Original French edition published by EDP Sciences, Les Ulis Cedex A, France, 2010
2014, XIV, 596 p. 114 illus.
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Provides a unified presentation of Morse theory and Floer homology that is unique in the English language
Explains all the required background on symplectic geometry, differential geometry, algebraic topology and analysis
This book is an introduction to modern methods of symplectic topology. It is devoted to explaining the solution of an important problem originating from classical mechanics: the 'Arnold conjecture', which asserts that the number of 1-periodic trajectories of a non-degenerate Hamiltonian system is bounded below by the dimension of the homology of the underlying manifold.
The first part is a thorough introduction to Morse theory, a fundamental tool of differential topology. It defines the Morse complex and the Morse homology, and develops some of their applications.
Morse homology also serves a simple model for Floer homology, which is covered in the second part. Floer homology is an infinite-dimensional analogue of Morse homology. Its involvement has been crucial in the recent achievements in symplectic geometry and in particular in the proof of the Arnold conjecture. The building blocks of Floer homology are more intricate and imply the use of more sophisticated analytical methods, all of which are explained in this second part.
The three appendices present a few prerequisites in differential geometry, algebraic topology and analysis.
The book originated in a graduate course given at Strasbourg University, and contains a large range of figures and exercises. Morse Theory and Floer Homology will be particularly helpful for graduate and postgraduate students.
Content Level »Graduate
Keywords »Arnold Conjecture - Floer Complex - Floer Homology - Gluing - Hamiltonian System - Maslov Index - Morse Complex - Morse Homology - Morse Inequalities - Morse Theory - Symplectic Group - Symplectic Manifold