By Michèle Audin, Mihai Damian
This publication is an creation to trendy tools of symplectic topology. it truly is dedicated to explaining the answer of an incredible challenge originating from classical mechanics: the 'Arnold conjecture', which asserts that the variety of 1-periodic trajectories of a non-degenerate Hamiltonian approach is bounded less than through the measurement of the homology of the underlying manifold.
The first half is an intensive creation to Morse idea, a basic software of differential topology. It defines the Morse complicated and the Morse homology, and develops a few of their applications.
Morse homology additionally serves an easy version for Floer homology, that is coated within the moment half. Floer homology is an infinite-dimensional analogue of Morse homology. Its involvement has been the most important within the fresh achievements in symplectic geometry and specifically within the evidence of the Arnold conjecture. The development blocks of Floer homology are extra problematic and suggest using extra subtle analytical tools, all of that are defined during this moment part.
The 3 appendices current a couple of must haves in differential geometry, algebraic topology and analysis.
The booklet originated in a graduate direction given at Strasbourg college, and incorporates a huge diversity of figures and workouts. Morse thought and Floer Homology can be fairly invaluable for graduate and postgraduate students.
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Extra info for Morse Theory and Floer Homology (Universitext)
Seventy nine seventy nine eighty one eighty three eighty four 87 ninety one ninety eight a hundred and one a hundred and ten 121 half II The Arnold Conjecture, Floer Homology advent to half II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 five What you must learn about Symplectic Geometry . . five. 1 Symplectic Vector areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . five. 2 Symplectic Manifolds, Definition . . . . . . . . . . . . . . . . . . . . . . . . five. three Examples of Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . . five. four Hamiltonian Vector Fields, Hamiltonian platforms . . . . . . . . . . five. five advanced buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . five. 6 The Symplectic team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 129 a hundred thirty 131 134 139 a hundred and forty four 6 The Arnold Conjecture and the Floer Equation . . . . . . . . . . 6. 1 The Arnold Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. 2 define of the facts, Floer Homology . . . . . . . . . . . . . . . . . . . . 6. three The motion practical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. four The Gradient, the Floer Equation . . . . . . . . . . . . . . . . . . . . . . . 6. five the gap of recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. 6 facts of the Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. 7 Appendix: features, Closed kinds, Covers . . . . . . . . . . . . . . . 6. eight Appendix: constitution of a Banach Manifold on LW . . . . . . . . 151 151 154 156 162 164 a hundred seventy five 183 186 7 The Geometry of the Symplectic team, the Maslov Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. 1 towards the Definition of the Index . . . . . . . . . . . . . . . . . . . . . . 7. 2 The Maslov Index of a course . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. three Appendix: building and homes of ρ . . . . . . . . . . . . . . 189 189 196 202 eight Linearization and Transversality . . . . . . . . . . . . . . . . . . . . . . . . . eight. 1 the implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eight. 2 The Banach Manifold P1,p (x, y) . . . . . . . . . . . . . . . . . . . . . . . . . eight. three the distance of Perturbations of H . . . . . . . . . . . . . . . . . . . . . . . . eight. four Linearization of the Floer Equation: Computation of the Differential of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 221 225 230 234 Contents xiii eight. five The Transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eight. 6 The recommendations of the Floer Equation Are “Somewhere Injective” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eight. 7 The Fredholm estate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eight. eight Computing the Index of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eight. nine The Exponential Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 269 285 296 nine areas of Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nine. 1 The areas of Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nine. 2 damaged Trajectories, Gluing: Statements . . . . . . . . . . . . . . . . . . nine. three Pre-gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nine. four development of ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nine. five houses of ψ: ψ Is an Immersion . . . . . . . . . . . . . . . . . . . . . . nine. 6 homes of ψ: area of expertise of the Gluing . . . . . . . . . . . . . . . . 305 305 311 313 315 333 334 10 From Floer to Morse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. 1 the implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. 2 The Linearization of the move of a Pseudo-Gradient box, evidence of Theorem 10. 1. three . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. three facts of Theorem 10. 1. 2 (Regularity) . . . . . . . . . . . . . . . . . . . . 10.