Félix, Yves; Oprea, John; Tanré, Daniel Algebraic models in geometry. (English) Zbl 1149.53002 Oxford Graduate Texts in Mathematics 17. Oxford: Oxford University Press (ISBN 978-0-19-920651-3/hbk; 978-0-19-920652-0/pbk). xxi, 460 p. (2008). Rational homotopy theory originated with the work of D. Quillen and D. Sullivan in the late 1960s. In particular, Sullivan defined tools and models for rational homotopy inspired by already existing geometrical objects. Moreover, he gave an explicit dictionary between his minimal models and spaces, and this facility of transition between algebra and topology has created many new topological and geometrical theorems in the last 30 years. An introduction to rational homotopy whose main applications were in algebraic topology was written some years ago. See for example Y. Felix, S. Halperin and J. C. Thomas [Rational Homotopy Theory, Volume 205 of Graduate Texts in Mathematics. Springer-Verlag, New York (2001; Zbl 0961.55002)]. Because of recent developments, it is clear that now is the time for a global presentation of some of the more representative geometrical applications of minimal models. That is the theme of book under review.The book has three parts. The first part (pp. 1–139), consisting of chapters 1–3, contains the classical theory and the main geometrical examples. Chapters 4–8 are the second part (pp. 145–348). Each of them is devoted to a particular topic in differential topology or geometry and they are, as the authors mention in the introduction, mostly independent. The third part is the florilege of chapter 9 (pp. 350–390) where the authors give short presentations of particular subjects, chosen to illustrate the evolution of applications of minimal models from the theory’s inception to the present day.There are three appendices about de Rham forms, spectral sequences and basic homotopy recollections. Each chapter encloses exercises. Reviewer: Grigori N. Milstein (Ekaterinburg) Cited in 93 Documents MathOverflow Questions: Equivariant cohomology of the complement to the arrangement \(\bigcup_{i\neq j}\vec x_i = \vec x_j\)? MSC: 53-02 Research exposition (monographs, survey articles) pertaining to differential geometry 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 58-02 Research exposition (monographs, survey articles) pertaining to global analysis 53C20 Global Riemannian geometry, including pinching 16E45 Differential graded algebras and applications (associative algebraic aspects) 53C22 Geodesics in global differential geometry Keywords:rational homotopy; minimal models; manifolds; Lie groups; symplectic geometry Citations:Zbl 0961.55002 PDFBibTeX XMLCite \textit{Y. Félix} et al., Algebraic models in geometry. Oxford: Oxford University Press (2008; Zbl 1149.53002)