Algebraic Models in Geometry (Oxford Graduate Texts in Mathematics, 17)

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. This Text Is Aimed At Both Geometers Needing The Tools Of Rational Homotopy Theory To Understand And Discover New Results Concerning Various Geometric Subjects, And Topologists Who Require Greater Breadth Of Knowledge About Geometric Applications Of The Algebra Of Homotopy Theory. Lie Groups And Homogeneous Spaces -- Minimal Models -- Manifolds -- Complex And Symplectic Manifolds -- Geodesics -- Curvature -- G-spaces -- Blow-ups And Intersection Products -- A Florilège Of Geometric Applications -- De Rham Forms -- Spectral Sequences -- Basic Homotopy Recollections. Yves Félix, John Oprea, Daniel Tanré. Includes Bibliographical References (p. [433]-450) And Index. Algebraic Models in Geometry......Page 4 Preface......Page 8 Contents......Page 16 1 Lie groups and homogeneous spaces......Page 24 2 Minimal models......Page 79 3 Manifolds......Page 127 4 Complex and symplectic manifolds......Page 168 5 Geodesics......Page 228 6 Curvature......Page 262 7 G-spaces......Page 294 8 Blow-ups and Intersection Products......Page 340 9 A Florilège of geometric applications......Page 373 Appendix A De Rham forms......Page 415 Appendix B Spectral sequences......Page 432 Appendix C Basic homotopy recollections......Page 446 References......Page 456 Index......Page 474