Download E-books Morse Theory and Floer Homology (Universitext) PDF

By Michèle Audin, Mihai Damian

This booklet is an advent to trendy equipment of symplectic topology. it truly is dedicated to explaining the answer of a tremendous challenge originating from classical mechanics: the 'Arnold conjecture', which asserts that the variety of 1-periodic trajectories of a non-degenerate Hamiltonian process is bounded lower than via the size of the homology of the underlying manifold.

The first half is a radical advent to Morse thought, a basic software of differential topology. It defines the Morse advanced and the Morse homology, and develops a few of their applications.

Morse homology additionally serves an easy version for Floer homology, that's 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 particularly within the facts of the Arnold conjecture. The development blocks of Floer homology are extra tricky and suggest using extra refined analytical equipment, 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 publication originated in a graduate path given at Strasbourg collage, and features a huge variety of figures and routines. Morse idea and Floer Homology should be really valuable for graduate and postgraduate students.

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Four. 6 Functoriality of the Morse Homology . . . . . . . . . . . . . . . . . . . . . four. 7 lengthy targeted series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . four. eight purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . four. nine Appendix: The Morse Homology is the mobile Homology . . workouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seventy nine seventy nine eighty one eighty three eighty four 87 ninety one ninety eight one hundred and one a hundred and ten 121 half II The Arnold Conjecture, Floer Homology creation to half II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 five What you want to find out 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 structures . . . . . . . . . . five. five advanced buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . five. 6 The Symplectic team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 129 one 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 evidence, Floer Homology . . . . . . . . . . . . . . . . . . . . 6. three The motion sensible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. four The Gradient, the Floer Equation . . . . . . . . . . . . . . . . . . . . . . . 6. five the gap of recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. 6 evidence of the Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. 7 Appendix: features, Closed varieties, 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 crew, the Maslov Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. 1 towards the Definition of the Index . . . . . . . . . . . . . . . . . . . . . . 7. 2 The Maslov Index of a direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. three Appendix: development and homes of ρ . . . . . . . . . . . . . . 189 189 196 202 eight Linearization and Transversality . . . . . . . . . . . . . . . . . . . . . . . . . eight. 1 the consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eight. 2 The Banach Manifold P1,p (x, y) . . . . . . . . . . . . . . . . . . . . . . . . . eight. three the gap 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 homes of ψ: ψ Is an Immersion . . . . . . . . . . . . . . . . . . . . . . nine. 6 houses of ψ: area of expertise of the Gluing . . . . . . . . . . . . . . . . 305 305 311 313 315 333 334 10 From Floer to Morse .

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