Lectures on Morse HomologyThis 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. |
Contents
II | 1 |
IV | 3 |
V | 4 |
VI | 5 |
VII | 6 |
VIII | 7 |
IX | 9 |
X | 10 |
XXXIV | 157 |
XXXVI | 165 |
XXXVII | 171 |
XXXVIII | 175 |
XXXIX | 195 |
XL | 196 |
XLI | 201 |
XLII | 207 |
XI | 11 |
XIII | 15 |
XIV | 20 |
XV | 21 |
XVI | 23 |
XVII | 31 |
XVIII | 45 |
XIX | 58 |
XX | 63 |
XXI | 73 |
XXII | 80 |
XXIII | 93 |
XXIV | 98 |
XXV | 111 |
XXVI | 116 |
XXVII | 127 |
XXVIII | 131 |
XXIX | 132 |
XXX | 134 |
XXXI | 137 |
XXXII | 143 |
XXXIII | 148 |
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Common terms and phrases
attaching maps bundle chain complex Chapter cofibration compact invariant set compact smooth Riemannian compute continuous map coordinates Corollary critical submanifolds CW-complex defined Definition deformation retract denote diffeomorphism dimension dimensional compact smooth eigenvalues exists finite dimensional compact finite dimensional smooth Floer homology function f Gn,n+k Gn,n+k(C gradient flow lines gradient vector field Grassmann manifold Hamiltonian Hence homotopy equivalence hyperbolic fixed point immersion implies index pairs intersection isolated compact invariant isomorphism Lemma Let f Lie algebra Lie group Manifold Theorem matrix metric g Morse function Morse Homology Morse Homology Theorem Morse-Bott function Morse-Smale function Morse-Smale-Witten chain complex orbit orientation points of f preceeding proof of Theorem Proposition prove Riemannian manifold Riemannian metric satisfies Show smooth function smooth manifold smooth map smooth Riemannian manifold subset tangent space Theorem Theorem topology transverse unstable manifolds zero
Popular passages
Page 312 - Helgason, Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics, vol. 80, Academic Press Inc.
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