Algebraic models in geometry

Rational homotopy is a very powerful tool for differential topology and geometry. This text aims to provide graduates and researchers with the tools necessary for the use of rational homotopy in geometry. Algebraic Models in Geometry has been written for topologists who are drawn to geometrical problems amenable to topological methods and also for geometers who are faced with problems requiring topological approaches and thus need a simple and concrete introduction to rational homotopy. This is essentially a book of applications. Geodesics, curvature, embeddings of manifolds, blow-ups, complex and Kähler manifolds, symplectic geometry, torus actions, configurations and arrangements are all covered. The chapters related to these subjects act as an introduction to the topic, a survey, and a guide to the literature. But no matter what the particular subject is, the central theme of the book persists, namely, there is a beautiful connection between geometry and rational homotopy which both serves to solve geometric problems and spur the development of topological methods.

1. Lie Groups and Homogeneous Spaces. 2. Minimal Models. 3. Manifolds. 4. Complex and Symplectic Manifolds. 5. Geodesics. 6. Curvature. 7. G-Spaces. 8. Blow-ups and Intersection Products. 9. A Florilege of Geometric Applications. Appendices. A. De Rham Forms. B. Spectral Sequences. C. Basic Homotopy Recollections.