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Rational Homotopy Theory

  • Yves Félix
  • Stephen Halperin
  • Jean-Claude Thomas
Textbook

Part of the Graduate Texts in Mathematics book series (GTM, volume 205)

Table of contents

  1. Front Matter
    Pages i-xxxii
  2. Homotopy Theory, Resolutions for Fibrations, and P-local Spaces

    1. Front Matter
      Pages xxxiii-xxxiii
    2. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 1-3
    3. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 4-22
    4. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 23-39
    5. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 40-50
    6. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 51-64
    7. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 65-67
    8. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 68-76
    9. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 77-87
    10. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 88-101
    11. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 102-114
  3. Sullivan Models

    1. Front Matter
      Pages N1-N1
    2. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 115-130
    3. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 131-137
    4. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 138-164
    5. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 165-180
    6. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 181-194
    7. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 195-222
    8. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 223-236
    9. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 237-259
  4. Graded Differential Algebra (continued)

    1. Front Matter
      Pages N3-N3
    2. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 260-267
    3. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 268-272
    4. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 273-282
  5. Lie Models

    1. Front Matter
      Pages N5-N5
    2. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 283-298
    3. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 299-312
    4. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 313-321
    5. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 322-336
    6. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 337-342
    7. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 343-350
  6. Rational Lusternik Schnirelmann Category

    1. Front Matter
      Pages N7-N7
    2. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 351-369
    3. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 370-380
    4. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 381-405
    5. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 406-414
    6. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 415-433
  7. The Rational Dichotomy: Elliptic and Hyperbolic Spaces and Other Applications

    1. Front Matter
      Pages N9-N9
    2. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 434-451
    3. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 452-463
    4. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 464-473
    5. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 474-491
    6. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 492-500
    7. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 501-510
    8. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 511-515
    9. Yves Félix, Stephen Halperin, Jean-Claude Thomas
      Pages 516-520
  8. Back Matter
    Pages 521-539

About this book

Introduction

as well as by the list of open problems in the final section of this monograph. The computational power of rational homotopy theory is due to the discovery by Quillen [135] and by Sullivan [144] of an explicit algebraic formulation. In each case the rational homotopy type of a topological space is the same as the isomorphism class of its algebraic model and the rational homotopy type of a continuous map is the same as the algebraic homotopy class of the correspond­ ing morphism between models. These models make the rational homology and homotopy of a space transparent. They also (in principle, always, and in prac­ tice, sometimes) enable the calculation of other homotopy invariants such as the cup product in cohomology, the Whitehead product in homotopy and rational Lusternik-Schnirelmann category. In its initial phase research in rational homotopy theory focused on the identi­ of these models. These included fication of rational homotopy invariants in terms the homotopy Lie algebra (the translation of the Whitehead product to the homo­ topy groups of the loop space OX under the isomorphism 11'+1 (X) ~ 1I.(OX», LS category and cone length. Since then, however, work has concentrated on the properties of these in­ variants, and has uncovered some truly remarkable, and previously unsuspected phenomena. For example • If X is an n-dimensional simply connected finite CW complex, then either its rational homotopy groups vanish in degrees 2': 2n, or else they grow exponentially.

Keywords

Algebraic topology CW complex Homotopy Homotopy group Loop group cofibration fibrations homology homotopy theory

Authors and affiliations

  • Yves Félix
    • 1
  • Stephen Halperin
    • 2
  • Jean-Claude Thomas
    • 3
  1. 1.Institut MathematiquesUniversite de Louvain La NeuveLouvain-la-NeuveBelgium
  2. 2.College of Computer, Mathematical, and Physical ScienceUniversity of MarylandCollege ParkUSA
  3. 3.Faculte des SciencesUniversite d’AngersAngersFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4613-0105-9
  • Copyright Information Springer Science+Business Media New York 2001
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-6516-0
  • Online ISBN 978-1-4613-0105-9
  • Series Print ISSN 0072-5285
  • Buy this book on publisher's site