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 (Theo 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 homol 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 Theo 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 en 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 conver sations we had with Bob Wells concerning a Morse function and its associated CW-complex.

Includes supplementary material: sn.pub/extras

Autorentext

Augustin Banyaga is a Professor of Mathematics and a Distinguished Senior Scholar at Penn State University in the Eberly College of Science and a Fellow of the African Academy of Sciences. He has authored at least 70 peer reviewed papers and 3 books, including Lectures on Morse Homology published by Springer. David Hurtubise is a Professor of Mathematics at Penn State Altoona. He has authored at least 14 peer reviewed papers, 140 Mathematical Reviews, 45 Zentralblatt Reviews, and the book Lectures on Morse Homology published by Springer. Peter Spaeth is a Senior Research Scientist at NASA's Langley Research Center. He has authored over 20 peer reviewed papers in mathematics, materials science, and nondestructive evaluation. In 2023 he was awarded the NASA Early Career Achievement Medal.

Inhalt

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.