Rational Homotopy Theory

1. Front Matter

   Pages i-xxxii

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

   1. Front Matter

      Pages xxxiii-xxxiii

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   2. Topological spaces

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 1-3

   3. CW complexes, homotopy groups and cofibrations

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 4-22

   4. Fibrations and topological monoids

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 23-39

   5. Graded (differential) algebra

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 40-50

   6. Singular chains, homology and Eilenberg- MacLane spaces

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 51-64

   7. The cochain algebra C ^*(X;\(\Bbbk \))

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 65-67

   8. (R, d)-modules and semifree resolutions

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 68-76

   9. Semifree cochain models of a fibration

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 77-87

   10. Semifree chain models of a G—fibration

       • Yves Félix, Stephen Halperin, Jean-Claude Thomas

       Pages 88-101

   11. p—local and rational spaces

       • Yves Félix, Stephen Halperin, Jean-Claude Thomas

       Pages 102-114

3. Sullivan Models

   1. Front Matter

      Pages N1-N1

      PDF

   2. Commutative cochain algebras for spaces and simplicial sets

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 115-130

   3. Smooth Differential Forms

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 131-137

   4. Sullivan models

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 138-164

   5. Adjunction spaces, homotopy groups and Whitehead products

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 165-180

   6. Relative Sullivan algebras

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 181-194

   7. Fibrations, homotopy groups and Lie group actions

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 195-222

   8. The loop space homology algebra

      • Yves Félix, Stephen Halperin, Jean-Claude Thomas

      Pages 223-236

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.

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

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

• 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

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