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.

every day, were transformed into objects and techniques of Floer homology. The charm, or one of vii viii Preface the charms, and the strength, of this theory, lie in the fact that in addition to geometry and topology, it uses much analysis, Fredholm operators and Sobolev spaces. Explaining this to genuine students is not an easy task. This is why we decided to continue writing lecture notes. Even though many works of research have used and still use these techniques, and many students need

have a vector ﬁeld Xr−1 such that the stable manifold of cr−1 is transversal to all the unstable manifolds. We apply the lemma to the vector ﬁeld Xr−1 and j = r. This gives a vector ﬁeld Xr that, in particular (this is the ﬁrst property given by the lemma), coincides with Xr−1 outside of the narrow strip where αr + ε ≤ f ≤ αr + 2ε. Moreover, since for every p ≤ r − 1, the stable manifold of cp for Xr−1 lies above this strip, the stable manifold is the same for Xr−1 as it is for Xr (see Figure

g(iZ) = ig(Z) for every Z (for a matrix A, this means that AJ0 = J0 A). (2) g ∈ Sp(2n) if and only if g preserves ω, that is, if and only if ω(gZ, gZ ) = ω(Z, Z ) for all Z and Z . For a matrix A, this means that t AJ0 A = J0 . (3) g ∈ O(2n) if and only if (gZ, gZ ) = (Z, Z ). For a matrix A, this means that tAA = Id. Any two of these conditions always imply the third one: • (2) and (3) imply that gZ, gZ = Z, Z and therefore that g ∈ U(n) ⊂ GL(n; C). • (3) and (1) imply that ω(gZ, gZ ) =

does not depend on s. . . and u is a solution of the Floer equation, that is, if and only if u is a critical point of the action functional AH . (3) If the solution u in question connects two critical points, that is, if there exist critical points x and y of AH such that lim us (t) = x, s→−∞ lim us (t) = y, s→+∞ then E(u) = AH (x) − AH (y) < +∞. The solutions that connect two critical points have ﬁnite energy. Let us therefore consider the space M deﬁned by M = u : R × S1 → W | u is a

Y · (X · f )(x) = [X, Y ]x · f = (df )x ([X, Y ]x ) = 0, M. Audin, M. Damian, Morse Theory and Floer Homology, Universitext, DOI 10.1007/978-1-4471-5496-9 1, © Springer-Verlag London 2014 7 8 1 Morse Functions the expression is a symmetric bilinear form in X and Y . The same computation also shows that this form is well deﬁned, that is, the result does not depend on the chosen extension Y . We will say that a critical point is nondegenerate if this bilinear form is nondegenerate. Moreover,