Floer ' S Infinite Dimensional Morse Theory and Homotopy Theory

x1 Introduction This paper is a progress report on our eeorts to understand the homotopy theory underlying Floer homology; its objectives are as follows: (A) To describe some of our ideas concerning what, exactly, the Floer homology groups compute. (B) To explain what kind of an object we think thèFloer homotopy type' of an innnite dimensional manifold should be. (C) To work out in detail how these ideas can be applied in some examples. We have not solved the problems posed by the underlying questions, but we do have a `programme' which we hope will lead to solutions of these problems. Thus it seems worthwhile to describe our ideas now, especially in a volume of papers dedicated to the memory of Andreas Floer. We plan to write a complete account of this approach tòFloer homotopy theory' in a future paper. Floer homology arises in two diierent contexts, and in each of them from two diierent perspectives, which one can think of as`Hamiltonian' and`Lagrangian'. Here we shall mainly connne ourselves to one context, the study of curves and surfaces in symplectic manifolds, though the other one, gauge theory on three-and four-dimensional manifolds, is closely parallel. The theory began with Floer's proof of the Arnold conjecture. On a compact symplectic manifold M a Hamiltonian ow is generated by a Hamiltonian function h : M ! R, and the stationary points of the ow are the critical points of h. Classical Morse theory tells us that there are at least as many such points as the dimension of the homology H (M; R). Arnold conjectured that the same is true of the number of xed points of a diieomorphism ' 1 : M ! M which arises from a time dependent Hamiltonian ow f' t g 0t1. The trajectories of such a ow are critical points of thèaction functional' S h on the space of paths : 0; 1] ! M where S h () = Z (pdq ? hdt); and h : M R ! Ris the varying Hamiltonian. The xed points of ' 1 are therefore the critical points of S h on the space LM of loops of length 1 in M. Thus Arnold's conjecture would follow from a version of Morse theory applicable to the function S h : LM ! R, but relating its critical points to the homology of M rather than LM. This was the theory …