Ebook: Rational Homotopy Theory

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.

---

While rational homotopy theory is remarkably computational and simpler than ordinary homotopy theory, it is exactly this simplicity which makes it possible to address a number of fundamental questions in geometry and homotopy theory. The three main objectives of this book are:

* to provide a coherent, self-contained, and userfriendly introduction to the tools and techniques of rational homotopy theory

* to provide an account of the main structural theorems with proofs that are often new or much simpler than the original versions in the literature

* to illustrate both the use of technology and the consequences of the theorems in a rich variety of examples.

It should be emphasized that this book is about topological spaces and that examples and applications given throughout the book are largely drawn from topology. The reader should have a basic knowledge of the fundamental group and singular homology.