Morse Theory, the Conley Index and Floer Homology

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In 1965 Arnold [1] conjectured that the number of fixed points of an exact symplectic diffeomorphism on a symplectic manifold M can be estimated below by the sum of the Betti numbers provided that the fixed points are nondegenerate. This estimate is, of course, much sharper than the Lefschetz fixed point theorem which would only give the alternating sum of the Betti numbers as a lower bound. Its proof is based on a Morse type index theory. If the symplectomorphism in question is C close to the identity then the problem can indeed be reduced to classical Morse theory using generating functions [2, 34]. The general case, however, represents a much deeper problem which has recently been addressed by many authors. (We do not attempt here to give a complete overview of the literature and instead refer to [3, 13] for a more extensive discussion of related works.) For the torus M = T a remarkable solution was given by Conley and Zehnder [7]. They used a variational principle on the loop space, unbounded on either side, and overcame the problem of an infinite Morse index by means of a finite dimensional reduction. An entirely different approach by Gromov [16] was based on the analysis of holomorphic maps and led to an existence proof for at least one fixed point. Recently, Floer [9-13] combined the ideas of Conley and Zehnder with those by Gromov and gave a beautiful proof of the Arnold conjecture for general symplectic manifolds, only assuming that every holomorphic sphere is constant. (In [13] Floer's assumption is somewhat more general but we will restrict ourselves to this case in order to avoid further complications.) He defined a relative index for a pair of critical points and generalized the Morse complex of critical points and connecting orbits (as described by Witten [35]) to the infinite dimensional situation of the loop space which led to the concept of Floer homology.