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Rational homotopy theory. (English) Zbl 0961.55002
Graduate Texts in Mathematics. 205. New York, NY: Springer. xxxii, 535 p. (2001).
Rational homotopy theory begins with the discovery of Sullivan in the 1960’s of an underlying geometric construction: simply connected topological spaces and continuous maps can be rationalized to topological spaces \(X_\mathbb{Q}\) and to maps \(f_\mathbb{Q}: X_\mathbb{Q}\to Y_\mathbb{Q}\) such that \(H_*(X_\mathbb{Q})= H_* (X;\mathbb{Q})\) and \(\pi_*(X_\mathbb{Q}) =\pi_*(X) \otimes\mathbb{Q}\). So, this theory replaces spaces with algebraic models, the rational homotopy theory having the advantage of being computational and simpler than ordinary homotopy theory.
This is a monograph written for graduate students who have already encountered the fundamental group and singular homology and the authors hope that the results described will be accessible to mathematicians in other parts of the subject, too.
This book is a coherent, self-contained, reasonably complete and usable description of the tools and techniques of rational homotopy, an account of many of the main structural theorems with proofs (often new and considerably simplified from the original versions) and illustrates the use of the technology and the consequences of the theorems in a rich variety of examples.
In this monograph the authors consider three differential graded categories: modules over a differential graded algebra \((R,d)\), commutative cochain algebras and differential graded Lie algebras (dgl) and construct a semi-free resolution of a module over \((R,d)\), a Sullivan model of a commutative cochain algebra and a free Lie model of a dgl which is a quasi-isomorphism from a dgl that is free as a graded Lie algebra. These models are the main algebraic tools of the subject.
The monograph is divided into forty sections grouped into six parts. Each section presents a single aspect of the subject organized into a number of distinct topics and described in an introduction at the start of the section.
Part I is on homotopy theory, resolutions for fibrations and \(P\)-local spaces, being a self-contained short course in homotopy theory. The authors present CW-complexes and (co)fibrations, constructing a CW-model for any topological space and establishing Whitehead’s homotopy lifting theorem. Then they introduce the first algebraic model: the semifree resolution of a module over a differential graded algebra (dga). For \(f\) a fibration, using a semifree resolution, they compute the cohomology of the fibre. Then they consider the case when the action is that of a principal \(G\)-fibration \(X\to Y\) and use a semifree resolution to compute \(H_*(Y)\).
The core of the monograph is Part II: “Sullivan models”. The authors identify the rational homotopy theory of simply connected spaces with the homotopy theory of commutative cochain algebras. They establish bijections between rational homotopy types of spaces and isomorphism classes of minimal Sullivan algebras and between homotopy classes of maps between rational spaces and homotopy classes of maps between minimal Sullivan algebras (they restrict to spaces and cochain algebras that are simply connected with cohomology of finite type).
Part III continues with graded differential algebras (spectral sequences, the bar and cobar constructions, projective resolutions of graded modules).
Part IV is on Lie models. The authors first introduce the graded Lie algebras and their universal enveloping algebras and give the examples of the homotopy Lie algebra \(L_X= \pi_*(\Omega X)\otimes k\) of a simply connected topological space \(X\) and the homotopy Lie algebra \(L\) of a minimal Sullivan algebra \((\Lambda V,d)\). If \((\Lambda V,d)\) is the Sullivan model for \(X\) then \(L_X\cong L\). If \((L,d)\) is a free Lie model for \(X\) then there is a chain algebra quasi-isomorphism \(U(L,d) \simeq C_*(\Omega X,k)\) which preserves the diagonals up to dga homotopy.
Part V “Rational Lyusternik-Shnirel’man category” begins with the presentation of the main properties of LS category for ‘ordinary’ topological spaces and continues with LS category of Sullivan algebras. A key point is the mapping theorem: “for a continuous map \(f:X\to Y\) between simply connected spaces, \(\pi_* (f)\otimes \mathbb{Q}\) injective \(\Rightarrow \text{cat}_0 X\leq\text{cat}_0 Y\)” and the Hess theorem. The Jessup theorem gives circumstances under which the rational LS category of a fibre must be strictly less than that of the total space of a fibration.
In the final Part VI: “Rational Dichotomy: Elliptic and Hyperbolic Spaces AND Other Applications” the authors use the rational homotopy theory to derive the results referred on the structure of \(H_* (\Omega X;k)\), when \(X\) is a simply connected finite CW-complex.
This interesting monograph on rational homotopy theory brings together the work of many researchers, accomplished as a co-operative effort for the most part over the last thirty years.

MSC:
55-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology
55P62 Rational homotopy theory
MathOverflow Questions:
Homology of homotopy fiber of inclusion
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