical Society, 2007)