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Rational Homotopy Theory

  • Textbook
  • © 2001

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

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About this book

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.

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Keywords

Table of contents (40 chapters)

  1. Homotopy Theory, Resolutions for Fibrations, and P-local Spaces

  2. Sullivan Models

Reviews

From the reviews:

MATHEMATICAL REVIEWS

"In 535 pages, the authors give a complete and thorough development of rational homotopy theory as well as a review (of virtually) all relevant notions of from basic homotopy theory and homological algebra. This is a truly remarkable achievement, for the subject comes in many guises."

 

Y. Felix, S. Halperin, and J.-C. Thomas

Rational Homotopy Theory

"A complete and thorough development of rational homotopy theory as well as a review of (virtually) all relevant notions from basic homotopy theory and homological algebra. This is truly a magnificent achievement . . . a true appreciation for the goals and techniques of rational homotopy theory, as well as an effective toolkit for explicit computation of examples throughout algebraic topology."

—AMERICAN MATHEMATICAL SOCIETY

Authors and Affiliations

  • Institut Mathematiques, Universite de Louvain La Neuve, Louvain-la-Neuve, Belgium

    Yves Félix

  • College of Computer, Mathematical, and Physical Science, University of Maryland, College Park, USA

    Stephen Halperin

  • Faculte des Sciences, Universite d’Angers, Angers, France

    Jean-Claude Thomas

Bibliographic Information

  • Book Title: Rational Homotopy Theory

  • Authors: Yves Félix, Stephen Halperin, Jean-Claude Thomas

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4613-0105-9

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 2001

  • Hardcover ISBN: 978-0-387-95068-6Published: 21 December 2000

  • Softcover ISBN: 978-1-4612-6516-0Published: 13 October 2012

  • eBook ISBN: 978-1-4613-0105-9Published: 06 December 2012

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 1

  • Number of Pages: XXXIII, 539

  • Topics: Algebraic Topology

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