This book is based on the lecture notes from a course we taught at Penn State University during the fall of 2002. The main goal of the course was to give a complete and detailed proof of the Morse Homology Theorem (Theoยญ rem 7.4) at a level appropriate for second year graduate students. The course was designed for students who had a basic understanding of singular homolยญ ogy, CW-complexes, applications of the existence and uniqueness theorem for O.D.E.s to vector fields on smooth Riemannian manifolds, and Sard's Theoยญ rem. We would like to thank the following students for their participation in the course and their help proofreading early versions of this manuscript: James Barton, Shantanu Dave, Svetlana Krat, Viet-Trung Luu, and Chris Saunders. We would especially like to thank Chris Saunders for his dedication and enยญ thusiasm concerning this project and the many helpful suggestions he made throughout the development of this text. We would also like to thank Bob Wells for sharing with us his extensive knowledge of CW-complexes, Morse theory, and singular homology. Chapters 3 and 6, in particular, benefited significantly from the many insightful converยญ sations we had with Bob Wells concerning a Morse function and its associated CW-complex
CONTENT
1. Introduction -- 2. The CW-Homology Theorem -- 3. Basic Morse Theory -- 4. The Stable/Unstable Manifold Theorem -- 5. Basic Differential Topology -- 6. Morse-Smale Functions -- 7. The Morse Homology Theorem -- 8. Morse Theory On Grassmann Manifolds -- 9. An Overview of Floer Homology Theories -- Hints and References for Selected Problems -- Symbol Index
Mathematics
Topological groups
Lie groups
Global analysis (Mathematics)
Manifolds (Mathematics)
Differential equations
Algebraic topology
Complex manifolds
Mathematics
Global Analysis and Analysis on Manifolds
Manifolds and Cell Complexes (incl. Diff.Topology)
